The Role of Gravitation in Physics

Report from the 1957 Chapel Hill Conference

ii

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for the History and Development
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Sources 5
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The Role of Gravitation in

Physics
Report from the 1957 Chapel Hill
Conference
Ccile M. DeWitt and Dean Rickles (eds.)
Communicated by
Jrgen Renn, Alexander Blum and Peter Damerow

Edition Open Access

2011

Max Planck Research Library

for the History and Development of Knowledge
Sources 5
Communicated by Jrgen Renn, Alexander Blum and Peter Damerow
Copyedited by Beatrice Gabriel
Editors:
Ccile DeWitt-Morette
Jane and Roland Blumberg Centennial Professor Emerita
Department of Physics
University of Texas at Austin
Dean Rickles
Senior Research Fellow
Unit for History and Philosophy of Science
University of Sydney

ISBN 978-3-86931-963-6
First published 2011
Printed in Germany by epubli, Oranienstrae 183, 10999 Berlin
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Edition Open Access
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Contents

The Chapel Hill Conference in Context

Dean Rickles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

The Original Chapel Hill Report

Session I Unquantized General Relativity
Chairman: B. S. DeWitt
3

The Present Position of Classical Relativity Theory

and Some of its Problems
John Wheeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

The Experimental Basis of Einsteins Theory

R. H. Dicke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Session II Unquantized General Relativity, Continued

Chairman: P. G. Bergmann
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
6

On the Integration of the Einstein Equations

Andr Lichnerowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Space-Time Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Cauchy Problem for the Gravitational Field . . . . . . . . . . . . .
Local Solutions for the Exterior Case . . . . . . . . . . . . . . . . . . . . . . . .
The Problem of the Initial Values . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Cauchy Problem for the Asymmetric Theory . . . . . . . . . . . .
Global Solutions and Universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
66
67
68
69
70
70

Remarks on Global Solutions

C. W. Misner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

viii

Contents

Solving The Initial Value Problem Using Cartan Calculus

Y. Fours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Some Remarks on Cosmological Models

R. W. Bass and L. Witten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Session III Unquantized General Relativity, Continued

Chairman: H. Bondi
9
9.1
9.2
9.3

Gravitational Waves
L. Marder, Presented by H. Bondi . . . . . . . . . . . . . . . . . . . . . . . . . .
Static Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Periodic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pulse Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97
97
98
99

10

Gravitational Field of an Axially Symmetric System

N. Rosen and H. Shamir, Presented by F. Pirani . . . . . . . . . . . . 107

11

The Dynamics of a Lattice Universe

R. W. Lindquist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Session IV Invited Reports on Cosmology

Chairman: F. J. Belinfante
12

12.1
12.2
12.3
12.4
13

Measurable Quantities that May Enable Questions

of Cosmology to be Answered
Thomas Gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurements and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cosmological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Observational Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121
121
121
122
122

Radio Astronomical Measurements of Interest to Cosmology

A. E. Lilley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Session V Unquantized General Relativity, Concluded

Chairman: A. Lichnerowicz
14

Measurement of Classical Gravitation Fields

Felix Pirani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Contents

ix

15

Correspondence in the Generalized Theory of Gravitation

Behram Kursunoglu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

16

Presentation of Work by T. Taniuchi

Ryoyu Utiyama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

17

Negative Mass in General Relativity

Hermann Bondi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Session VI Quantized General Relativity

Chairman: J. A. Wheeler
18

The Problems of Quantizing the Gravitational Field

P. G. Bergmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

19

Conceptual Clock Models

H. Salecker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

20

The Three-Field Problem

F. J. Belinfante . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Session VII Quantized General Relativity, Continued

Chairman: A. Schild
21

Quantum Gravidynamics
Bryce DeWitt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Session VIII Quantized General Relativity, Concluded

Chairman: V. Bargmann
22

The Possibility of Gravitational Quantization . . . . . . . . . 243

23

The Necessity of Gravitational Quantization . . . . . . . . . . 247

Closing Session
Chairman: B. S. DeWitt
24

Divergences in Quantized General Relativity

S. Deser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

25

Critical Comments
R. P. Feynman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

26

Summary of Conference
P. G. Bergmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

27

An Expanded Version of the Remarks by R.P. Feynman on the Reality of Gravitational Waves . . . . . . . . . . . . 279
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Dedicated to the memory of Bryce DeWitt

No one man lives long on this earth. It will take the concerted eorts of
many men to pry forth one of the deepest and most obstinate, but one
of the most important and potentially useful secrets of nature ... In this
quest we are reaching for the stars, and beyond.
Agnew H. Bahnson
The Stars Are Too High

Preface

This book contains the original report from the Conference on the Role
of Gravitation in Physics, which took place at the University of North
Carolina, Chapel Hill, over six days in 1957. The report was taken down
by Ccile DeWitt and several other reporters, as part of a conference
funding agreement with the Wright Air Development Center, a U.S. Army
(Air Force) funding body (the reports ocial designation is: WADC
Technical Report 57-216). Ccile DeWitt then edited the recorded material
into its nal form. The report, though publicly available as a government
document, has not previously been published in book form, and there
are not many copies of the report left in existence. Given the immense
historical signicance of the conference - giving gravitational research some
much needed impetus at a time when it was in a state of dire neglect we thought it was high time to produce a version of the report for the
masses as it were. The version presented here is almost entirely faithful to
the original, and aside from the correction of a few spelling mistakes (and
the possible addition of some entirely new typos!) features no substantive
alterations or annotations. However, in order to make the document more
navigable and more useful as a research tool, we have added an index (of
both names and subjects) and also imposed a little more structure on the
sessions, by setting some of the meatier contributions as chapters. This in
no way interferes with the ordering, and simply amounts to the addition,
in several places, of a title to the presentations and discussions that follow.
In addition to classic debates over cosmological models and the reality
of gravitational waves, many of the still-pressing issues in quantum gravity
were formulated in the discussion periods and interjections reproduced in
the following pages. (Philosophers of physics might be interested to see
Thomas Gold, p. 244, pressing those who assume that the gravitational
eld had to be quantized to prove it!) One can also nd the roots of many
current research programs in quantum gravity, as well as early intimations of what would become classic thought experiments, oering some
guidance to a subject without genuine experiments. We also nd, fully
staked out, three central approaches to the quantization of gravity: canon-

Contents

ical (focusing on the observables and constraints of the classical theory3 );

the Feynman functional-integral approach; and the covariant perturbative
approach. We also nd what is, I believe, the rst presentation (albeit
very briey: see p. 270) of Hugh Everetts relative-state interpretation of
quantum mechanics - Feynman gives an explicitly many-worlds characterization of Everetts approach (used, by Feynman, in fact, as a reductio
of the interpretation).
As Ccile DeWitt notes in the foreword to the original report (also
reproduced here), the document constitutes a somewhat incomplete representation of the actual proceedings in various ways, and does not amount
to an exact transcription of all that went on at the conference.4 However,
the discussions that were captured are often very rich, and of such an interesting (and, I would say, still highly relevant) nature, that the report fully
deserves its present resurrection if only to bring these discussions alone to
a wider audience. However, the transcribed presentations also often reect
a research area on the cusp of various exciting discoveries, in astrophysics,
cosmology, and quantum gravity. We are sure that both physicists and
historians of physics will nd much to interest them in the following pages.
By way of placing this report in the context of its time, an introductory
chapter provides a brief account of how the conference came to be, for it
is a rather remarkable story in itself. We then reproduce the original front
matter from the report, followed by the report itself.
DR
Max Planck Institute for the History of Science, Berlin
September 2010

3 As Peter Bergmann puts it, on p.192, once the classical problems are solved, quantization would be a walk.
4 How could it? As Agnew Bahnson (about whom, see Chapter 1) noted in his postconference report, there were some 57 half-hour tapes recorded in total.

Acknowledgements

We would like to express our thanks to all of those involved in the original
Chapel Hill conference (or their estates) who very kindly gave their permission to reproduce their papers and comments, without which this book
would not have been possible. We would also like to thank those involved
in the smooth, speedy publication of the book, especially Beatrice Gabriel
for her excellent proof reading.
DR is grateful to the the Dolph Briscoe Center for American History
at the University of Texas at Austin for allowing him access to the Bryce
DeWitt Papers; to Ccile DeWitt, rstly for allowing access to her own
archive of papers and letters, from which much of the historical material
in the introductory chapter was drawn, but also for allowing me to take
part in this wonderful project; to Jrgen Renn, Max Planck Institute for
the History of Science, and Don Salisbury, Austin College, for some much
needed motivation, resulting in a signicant speed up in the books production; and to the Australian Research Council for their generous support
of the History of Quantum Gravity Project (DP0984930) that the present
book contributes to; and to the Max Planck Institute for the History of
Science for hosting me as a visiting fellow during the bookss completion.
Finally, DR would like to thank his wife, Kirsty, and children, Sophie and
Gaia, for putting up with his very many absences while this work was
being completed.

Chapter 1
The Chapel Hill Conference in Context
Dean Rickles

The Conference on the Role of Gravitation in Physics was the (ocial) inaugural conference of the Institute of Field Physics [IOFP] which had only
just been established at Chapel Hill, with Ccile and Bryce DeWitt at the
helm.1 The IOFP received its certicate of incorporation on September
7, 1955. In fact, it nearly had a very dierent name, The Research Institute of the University of North Carolina, which, quite naturally, won the
unanimous approval from the University (and Bryce DeWitt), but not,
it transpires, the approval of the primary funder, Mr Agnew Bahnson.
Bahnson was a wealthy North Carolinian industrialist with a passion for
physics (especially gravitational); he made his fortune from industrial air
conditioning systems.
The initial meeting between Bryce DeWitt and Bahnson took place
on July 9, 1955, in Raleigh, N.C. They were joined by Cliord Beck,
who was head of physics at State College in Raleigh - Bahnson originally
planned to have the IOFP at State College since they had a nuclear reactor there, which, as Bryce DeWitt puts it, Bahnson felt might be useful
for anti-gravity [3]. Curiously, at the same time, the Glenn L. Martin
Company (now Lockheed-Martin) was setting up its own research institute
that would do work for the U.S. National Defense Service2 though of a
1 An initial get to know each other meeting of the IOFP was held June 8-10, 1956,
at Roaring Gap in North Carolina, where Bahnson had a summer house. This was
open to all members of the IOFP and a few select others, including Freeman Dyson and
Lothar Nordheim, potential funders and a reporter from the Winston-Salem Journal &
Sentinel. As Bahnson put it, in his 4th Memorandum (of June 20, 1956) the purpose
of the meeting was to introduce members of the Institute and their guests to Mr and
Mrs DeWitt and to dene more clearly the problems to be dealt with at the IOFP
(with gravity as the focal point of interest (letter from Ccile DeWitts own archive;
henceforth, unless otherwise specied, references will be to documents contained in this
archive).
2 This would become RIAS, or the Research Institute for Advanced Study. It is possible
that DeWitts meeting was with the director of RIAS, Welcome Bender, who was in
charge of recruitment, and who also attended the Roaring Gap conference on behalf

1. The Chapel Hill Conference in Context

rather dierent sort, industry sponsored, and not ensconced in a university. DeWitt had a meeting with its vice-president, George Trimble (or
Bender - see footnote 2), to discuss a potential project involving gravitational physics research (more on this below). DeWitt ew to his meeting
with Bahnson immediately after visiting the Glenn Martin Company in
Baltimore - indeed, the airplane was owned by a friend of Bahnsons, one
Earl Slick of Slick Airways!
Bahnson had originally written to DeWitt on May 30, 1955. Just
prior to this, DeWitt had already also been in correspondence with Roger
Babson, another wealthy industrialist with a passion for gravity research.
Babson had established the Gravity Research Foundation (GRF) in Salem,
and had established an essay competition (governed by George Rideout)
for the best two thousand word essays on the possibilities of discovering
some partial insulator, reector or absorber of gravity waves ([2], p. 344)!
DeWitt won rst prize in this competition (in 1953) with an essay dismissing the whole idea; or, as he put it: [E]ssentially giving them hell for such
a stupid - the way it had been phrased in those early years [3]. DeWitt
wrote the essay in a single evening: ... the quickest $1000 I ever earned!
(ibid).3 Given that the essay led to the original exchange between Rideout
and Bahnson over Bryce4 , it seems that this single nights work might in
fact have earned him rather more than $1000!
Babson clearly saw in DeWitt one who could lift the respectability of
the GRF. Indeed, this seemed to be the case, for whereas the prize was
previously avoided by serious physicists, after DeWitt won, the oodgates
opened. The next year, 1954, saw Arnowitt and Deser take rst prize.5
of the Glenn Martin Company. However, there certainly was correspondence between
DeWitt and Trimble.
3 In a section entitled Purposes, Objects and Powers of the document of incorporation of the Gravity Research Foundation formed by Babson, Clause (1) states that the
purpose of the corporation is to Observe the phenomena of nature and encourage, promote and support investigations in search of underlying knowledge of these phenomena.
Conduct theoretical and experimental studies to discover the laws which aect them
and evolve new technological concepts for the improvement and welfare of mankind
(Bryce DeWitt Archives, Center for American History, University of Texas at Austin).
Babson had a curious view of gravity as a kind of natural evil ( a dragon), caused
by his sisters and grandsons drowning by, as he saw it, the downward pull of gravity
- see [1].
4 Note that Bryce concluded his GRF essay with the words External stimuli will
be urgently needed in the near future to encourage young physicists to embark upon
gravitational research in spite of the odds, which clearly resonated with both Babson
and Bahnson.
5 Much to the chagrin of Robert Oppenheimer who was supervising both of them at
the Princeton Institute for Advanced Study. Oppenheimer thought that entering the

1. The Chapel Hill Conference in Context

The GRF could never really gain the prestige it so desired. Babson
held too much control over what lines of research were investigated. Since
he was no scientist, these tended to be crankish - in one GRF bulletin the
biblical miracle of Jesus walking on water was oered up as evidence in
the possibility of antigravity shields, as was the ability of angels to defy
gravity! The GRF stood no chance (at least not in this form). Martin
Gardner famously mocked Babsons own gravity ideas in an article entitled
Sir Isaac Babson.6 . Gardner called the GRF perhaps the most useless
scientic project of the twentieth century ([6], p. 93). He was referring to
the stated aims of the GRF and its essay competition; namely to discover
some kind of gravity screen (the right kind of alloy - cf. [2], p. 343).
Gardner rightly points out that the concept of a material that is opaque
to gravitational interactions was made obsolete by the shift to the general
theory of relativity.
Agnew Bahnson was a close friend of George Rideout, the president
of Babsons GRF, and the one who had initially suggested the idea of a
GRF to Babson. On the basis of how the initial organizational meetings
went, Gardners critique perhaps oered up a recipe for a more successful
venture. Whereas Babson promoted a vision of spectacular technologies
as a result of its gravitational research, Bahnson adopted a more sober approach. Thus, in the foreword of an early draft (dated November 17, 1955)
for the IOFPs promotional brochure Bahnson wrote (in stark contrast to
Babsons statements):
competition and accepting the prize brought the IAS into disrepute. He believed that
Arnowitt and Deser exploited their positions at the institute (see [3]). Note that Pascual Jordan was placed in 5th position this same year, with his essay on his theory
of a variable gravitational constant. James L. Anderson submitted an essay on the
Measurability of Gravitational Fields, but was not placed at all - the same is true of
Dennis Sciama who wrote on a possible method of shielding gravitation using Machs
principle. Frederik Belinfante took 1st prize in 1956 for his paper on gravitational absorbers and shields. In the 1957 competition, Thomas Gold took 1st prize for a paper
on the gravitational interaction of matter and anti-matter, that he co-authored with
Philip Morrison (John Wheeler took second prize, and Felix Pirani took 3rd).
6 This is a reference to Babsons penchant for all things Newtonian. This penchant led
to the establishment of one of the largest collections of Newtonia (as he calls it, [2] p.
340) in the world, on the campus of the Babson Institute in Wellesley, Massachusetts.
One of the librarys rooms is an actual room used by Newton while in his nal years
in London. This was purchased by Babsons wife when she discovered the building was
being demolished, shipped over from the UK, and rebuilt on site as Newton would have
used it with the same walls, doors and even the identical shutters containing the hole
through which he carried on his rst experiments in connection with the diusion of
light (loc. cit., p. 340). David Kaiser has a useful discussion of this curious episode in
his PhD thesis, Making Theory: Producing Physics and Physicists in Postwar America
(Harvard University, 2000). See also Kaiser [9]

10

1. The Chapel Hill Conference in Context

In the minds of the public the subject of gravity is often

associated with fantastic possibilities. From the standpoint of
the institute no specic practical results of the studies can be
foreseen at this time.
In many ways, then, Babsons GRF was used as a foil, highlighting things
to adopt but (more importantly) things to avoid. A lengthy exchange
of letters between Bahnson and several senior physicists (especially John
Wheeler, whom Bahnson clearly admired a great deal) set to work on
eradicating any aspects that might lead to claims that the institute was
for crackpot research.
In 1955, G. S. Trimble, vice president of the Glenn Martin Company,
wrote to Bryce DeWitt7 that:
During a recent conversation with Mr. George Rideout,
president of Roger Babsons Gravity Research Foundation, we
were commiserating on the unfortunate state of the aairs that
knowledgable folks do not wish to get mixed up in the eld
of gravity research. During the course of the conversation he
reviewed with me your suggestion that perhaps his Gravity Research Foundation might be transformed from its present function into an active center of research concentrating on the eld
of gravity. He also told me that the foundation was not able
to undertake such an expansion. (Letter from G. S. Trimble
to Bryce DeWitt, dated, June 10, 1955)
It seems that DeWitt had suggested to Babson something along the lines
of the Institutes for Advanced Study - a model he was very familiar with
since both he and his soon to be wife Ccile Morette, had both spent time
in several of them. This model tted with the Glenn Martin Companys
plans. However, their goal was not pure research, but something grander,
it seems - Trimble describes the proposed activity as an industrial version
of the Institute for Advanced Study.8 The letter goes on:
7 At this time senior research physicist, Radiation Laboratory, University of California,
Berkeley and Livermore, then working on detonation hydrodynamics.
8 Trimble makes an interesting remark concerning the tight relationship between scientic research and society: [W]e feel morally obligated to push forward in the basic
sciences and we believe as a dynamic industry we can provide the motivation for advances that can be obtained in no other way. In other words, for better or for worse,
the pursuit of certain areas of basic research demand some kind of motivation beyond
the search for deeper knowledge. Practical applications are one way to motivate such
study.

1. The Chapel Hill Conference in Context

11

It occurred to us sometime ago that our industry was vitally

concerned with gravity. As time goes on we become more and
more concerned because we feel certain that sooner or later
man will invade space and we see it as our job to do everything possible to speed this event. At least one category of the
things one must study, when he desires to bring space ight to
a reality, is the laws of nature surrounding the force of gravity.
(ibid.)
One wonders what Roger Babson would have made of the fact that it was
Newtons equations that got man into space!9 Trimble bemoans the fact
that most of those working on gravitation are mad men and quacks perhaps he has those connected with Babsons own endeavour in mind?
Indeed, notes Trimble, any relevant work that they had done on space ight
had been contracted out to German scientists working within Germany.
Louis Witten (the father of Ed Witten, and participant in the Chapel
Hill conference)10 was the rst researcher hired by RIAS, of the Glenn
Martin Company. He approached DeWitt in a bid to have him join RIAS,
but DeWitt was then already being approached by Bahnson, oering more
lucrative terms (both for him and Ccile): the promise of a position in a
more traditionally academic environment, but with the freedom of an externally funded position, would ultimately win out. However, Trimble, and
the Glenn Martin Company nonetheless played a role in helping the IOFP
get o the ground. Not only did they purchase a Founders Membership
for the Institute of Field Physics, for the considerable sum of $5000, they
also oered their support to solicit further funding (letter from Trimble
to Bahnson). The letter reproduced in Fig. 1.1, from Bahnson to Bryce
and Ccile DeWitt, shows just how tightly bound industry, military, and
research were at this time, and also how much inuence a single individual
could possess.
The plan of the IOFP was to house an institute within an academic
institution, so as to avoid the conict that physicists felt working in an
industrial setting. The chosen location was the physics department at the
University of North Carolina, Chapel Hill. In order to lend further prestige to the IOFP, Bahnson secured letters of comment from several of the
most prominent physicists of the day, including Belinfante, Oppenheimer,
9 Curiously,

DeWitt later did a report on The scientic uses of large space ships, for
the Department of Defense (General atomic report GAMD 965, 1959).
10 Bob Bass, who coauthored the paper with Witten in this conference report, was the
rst person to be hired by Witten, in 1956. Bass, together with Witten, would later
manage to hire R. E. Kalman and Solomon Lefschetz, in 1956 and 1957 respectively.

12

1. The Chapel Hill Conference in Context

Figure 1.1: Letter from Agnew Bahnson to the DeWitts from the early
phase of the development of the Institute of Field Physics,
December 29, 1955.

1. The Chapel Hill Conference in Context

13

Dyson, Teller, Feynman, and Wheeler. Wheeler did much behind the
scenes sculpting of the IOFP, and was, next to Bahnson, perhaps most responsible for the bringing about of the IOFP. For example, it was Wheeler
who tempered Bahnsons own (somewhat Babsonesque) proposal entitled
The Glorious Quest, to make it more attractive to funding agencies:
Ebullient as you and I are, I suspect sober going may go further when it
comes to getting money from a foundation (Wheeler to Bahnson, August
29, 1955). Then, writing to the acting president of the University of North
Carolina, Harris Purks, November 25, 1955, Wheeler writes of:
the absolute necessity to avoid identication with so-called
anti-gravity research that may be todays version of the last
centurys search for a perpetual motion machine. [...]
Unfortunately, there are sensationalists only too willing to
confuse in the public mind the distinction between so-called
anti-gravity research ... and responsible, well informed attempts to understand eld physics and gravitational theory at
the level where it really is mysterious, on the scale of the universe and in the elementary particle domain.
He goes on to applaud the step (in fact suggested by Wheeler himself,
earlier) of attaching to every piece of IOFP publicity a disclaimer to the
eect that the IOFP is in no way connected to anti-gravity research. This
Protection Clause would be attached to each IOFP statement:
The work in eld physics and gravitation theory carried on at
the University of North Carolina at Chapel Hill, and nanced
by the Institute of Field Physics, as fund raising agency, has no
connection with so-called anti-gravity research of whatever
kind and for whatever purposes. Its scientists, basing their
investigations upon veriable data, accept the Newton-Einstein
analysis of gravity as free of a single established exception, and
as the most comprehensive physical description we have today.
They seek the implications of gravity and other elds of force
at the level of the elementary particles. More generally, the
Chapel Hill project is a modest attempt to learn more about
the nature of matter and energy.
This expedient, Wheeler argued, is necessary to avoid discouraging both
sponsors and scientists. The message that anti-gravity connotations must

14

1. The Chapel Hill Conference in Context

be avoided at all costs runs through much of the correspondence and foundation documents like a mantra. It clearly played a vital role (in the minds
of physicists) in establishing the legitimacy of the enterprise.
Wheeler did not hold back on the need for the IOFP, though his
claims were moderated somewhat by a knowledge that progress might well
be very slow:
It is hard to see how one can get to the bottom of the elementary particle problem - the central issue of modern physics without coming to the very foundations of our physical world
and the structure of space and time. Gravity, elds and particles must in the end be all one unity. The absence of any paradox or discrepancy in gravitation theory at the human and
astronomical levels creates an obligation to apply Einsteins
ideas down to smaller and smaller distances. One must check
as one goes, until one has either a successful extension to the
very smallest distances, or a denite contradiction or paradox
that will demand revision. ... The challenge cannot be evaded.
Exactly how to proceed is a matter of wisdom, skill, judgement,
and a good idea. Nobody guarantees to have a good idea, but
the DeWitts, fortunately, have a very sound plan of what to
do while searching for a good idea. They propose to do something that has long needed doing - help make clear the fundamental facts and principles of general relativity so clearly and
inescapably that every competent worker knows what is right
and what is wrong. They can do much to clear away the debris
of ruined theories from the rocklike solidity of Einsteins gravitation theory so its meaning and consequences will be clear to
all. This is a great enterprise. Einsteins theory of the spacetime-gravitation eld is even richer than Maxwells theory of
the electromagnetic eld. That eld has been investigated for
many years, and now forms the foundation for a great science.
One cannot feel physics has done its job until a similarly complete investigation has been made for the gravitational eld.
(John Wheeler, letter to Bahnson, November 25, 1955)
Though there is, of course, a good deal of colourful rhetoric in this passage,
it nonetheless shows the importance in Wheelers mind of the role that the
IOFP (and the DeWitts) would play. (Note that Wheeler had only just
recently had to intervene in a proposal of Samuel Goudschmidts to impose

1. The Chapel Hill Conference in Context

15

an embargo on all papers dealing with general relativity and unied eld
theory from the pages of Physical Review - cf. [5], p. 414.)
The various letters of support (dating from between October 1955 and
January 1956), for which the preceding letter from Wheeler to Purks provides a cover letter, highlight the recognition that general relativity and
gravitational research had been unfairly neglected, and the need for a renewal of interest. Oppenheimer writes that he shares with most physicists
the impression that this eld has been rather neglected by us. Dyson seconds this (as does Nordheim), but adds some conditions for success, more
or less reiterating what Wheeler had already said: that immediate results
should not be expected, and that (to avoid becoming isolated and sterile) the institute should be settled as rmly as possible in the framework
of normal university life. Edward Teller remarks in his letter that a
comprehensive examination of general relativity and high-energy physics,
together with an investigation of the interaction between these two elds
may very well lead to the essential advance for which we are all looking.
Feynman too voiced the opinion that the problem of the relation
of gravitation to the rest of physics is one of the outstanding theoretical
problems of our age. However, he was less positive about the chances of
the proposed institute in (what he thought was) its original form. Feynman was not convinced that an industrially funded institute, detached
from a university, could possibly deliver the requisite exibility to develop
new fundamental knowledge: that required absolute freedom to bounce
around between topics, as one chose. On learning that the institute was to
be housed in a university, Feynman was unreservedly positive about the
proposal (letter to Wheeler, dated December 2, 1955).
John Toll, head of physics at the University of Maryland, writes, directly discussing the other letters:
Most of my colleagues have pointed out in their comments
that the eld of general relativity has not received the attention which it deserves and that it is particularly important
to attempt to obtain some synthesis of the methods and concepts used in general relativity with the ideas now employed
to discuss elementary particles. One reason for the neglect of
general relativity has been the great diculty of work in this
eld which challenges even the best theoretical physicists; solution of the major problems involved will probably require a
determined program which may extend over many years. A
second and related reason has been the diculty of obtaining
adequate support for this eld; the problems are not of the

1. The Chapel Hill Conference in Context

16

type which are supported by federal agencies which nance so

much of the research in physics in the United States by short
term contracts, mostly in elds which appear to have more immediate applicability to defence problems. (Letter from J. S.
Toll to John Wheeler, dated December 28, 1955.)
This was all written towards the end of 1955.11 By 1957 the picture looked
remarkably rosier. Whether it was due to some degree of inuence of the
IOFP12 and/or RIAS (or the beginnings of the Space Race and the Cold
War), the Air Force and the Department of Defense in fact soon began to
fund fundamental research in gravitational physics. Rather fortuitously,
Joshua Goldberg was, at the time of the establishment of the IOFP and
the conference, in charge of aspects of the Air Forces funding of general
relativity, in the modern physics research branch (based at the Aeronautical Research Laboratory at Wright Patterson Air Force Base). In
addition to a $5000 grant, one of the (very crucial) things he was able
to do was secure MATS (Military Air Transport Service) transportation
to and from the USA (initially just for Ghniau, Rosen, and Laurent,
but later this would include transportation for Behram Kursunoglu from
Turkey and Ryoyu Utiyama from Japan, extending to 11 nations in all).
In a letter to Bryce DeWitt (dated October 3, 1956), Goldberg notes that
the suggestion to use Air Force transportation to bring over physicists
11 December

7, 1955 saw DeWitt deliver a paper focusing on current research in gravitational physics to the American Astronautical Society (published as [4]). By this time he
was able to give his position as Director of the IOFP. The talk was clearly intended as
a piece of propaganda for the IOFP. DeWitt opened by distancing his work from any
foreseeable practical applications. He then notes the lack of serious research being carried out, counting just seven institutions with gravitation research projects: Syracuse,
Princeton, Purdue, UNC, Cambridge, Paris, and Stockholm - with RIAS, Inc, on the
industrial side. DeWitt mentions even at this early stage of quantum gravity history
the problems that would plague the quantum geometrodynamical approach throughout
its existence (until it transformed into loop quantum gravity): these are the problems
of dening the energy and the quantities that are conserved with respect to it (i.e. the
observables), and the factor ordering problem. (This problem refers to an issue caused
by the straightforward canonical quantization of general relativity, based on the metric
variables. According to the standard quantization algorithm, when one meets a momentum term, one substitutes a derivative. However, when this procedure is applied in
general relativity, one faces situations were one has products, and so one has to multiply
as well as dierentiate. The order in which one does this matters for the form of the
nal wave equation.) The former was studied by Bergmanns group at Syracuse, while
the latter problem was studied by DeWitts own group at the IOFP.
12 Not so far fetched as it might sound. Bahnson notes in a letter to Bryce and Ccile
of December 29, 1956, that by then the Air Force had expressed interest in their work
(this he heard directly from Glenn Martin) - see Fig.1.1.

1. The Chapel Hill Conference in Context

17

from Europe had been Peter Bergmanns idea - Goldberg gives a personal
account of his role in the Air Forces support of general relativity (and the
possible reasons behind the military support of research in gravitation) in
[7]. Such free transportation became commonplace for the IOFP at this
time, and since many commentators who lived through this experience
have suggested that the ability to be able to network, made possible by
the availability of easy transportation, played a key role in the reemergence of gravitational physics. Later funding would also take the form of
free computing time on IBMs best machines13 , and (in limited cases) free
ights on TWAs line.
Securing additional funding for the conference was time-consuming.
In May 1956, the DeWitts visited the National Science Foundation in
Washington, to explain the nature of their project - a visit that met
with success. The same week Bryce DeWitt gave a laymans talk to the
Winston-Salem Rotary Club - at which various industrialists and wealthy
interested parties were present - in which he described the various technological innovations that have emerged from pure research. It seems (from
a memorandum Bahnson sent to his fellow funders) that the Chapel Hill
conference was virtually entirely externally funded (i.e. independent of
the IOFPs own funds). This, he notes, is almost entirely thanks to the
work of Ccile DeWitt (Bahnson, Memorandum No. 9, May 7, 1957) in a letter to Bahnson dated November 5, 1956, Bryce DeWitt notes that
in the space of two weeks, Ccile composed 52 letters and placed 10 long
distance calls, chasing potential funding for the conference.
Though not quite a cascade, the IOFP had enough funding in its heyday to attract several rst-rate postdoctoral fellows. These included Peter
Higgs, Heinz Pagels, and Ryoyu Utiyama. Among the rst postdoctoral
fellows at the institute was Felix Pirani, who had previously belonged (and
would later return) to Hermann Bondis Relativity and Gravitation group
at Kings College, London (Clive Kilmister was another long term member of this group).14 Pirani received his (rst) doctorate under Arthur
Schild (who would later head the Center for Relativity) at The Carnegie
13 This

possibility was initially raised by IBMs representative, Dr John Greedstadt,

at the Roaring Gap meeting, though apparently trying to dene problems to run on
the machines was not an easy task. It is, however, worth mentioning that Bryce DeWitt would later be a pioneer of numerical relativity; also, the issue of putting general
relativity on a computer arises in the Chapel Hill report: p. 83.
14 As Ezra Newman pointed out in his recollections of the early history of general relativity ([11], p. 379), Pirani completely abandoned physics for a life as an author of
childrens books. See also [14].

18

1. The Chapel Hill Conference in Context

Institute of Technology - he received a second doctorate from Cambridge

University, under Hermann Bondi.
Peter Higgs too had been part of Hermann Bondis Relativity and
Gravitation group at Kings College, London, since 1956. It was Pirani
who urged him to take more interest in quantum gravity, prompting him
to take up the position at the IOFP. Though invited to the institute
to study gravitation, Peter Higgs ruefully admits that he spent his time
there working on symmetry breaking in quantum eld theory.15 Higgs
rst encountered Bryce DeWitt in 1959, in Royamont France. This was
the second GRG conference16 , and it was shortly after that the International Committee on General Relativity and Gravitation was formed (see
Kragh [10], p. 362). He met him again at the GRG3 conference in Warsaw, in 1962. After 1956, following Piranis advice, Higgs began looking
at quantum gravity - at the time he was working with Abdus Salam at
Imperial College. Here he wrote on the constraints in general relativity
[8]. This led, in 1964, to DeWitts invitation to Higgs to spend a year
at the institute, which he did, arriving in September 1965, after a years
postponement. Bahnson died tragically in an airplane crash the year prior
on June 3, 1964 - a chair was established at UNC in his honour, to be
occupied by Bryce DeWitt.
A follow up meeting focusing purely on Exploratory Research on the
Quantization of the Gravitational Field was held in Copenhagen, June 15
to July 15, the same year as the Chapel Hill conference. This meeting involved DeWitt, Deser, Klein, Laurent, Misner, and Mller. Again, MATS
was utilised for this meeting, courtesy of Goldberg and the ARL. Mller
and Laurent would return to Chapel Hill as visiting fellows for two months,
starting in February. Their brief was to work on the role of gravitation in
articial Earth satellites and also atomic clocks (Bahnson, Memorandum
#11, Feb. 3, 1958).17
There are similarities between the Chapel Hill conference and the 1955
conference to celebrate the 50th anniversary of Einsteins theory of special relativity, held in Berne. Einstein was, of course, to have attended
15 Letter

to Ccile DeWitt.
the third, depending on whether one counts the conference in Berne in 1955 to
mark the Jubilee of Einsteins theory of special relativity. This conference is often
referred to as GR0.
17 Cold War paranoia can be clearly seen in this memorandum. Bahnson mentions a
recent report (apparently reported in American Aviation magazine) of a graviplane
about to be produced by the Russians, based on the extension of Einsteins theories
by Dr. Foch (sic.) of Leningrad - though Bahnson admits it is likely a propaganda
trap.
16 Or

1. The Chapel Hill Conference in Context

19

but he fell ill during the planning and died just before the event took
place, prompting Pauli to declare that This important moment in history
is a turning point in the history of the theory of relativity and therefore
physics ([12], p. 27). Pauli himself died not long after, in 1958. Somewhat
surprisingly, this jubilee celebration was in fact the rst ever international
conference devoted solely to relativity. As it would turn out, the conference dealt almost exclusively with general relativity, special relativity
being more or less a nished enterprise, formally, experimentally, and conceptually. The conference would later come to be known as GR0, the
zeroth conference in a series which continues to this day, and of which
Chapel Hill was the rst, GR1.18
The proceedings (replete with post-talk discussions) were quickly edited by Andr Mercier and Michel Kervaire. This proceedings volume, and
the conference itself, played a central role in the future evolution of classical and quantum gravity. However, it was no Shelter Island. Whereas
that conference had been driven by the younger generation of physicists Feynman, Schwinger, Wheeler, and others - the Berne Jubilee conference
was dominated by older, more established physicists. In his own report
on the Chapel Hill conference, Bahnson noted that he had talked to a
physicist (unnamed) who had also participated in the Berne conference,
who had remarked that the Chapel Hill conference had greater informality and that the younger participants contributed to more discussion
and exchange of information. The Chapel Hill conference on the Role
of Gravitation in Physics, that would happen just two years later, did
18 Schweber

traces the reawakening of interest in the eld [of GR] ([13], p. 526) to
the Berne conference, GR0, in 1955. He also notes that interest in quantum gravity
was made respectable as a result of Feynmans course at Caltech between 1962 and
1963 (loc. cit., p. 527). The Berne conference was important. But it was distinctly a
European aair. In the United States, as we have seen, there were several converging
lines of attack leading to a reawakening of interest. Indeed, I would argue that since the
Berne conference consisted mostly of an older generation who had persistently thought
about general relativity and quantum gravity for decades, the phrase reawakening
of interest is not really appropriate. Klein, Pauli, and Rosenfeld, for example, were
veterans when it came to the study of both. In fact, there was an earlier conference in
honour of Bohr to which many of the same people contributed, and gave very similar
talks. Further, to trace the respectability of quantum gravity research to Feynmans
course is over-stretching. Wheeler had been including material on quantum gravity
from the time he began teaching his general relativity course at Princeton. One can
even nd quantum gravity problems posed within his earlier advanced quantum theory
course. In addition to this, there were, as Schweber himself notes, several strong schools
concentrating on gravitation research by the end of the 1950s. That Feynmans course
happened was as a result of the increased respectability already in operation - moreover,
Feynmans interest was surely stimulated as a result of his participation in the Chapel
Hill conference, though he seems to have already been investigating the subject by 1955.

20

1. The Chapel Hill Conference in Context

for general relativity and gravitation what Shelter Island did for quantum
electrodynamics. The Chapel Hill conference was a genuine break from
the Berne conference, both in terms of its organization, its content, but
more so its spirit.
References
[1] Babson, R. W. (1948) Gravity - Our Enemy Number One. Reprinted
in H. Collins Gravitys Shadow: The Search for Gravitational Waves
(pp. 828-831). University of Chicago Press, 2004.
[2] Babson, R. W. (1950) Actions and Reactions: An Autobiography of
Roger W. Babson. (2nd revised edition). Harper & Brothers Publishers.
[3] Interview of Bryce DeWitt and Ccile DeWitt-Morette by Kenneth
W. Ford on February 28, 1995, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA: http://www.aip.
org/history/ohilist/23199.html
[4] DeWitt, B. S. (1957) Principal Directions of Current Research Activity in the Theory of Gravitation. Journal of Astronautics 4: 23-28.
[5] DeWitt, B. S. (2009) Quantum Gravity: Yesterday and Today. General Relativity and Gravitation 41: 413-419.
[6] Gardner, M. (1957) Fads & Fallacies in The Name Of Science. Dover.
[7] Goldberg, J. M. (1992) US Air Force Support of General Relativity:
1956-1972. In J. Eisenstaedt and A. J. Kox (eds.), Studies in the
History of General Relativity (pp. 89-102). Boston: Birkhuser.
[8] Higgs, P. W. (1958) Integration of Secondary Constraints in Quantized General Relativity. Physical Review Letters 1(10): 373-374.
[9] Kaiser, D. (1998). A is just a ? Pedagogy, Practice, and the Reconstitution of General Relativity, 1942-1975. Studies in the History
and Philosophy of Modern Physics 29 (3): 321-338.
[10] Kragh, H. (2002) Quantum Generations: A History of Physics in the
Twentieth Century. Princeton University Press.
[11] Newmann, E. (2005). A Biased and Personal Description of GR. In
A. Kox and J. Eisenstaedt, The Universe of General Relativity (pp.
373-383). Boston: Birkhuser, 2005.

1. The Chapel Hill Conference in Context

21

[12] Pauli, W. (1956) Opening Talk. In A. Mercier and M. Kervaire (eds.),

Fnfzig Jahre Relativittstheorie, Bern, July 11-16, 1955. (p. 27).
Helvetica Physica Acta, Suppl. 4. Basel: Birkhuser Verlag.
[13] Schweber, S. (2006) Einstein and Oppenheimer: Interactions and Intersections. Science in Context 19 (4): 513-559.
[14] Trautman, A. (2009) Editorial Note to J. L. Synge, On the Deviation
of Geodesics and Null-Geodesics, Particularly in Relation to the Properties of Spaces of Constant Curvature and Indenite Line-Element
and to F. A. E. Pirani, On the Physical Signicance of the Riemann
Tensor. General Relativity and Gravitation 41 (5): 1195-1203.

Chapter 2
The Authors

In this chapter we provide brief biographical notes on the participants,

focusing on those who made a contribution that is represented in the report. Many are, of course, very well-known gures, but others may not
be familiar. For convenience we have arranged them according to their respective host countries (that is, the countries in which they were working
at the time of the conference). Rather than provide any kind of extensive
biography, we simply focus on key details and material that is relevant to
the conference theme.
USA
J. L. Anderson James Anderson was born in Chicago in 1926. He received his PhD in 1952, under Peter Bergmann at Syracuse University, with a thesis entitled: On the Quantization of Covariant Field
Theories. He was Bergmanns research assistant between 1951 and
1952 and continued to work with him on the canonical approach to
quantum gravity.
V. Bargmann Valentine Bargmann was born in 1908 in Berlin. He obtained his PhD in 1936 from the University of Zurich under the supervision of Gregor Wentzel, on the subject of electron scattering in
crystals - ber die durch Elektronenstrahlen in Kristallen angeregte
Lichtemission. He had worked with Peter Bergmann, and was also
Einsteins assistant, at the Institute for Advanced Study, working on
the problem of motion in general relativity and unied eld theory.
He also did much important work on the application of group theory
to physics in the years prior to the Chapel Hill conference.
R. Bass Robert Bass was born in 1930. He received his PhD in mathematics from Johns Hopkins University in 1955, under Aurel Wintner,
on the subject: On the Singularities of Certain Non-Linear Systems
of Dierential Equations. Following a postdoc at Princeton University, he was a sta scientist at RIAS [Research Institute for Advanced

24

2. The Authors

Study] (a division of the Martin-Marietta Company) at the time of

the Chapel Hill conference.
F. J. Belinfante Frederik Belinfante was born in the Hague, in the Netherlands, in 1913, and received his PhD under the supervision of Kramers
at the University of Leiden. After World War II he moved to Purdue
University (via the University of British Columbia). At the time of
the Chapel Hill conference he was one of the few physicists pursuing
an active research program in quantum gravity research, including
the supervision of PhD students on this topic.
P. G. Bergmann Peter Bergmann was born in Berlin in 1915. He received
his doctorate in 1936, under Philipp Frank, at the Deutsche Technische Hochschule in Prague, with a thesis entitled: Der harmonische
Oszillator im sphrischen Raum. He was Einsteins research assistant until 1941, working on unied eld theory. He began working
on the direct quantization of gravity in 1949, and soon established a
group that specialized in the canonical quantization of gravity. By
the time of the Chapel Hill conference he was responsible for one of
the few schools of relativity, based at Syracuse.
D. R. Brill Dieter Brill was born in 1933, and received his PhD from
Princeton in 1959, on the subject of the positivity of mass in general
relativity. He was a student of John Wheeler during the conference.
He would later go on to write one of the earliest review articles on the
quantization of general relativity, with his student Robert Gowdy.
M. J. Buckingham Michael Buckingham was born in Sydney in 1927. He
gained his PhD in Liverpool, in the UK, under Herbert Frhlich,
on the subject of superconductivity. At the time of the Chapel Hill
conference he was a postdoctoral fellow at nearby Duke University,
also in North Carolina. He returned to Australia shortly after the
conference.
W. R. Davis William Davis received his PhD from the University of Gttingen in 1956. He was hired by the University of North Carolina in
1957. He later became one of the worlds foremost authorities on the
works of Cornelius Lanczos, and is the editor of Lanczos complete
works.
B. S. DeWitt Bryce DeWitt was born in California in 1923. He received
his PhD at Harvard under the supervision of Julian Schwinger in

2. The Authors

25

1950. Entitled I. The Theory of Gravitational Interactions. II.

The Interaction of Gravitation with Light, it is one of the earliest
examples of a thesis devoted to quantum gravity. After a string
of research fellowships, including a Fullbright, he took up (with his
wife Ccile) directorship of the Institute for Field Physics at the
University of North Carolina.
C. M. DeWitt Ccile DeWitt-Morette was born in Paris in 1922. She
completed her doctoral research at the Dublin Institute for Advanced
Study, under Walter Heitler, in 1947, on the physics of mesons, entitled Sur la Production des Msons dans les Chocs entre Nuclons though ocially she received her degree from the Universit de Paris.
She traveled to the Institute for Advanced Study at the request of
Robert Oppenheimer were she rst encountered Feynmans path integral theory, a subject she would specialize in. She founded the Les
Houches summer school series in 1951, and directed the Institute of
Field Physics with her husband Bryce DeWitt. She was the chief
organizer of the Chapel Hill conference.
R. H. Dicke Robert Dicke was born in Missouri in 1916. He received his
PhD from the University of Rochester in 1941, under Victor Weisskopf, with a thesis on the experimental investigation of the inelastic
scattering of protons. He was based at Princeton in 1957, and was
already gaining a reputation for his research on new experiments in
the theory of gravitation as well as controversial interpretations of
gravitational phenomena.
F. J. Ernst Frederick Ernst was born in 1932 near Manhattan. He was
among those students to attend John Wheelers rst ever general relativity course at Princeton in 1953. He would later complete a thesis
on geons under Wheelers supervision (between 1951 and 1955), the
contents of which were presented in his talk at the Chapel Hill conference. He gained his doctoral thesis in 1958 under Robert Sachs,
on quantum eld theory.
R. P. Feynman Richard Feynman was born in 1918 in New York City. He
received his PhD under the supervision of John Wheeler, at Princeton, in 1942, with his thesis on The Principle of Least Action in
Quantum Mechanics. Even by 1955 Feynman claimed to have spent
a great deal of eort on the problem of gravitation (letter from Bryce
DeWitt to Agnew Bahnson, November 15, 1955).

26

2. The Authors

T. Gold Thomas Gold was born in Austria in 1920. After a circuitous

trajectory through engineering and the theory of hearing, Gold was,
in 1947, elected to a Fellowship at Trinity College, Cambridge (without having obtained a PhD). Soon after he devised, with Hermann
Bondi and Fred Hoyle, the idea of continuous creation of matter on
the basis of a conict between Hubbles time constant and empirical
evidence. By the time of the Chapel Hill conference he was working
on a range of problems in and around astronomy at Harvard.
J. N. Goldberg Joshua Goldberg was born in New York in 1925. He was a
graduate student at Syracuse, where he received his PhD under Peter
Bergmann in 1952. He took up a position as research physicist at the
Wright-Patterson Air Force Base and, as part of his administrative
duties at Wright-Patterson, was crucial in obtaining funding for the
Chapel Hill conference.
A. E. Lilley Arthur Lilley was born in 1928. He received his PhD in 1956
in astronomy from Harvard University under the direction of Bart
Bok. At the time of the conference he was a radio astronomer with
the U.S. Naval Research Laboratory, though aliated to Harvard.
He was a pioneer in radio astronomy and its uses in the testing of
relativity.
R. W. Lindquist Richard Lindquist was a graduate student of Wheelers,
and received his PhD under Wheelers direction in 1962, with a thesis on The Two Body Problem in Geometrodynamics. He later
extended this work into numerical relativity in which he was one of
the rst to develop computer simulations of colliding black holes.
C. W. Misner Charles Misner was born in Michigan in 1932. He received
his PhD under the supervision of John Wheeler the same year as
the Chapel Hill conference, on the subject of the Feynman quantization of general relativity. He was already an instructor at Princeton
from 1956, and would often give guest seminars in John Wheelers
courses. He was very closely involved in Wheelers geometrodynamical approach at the time of the conference.
R. Mjolsness Raymond Mjolsness was a beginning graduate student of
John Wheelers at Princeton (on the NSF graduate fellowship) though he later gained his PhD under Edward Freiman in 1963, with
a thesis entitled: A Study of the Stability of a Relativistic Particle
Beam Passing Through a Plasma.

2. The Authors

27

E. T. Newman Ezra Newman was born in New York in 1929. He was

a student of Peter Bergmann at Syracuse and gained his PhD under his supervision in 1956 - Newman initially began working with
Bergmann as an undergraduate in 1951.
A. Schild Alfred Schild was born in Istanbul in 1921. He was interned
in Canada during World War II, and obtained his PhD there, from
the University of Toronto, in 1946, under the supervision of Leopold
Infeld, with a thesis entitled: A New Approach to Kinematic Cosmology. He moved to the University of Austin, Texas, just after
the Chapel Hill conference and established the very successful Center for Relativity Theory. Together with his student Felix Pirani, he
had written one of the rst papers on the canonical quantum of the
gravitational eld.
R. S. Schiller Ralph Schiller was a graduate student with Peter Bergmann
at Syracuse and had worked on translating results from the canonical formulation of general relativity into the Lagrangian formulation.
He later switched to biophysics after spending some time as David
Bohms research assistant in Brazil.
J. Weber Joseph Weber was born in New Jersey in 1919. His interests
spanned laser physics and gravitational physics. He received his PhD
in 1951 under Keith Laidler on the subject of Microwave Technique
in Chemical Kinetics at the Catholic University of America. In
1955 he began investigating gravitational radiation, and accompanied John Wheeler to Leiden in 1956. Weber began the construction
of the gravitational wave detector that bears his name the year following the Chapel Hill conference.
J. A. Wheeler John Wheeler was born in Florida in 1911. He obtained his
PhD from Johns Hopkins University (under Karl Herzfeld) in 1933.
He had a range of postdoctoral positions following this, including
one under Bohrs supervision. He was a professor at Princeton when
the Chapel Hill conference took place, and had only fairly recently,
in the fall of 1952, begun to teach and learn general relativity. Many
of his doctoral students were present at the conference. Just prior to
the conference, Wheeler was Lorentz Professor in Leiden - he took
three of his brightest doctoral students with him, including Charles
Misner and Joseph Weber.
L. Witten Louis Witten was born in Baltimore in 1921. He received his
PhD in 1951, from Johns Hopkins University, under the supervision

28

2. The Authors

of Theodore Berlin. His interests shifted to general relativity during

postdoctoral stints at Princeton and Maryland. At the time of the
conference he was working at RIAS.
UK
H. Bondi Hermann Bondi was born in Vienna in 1919. He did his PhD at
Cambridge under Arthur Eddington. In 1948 Bondi had, together
with Thomas Gold, published a theory of cosmology (the steady
state theory) based on the perfect cosmological principle, which
was still being debated at the time of the Chapel Hill conference. By
1957, Bondi had established (at Kings College, London) one of the
few schools of general relativity.
F. Pirani Felix Pirani was born in 1928. He received his rst PhD under Arthur Schild, at the Carnegie Institute of Technology, in 1951,
where he had worked (together with Schild) on the canonical quantization of the gravitational eld. He obtained a second PhD under
Hermann Bondi at Cambridge in 1956, on The Relativistic Basis Of
Mechanics. Just prior to the Chapel Hill conference he had proposed
the geodesic deviation equation as a tool for designing a workable
gravitational wave receiver. He later became an author of childrens
books, a shift which seems to have been triggered by his experience
of racism in the United States during a postdoctoral fellowship at
Chapel Hill in 1959.
L. Rosenfeld Lon Rosenfeld was born in Belgium in 1904. He received
his PhD in 1926, at the University of Lige under Marcel Dehalu and
Niels Bohr. Following this he undertook a series of fellowships, with
de Broglie, Born, Pauli and Bohr. He had done pioneering work on
constrained systems and the quantization of the gravitational eld
in the 1930s. He succeeded Hartree at Manchester in 1947 and was
still based there at the time of the Chapel Hill conference.
D. Sciama Dennis Sciama was born in Manchester in 1926. He received
his PhD from Cambridge under Dirac in 1953, with a thesis on
Machs Principle. He followed his PhD with postdoctoral fellowships at the Institute for Advanced Study and Harvard. He later
returned to Trinity College in Cambridge on an earlier fellowship,
before joining Bondis group at Kings College in 1958.

2. The Authors

29

France
Y. Fours Yvonne Fours (ne Bruhat; now Choquet-Bruhat) was born
in Lille in 1923. She received her PhD (Docteur des Sciences) from
the Universit de Paris in 1951 on the Cauchy problem for a system
of second order partial dierential equations (with application to
general relativity). She spent 1951-1952 at the Institute for Advanced
Study in Princeton. She was a member of the Facult des Sciences
de Marseille at the time of the Chapel Hill conference.
Andr Lichnerowicz Lichnerowicz was born in Bourbon lArchambault in
1915. He received his PhD in 1939, on dierential geometry in general relativity, under the supervision of Georges Darmois. He was
based at the Collge de France, in Paris when the conference took
place. He was a full professor of mathematics, rather than physics.
As we have seen, at the time it was common for general relativists
to be found in mathematics departments. Lichnerowicz had earlier
been to the States, where he had spent some time as a visiting professor at Princeton. The same year as the conference, Lichnerowicz
founded (along with John Wheeler and Vladimir Fock) the International Society for General Relativity and Gravitation.
M. A. Tonnelat Marie-Antoinette Tonnelat-Baudot was born in 1912 in
Southern Burgundy. She gained her doctorate in 1939, under the supervision of de Broglie. In this work (which includes a little-known
paper coauthored with de Broglie) she investigated the relationship
between the wave mechanics of spin-2 particles (gravitons) and linearized general relativity. By the time of the conference she had a
long history of research in general relativity and unied eld theory
and was holding a position at the Henri Poincar Institute at the
Universit de Paris.
Germany
H. Salecker Helmut Salecker had been a research assistant of Eugene
Wigner, and was reporting on their joint work at the Chapel Hill
conference. He spent part of 1957 based in Princeton with Wigner,
working on general relativistic invariance and quantum theory, and
the work on quantum limitations of measurements of spacetime distances. Following this he took up a position at the Institut fr Theoretische Physik at Freiburg.

30

2. The Authors

Japan
R. Utiyama Ryy Utiyama was born in Shizuoka, Japan in 1916. He
received his PhD from Osaka University under the supervision of
Minoru Kobayashi in 1940. He had been a visiting fellow at the
Institute for Advanced Study at Princeton between 1954 and 1956,
where he was working on general gauge theory. He had done earlier
quite pioneering work on quantum gravity. He would be among the
rst postdoctoral fellows at the Institute for Field Physics at Chapel
Hill.
Denmark
S. Deser Stanley Deser was born in 1931, in Poland. He obtained his PhD
from Harvard in 1953, under the supervision of Julian Schwinger
with a thesis on Relativistic Two-Body Interactions. He had been
an NSF Jewett Postdoctoral Fellow at the Institute of Advanced
Study (IAS) between 1953 and 1955 under the supervision of Robert
Oppenheimer - he was also associated with the Rad lab at Berkeley
during 1954. At the time of the conference he was a member of the
Niels Bohr Institute in Copenhagen.
Sweden
B. Laurent Bertel Laurent had been Oskar Kleins last student. In the
years immediately preceding the Chapel Hill conference he had been
working on the rst attempt at a Feynman-style quantization of general relativity.
Turkey
B. Kursunoglu Behram Kursunoglu was born in Turkey in 1922. He received his PhD from Cambridge, where he was part of a group including Kemmer, Salam, and Dirac - Kursunoglu also attended Diracs
quantum mechanics lectures. He was the chief scientic advisor to
the Turkish general army sta at the time of the conference, and was
working on a unied eld theory.

The Original Chapel Hill Report

Foreword
Ccile DeWitt

A conference on The Role of Gravitation in Physics was held at the University of North Carolina, Chapel Hill, from January 18 to January 23, 1957.
It was planned as a working session to discuss problems in the theory of
gravitation which have recently received attention.
The present report was undertaken as a necessary requirement to
obtain conference funds. However, as the conference progressed, it became
more and more apparent that a report of the discussions would have a
scientic interest, partly because of an increasing number of requests for a
report from physicists unable to attend the conference, and partly because
of the nature of the discussions.
Research in gravitational theory has been relatively neglected in the
past two or three decades for several good reasons: (1) the lack of experimental guideposts, (2) the mathematical diculties encountered in the
study of non-linear elds, and (3) the experience of repeated early failures
to extend general relativity theory in a permanently interesting fashion.
A renewed interest in the subject has recently begun to develop, and the
Chapel Hill conference gave an opportunity to the few physicists actively
working in the eld - some having kept up an interest in it in spite of its
diculties, others having lately engaged in its study, often from a new
point of view - to discuss the preliminary results obtained and to present
new lines of approach.
This situation gave rise to very lively discussions. Obviously, most of
the material discussed is not ready for publication yet and - owing to the
diculty of the problems under consideration - may not be for a long time
to come. That is why an informal report of the proceedings is valuable, but
at the same time delicate to write up. An eort has been made to check
with the authors the report of their contributions to the conference, and
some have very kindly rewritten the draft proposed to them. However, the
time allotted to the preparation of the report had to be limited, because
the usefulness of such a report decreases with time more rapidly than its
quality increases; moreover, the reporters should not be asked to spend an
undue amount of time working on the report, which could be better spent

on original research. Consequently, no statement from this report

can be quoted without explicit permission from the author.
The papers which are ready for publication will appear in the July
1957 issue of Reviews of Modern Physics. A brief summary of the conference intended for non-specialists has been sent to Science.
It can hardly be said that the report gives a perfectly true picture
of the conference. The report has been prepared from notes taken during
the session, from material given by the authors, and from tape recordings.
(The reporters had hoped to have a stenographic transcript available, but
the cost of this transcript was beyond common sense.) Some contributions
have been very appreciably abridged, some are reproduced practically verbatim, some are extended, and some have not been recorded, depending
largely on the communication (both material and intellectual) between
authors on the one hand and reporters and editors on the other.
At the end of the conference, the participants expressed - together
with their belief in the importance of the subject matter - the wish to meet
again, if possible in Europe in the summer of 1958. Such meetings should
truly give an opportunity for discussions and for contacts between senior
physicists who have a broad and thorough understanding of the subject
and younger physicists of various backgrounds who have an interest in it.
The need is also felt for discussions and contacts between workers in the
eld for periods of time longer than conferences.
Thanks are due to the sponsors, the members of the steering committee, and the members of the Physics Department of the University of
North Carolina. The interest shown in the conference by Governor Luther
H. Hodges and by the ocers of the University was very gratifying. Special mention should be made of the cooperation of the University Extension
Division and of the many individuals who assisted in hospitality arrangements for the conferees. To all who have made the Chapel Hill conference
possible and have helped in its organization, this report is dedicated as a
token of appreciation.

Ccile M. DeWitt
Chapel Hill, North Carolina
March 15, 1957

Abstract
From January 18-23, 1957, a group of physicists from several countries
met at the University of North Carolina to discuss the role of gravitation
in physics. The program was divided into two broad headings: Unquantized and quantized general relativity. Under the former came a review
of classical relativity, its experimental tests, the initial value problem,
gravitational radiation, equations of motion, and unied eld theory. Under quantized general relativity came a discussion of the motivation for
quantization, the problem of measurement, and the actual techniques for
quantization. In both sections the relationship of general relativity to fundamental particles was discussed. In addition there was a session devoted
to cosmological questions. A large part of the relevant discussions is reproduced in this report in a somewhat abridged form. A conference summary
statement is presented by Professor P. G. Bergmann.

PROCEEDINGS
OF THE
CONFERENCE ON THE ROLE OF GRAVITATION IN PHYSICS
University of North Carolina, Chapel Hill, January 18-23, 1957

UNDER THE SPONSORSHIP

OF THE
International Union of Pure and Applied Physics, with nancial support
from
UNESCO
National Science Foundation
Wright Air Development Center, U.S. Air Force
Oce of Ordnance Research, U.S. Army
This conference was an activity of the North Carolina Project of the Institute of Field Physics, established in 1956 in the Department of Physics
of the University of North Carolina, Chapel Hill.
Its organization has been carried out through the Institute of Natural
Science of the University of North Carolina, Chapel Hill.
STEERING COMMITTEE
Frederick J. Belinfante, Purdue University
Peter G. Bergmann, Syracuse University
Bryce S. DeWitt, University of North Carolina
Ccile M. DeWitt, University of North Carolina
Freeman J. Dyson, Institute for Advanced Study
John A. Wheeler, Princeton University
MARCH 19571
Wright Air Development Center
Air Research and Development Command
United States Air Force
Wright-Patterson Air Force Base, Ohio
1 Ccile M. DeWitt mailed this report to the Air Force on March 18; on March 19, she
gave birth to her third daughter.

Reporters
W. A. Bowers
B. S. DeWitt
Ccile M. DeWitt
M. M. Duncan
J. M. Ging
J. R. Herring
E. M. Lynch
A. V. Masket
Eugen Merzbacher

Technical editor and typist

Ouida C. Taylor

Scientic editors
Ccile M. DeWitt
Bryce S. DeWitt

PARTICIPANTS IN THE CONFERENCE2

Anderson, J. L.
Bargmann, V.
Bass, R.W.
Belinfante, F. J.
Bergmann, P. G.
Bondi, H.
Brill, D. R.
Buckingham, M. J.
Davis, W. R.
Deser, S.
DeWitt, B. S.
DeWitt, C. M.
Dicke, R. H.
Ernst, F. J.
Estabrook, F. B.
Euwema, R. N.
Feynman, R. P.
Fours, Y.
Gold, T.
Goldberg, I.
Goldberg, J. N.
Havas, P.
Janis, A.
Kervaire, M.
Kursunoglu, B.

Stevens Institute of Technology

Princeton University
RIAS , Inc., Baltimore
Purdue University
Syracuse University
Kings College, London
Princeton University
Duke University
Oklahoma City, Oklahoma
Institute for Theoretical Physics, Copenhagen
University of North Carolina
University of North Carolina
Princeton University
University of Wisconsin
Oce of Ordnance Research
Princeton University
California Institute of Technology
University of Marseille
Cornell University
Syracuse University
Wright Air Development Center
Lehigh University
Syracuse University
Massachusetts Institute of Technology
Maltepe, Ankara, Turkey

2 Note that in the original report the front matter featured an index of (the more
substantial) contributions of the various participants. In the present version, for convenience, this has been incorporated into a more general index. I have also included M.
J. Buckingham who had been omitted in the original version.

2. The Authors

40

Laurent, B.
Lichnerowicz, A.
Lilley, A. E.
Lindquist, R. W.
Mace, R.
Misner, C. W.
Mjolsness, R.
Newman, E. T.
Pirani, F. A. E.
Rosen, N. (in absentia)
Rosenfeld, L.3
Sachs, R.
Salecker, H.
Schild, A.
Schiller, R. S.
Sciama, D.
Tonnelat, M. A.
Utiyama, R.
Weber, J.
Wheeler, J. A.
Witten, L.

Institute for Mechanics

& Mathematical Physics, Stockholm
Collge de France, Paris
Naval Research Laboratory, Washington
Princeton University
Oce of Ordnance Research
Princeton University
Princeton University
University of Pittsburgh
Kings College, London
Israel Institute of Technology, Haifa
University of Manchester
Syracuse University
Institute of Theoretical Physics,
Freiburg
Westinghouse Research Laboratories,
Pittsburgh
Stevens Institute of Technology
Trinity College, Cambridge
Henri Poincar Institute, Paris
University of Osaka, Japan
University of Maryland
Princeton University
RIAS, Inc., Baltimore

3 Rosenfeld had originally been denied a visa to attend the 1957 Chapel Hill conference
by the US Consul in Manchester (FBI le January 24, 1957). Ccile DeWitt telephoned
the US Attorney General and was able to convince him to reverse the Consuls decision.

Session I Unquantized General Relativity

Chairman: B. S. DeWitt

Chapter 3
The Present Position of Classical Relativity Theory and
Some of its Problems
John Wheeler

We are here to consider an extraordinary topic, one that ranges from

the innitely large to the innitely small. We want to nd what general
relativity and gravitation physics have to do with the description of nature.
This task imposes a heavy burden of judgement and courage on us, for
never before has theoretical physics had to face such wide subject matter,
assisted by so comprehensive a theory but so little tested by experiment.
Our problems fall under three heads. First: What checks do we have
today of Einsteins general relativity theory, and what can be done to
improve our tests? Second: What ways can we see to draw new richness
out of the theory? And third: What connection can we foresee between
this theory and the quantum world of the elementary particles?
Without going over all the evidence from observational tests of general
relativity, it is perhaps worthwhile to set down a few numbers relating to
the three traditional tests of the theory.
Table 3.1: Precession in seconds of arc per century

Planet
Mercury
Earth
Mars

Observed Value
42.56 0.94
4.6 2.7

Prediction from General Relativity

43.15
3.84
1.35

Table 3.1 compares the predicted and observed values of the precession of
the perihelion of Mercury and the Earth; for Mars there are not yet data
to give a check. The predicted eects are well within the experimental
error.

44

3. Present Position of Classical Relativity and Some of its Problems

The second well-known test is the deection of light in passing the

sun, which shows itself as the dierence between photographs of the star
eld taken when the sun is in its midst and those taken when the sun is
elsewhere. The expected eect is 1.751 seconds of arc deection for a ray
which passes the sun at grazing incidence. Although there has been some
disagreement among astronomers, the careful review of the evidence by
McVittie leads to acceptance of Mitchells result, based on the 1919, 1922,
and 1929 eclipses, of 1.79 .06 seconds of arc, which value has since been
conrmed by observations of the 1952 eclipse. The theoretical prediction
is thus in agreement with the data.
The third observational test of general relativity, the gravitational
red shift of spectral lines emitted on the surface of a massive star, presents
insuperable diculties in the case of the sun, owing to the fact that spectral
shifts originating in atmospheric currents and turbulence completely mask
the eect. On the other hand, white dwarf stars have mass to radius ratios
large enough to make these Doppler eects negligible in comparison with
the gravitational red shift. Observations are available on two stars, and
are summarized in Table 3.2.
Table 3.2: Observed and Predicted Gravitational Red Shifts

40 Eri B
Sirius B

Mass
SunMass

0.43 0.04
1

Radius
Sun Radius

0.016 0.002
0.008

Red Shift
Predicted Observed

17 3 21 4
79 60 80

It appears from these three traditional tests that general relativity is not
in disagreement with the observations.
The chief possibilities of improvement exist in the second and third
eects - that is, the bending of light and the gravitational red shift. We
hope to hear from Professor Dicke later what prospects there are for new
experiments. In particular, the development of the atomic clock may
begin to make feasible detection of gravitational red shift on the earth
itself.
To go on to the second general problem, namely, how to draw new
richness out of the theory on the classical level: there is a far-reaching
analogy between the equations of general relativity on the one hand and
those of Maxwells theory on the other; they are both second order partial

3. Present Position of Classical Relativity and Some of its Problems

45

dierential equations, and the future can be predicted from the knowledge
of initial conditions. From Maxwells equations, which are quite simple, a
wealth of information can be obtained: dielectric constants, scattering of
waves, index of refraction, and so on; the equations of general relativity
have, however, scarcely been explored, and much remains to be done with
them. Another task ahead of us is the construction of the curve giving
the spectrum of gravitational radiation incident on the earth, analogous
to the known curve of the electromagnetic radiation spectrum; or at least
the determination of upper limits on it. Until something is known about
this, we can scarcely be said to know what we are talking about.
We can, however, go on to explore some of these analogies on a deeper
level. First among the problems is the initial value problem. If I make
a diagram similar to a shueboard court, with dierent squares corresponding to the various unsolved problems, and with scores attached to
the various squares, we should assign a very high score - say 100 points to this problem. We know that in electromagnetic theory, if E and H are
given at time t = t0 , their values at later times can be predicted. However,
they may not be specied arbitrarily on the initial time-like surface, but
must obey certain conditions - namely the vanishing of their divergences.
Through the work of Professor Lichnerowicz, we now know the analogous
conditions in the case of relativity theory. They are non-linear conditions,
rather than linear ones as in the Maxwell theory. Unfortunately, however,
we do not know yet any super-potentials, analogous to A in electromagnetic theory, which can be specied arbitrarily and whose derivatives give
the eld quantities while automatically satisfying the supplementary conditions. This is one of the most important problems of relativity theory,
and one may well believe that it must be solved before further progress
with quantization of the theory can be made. The further problem of
the time-symmetric specication of initial conditions, which we know
can be done in electromagnetic theory by specifying H on two time-like
surfaces instead of specifying both E and H on one surface, and which is
the appropriate form for quantization, is also yet to be solved in relativity
theory.
Now this is a local condition; but boundary conditions in the large
must be considered, especially when considering systems with closed topology - and to this problem we may assign a score of, say, 25.
A third problem is that of translating Machs principle - that the distribution of matter in the universe should specify the inertial properties at
any given point of space - into concrete, well dened form. We know from
the work of Dr. Sciama that one can translate the dierential equations of

46

3. Present Position of Classical Relativity and Some of its Problems

the Einstein theory into integral equations, and that one has an analogy
between the solutions of the equations of static and radiative type, respectively, and the corresponding types of solutions of Maxwells equations:
namely, inverse square behavior of the static solutions, but inverse rst
power behavior at large distances, arising from the radiative terms. If one
sums these interactions over all masses in the universe, one sees something
of the connection between the inertia of one particle and the distribution of
the rest of matter in space. This problem of spelling out Machs principle
in a better-dened way we may assign a score of, say, 25. It is of course
related also to the problem of specication of conditions in the large, which
we mentioned earlier; it is also related to another issue: that of uniqueness
of solutions of the gravitational eld equations, to which we may assign,
say, 10 points.
To pass on from these issues, let us consider some further analogies
with electromagnetic theory. We know that if we have in a given frame of
reference a specied eld, there exists another frame of reference moving
with velocity
v 2(E H)
= 2
c
E + H2
in which E and H have been reduced to parallelism. We then have a
canonical situation in which, if we chose the z-axis as the direction of E
and H, any rotation in the x-y plane leaves the situation unchanged, and
any Lorentz transformation in the z-t plane also leaves the situation unchanged. Thus what Schouten has called a two-bladed structure of the
space-time continuum at any point is dened by the electromagnetic eld.
This is then a geometrical interpretation of what we have to deal with
in the electromagnetic case, and we would like to understand what the
analogous situation is in the gravitational case. Here we have to deal with
the quantities Ri jklm that measure the curvature of space - geometrical
quantities much more elaborate than the electromagnetic eld tensor, of
course. We would like to understand what kinds of invariance this quantity denes. We have the remarkable work of Ghniau and Debever on
this problem, but there are still issues to be dened in giving a clear understanding of what geometrical quantities are specied by the existence
of the gravitational eld. I would attach a score of, say, 15 to this problem
of the geometrical interpretation of the gravitational eld quantities.
This in turn leads us to another issue. Einstein has taught us to
think in terms of closed continua, rather than the open continua such as
Euclidean space; but since the dierential equations are purely local and

3. Present Position of Classical Relativity and Some of its Problems

47

say nothing about the topology in the large, we are obliged to consider
what the possibilities are. Recently we have become aware, for example,
of one way of understanding, in terms of topological notions, such a concept
as that of electric charge. Figure 3.1 below shows one intriguing kind of
topological connectedness in the small.

A
P1

P2

Figure 3.1
If one thinks of the upper region A as a two-dimensional space, there exists
the possibility of connecting two regions P1 and P2 by a wormhole so that
for instance an ant coming to P1 , would emerge at P2 without ever having
left the two-dimensional surface on which he started out. Now ordinarily,
if we have electric lines of force converging on some point in space, we
think of two possibilities: either Maxwells equations break down near the
point, or there exists a mysterious entity called electric charge at the point.
Now a third possibility emerges: the topology in the neighborhood of the
point may be such as to give a space-like binding to the lines of force; they
may enter the wormhole at P1 and emerge at P2 (see Fig. 3.2). Misner
has shown, in fact, that the ux through such a wormhole is invariant,
giving one a possibility of identifying this ux with the notion of charge.
Actually, there are strong reasons why this cannot be identied with an
elementary charge; our purpose is to classify and try to understand the

48

3. Present Position of Classical Relativity and Some of its Problems

topological possibilities of the classical theory. I would give this problem

a rather high score, say 50 points.

Figure 3.2
An objection one hears raised against the general theory of relativity is that the equations are non-linear, and hence too dicult to correspond to reality. I would like to apply that argument to hydrodynamics rivers cannot possibly ow in North America because the hydrodynamical
equations are non-linear and hence much too dicult to correspond to
nature! We need to explore the analogies between the gravitational and
the hydrodynamic equations; are there gravitational counterparts of such
hydrodynamic phenomena as shock waves, cavitation, and so on? A fairly
high score, say 40 points, should be given to this problem of the physics
of non-linear elds, and analogies with hydrodynamics.
The non-linear couplings between gravitation and electromagnetism
also give rise to new structures such as geons, which should be studied
further to enrich ones understanding on the physical side.
The last topic to be covered under the classical aspects of general
relativity is the topic of unied eld theory - namely, what can one nd in
the way of a description of electromagnetism which has the same purely
geometrical character as that which Einstein gave to gravity? Although
the various attempts to modify Einsteins theory in such a direction are
well known, it does not seem to be so well known that within the framework
of Einsteins theory one can construct, so to speak, an already unied
theory, as Misner and earlier Rainich have shown, based on the role of
the electromagnetic eld as a source of the gravitational eld through its
stress-energy tensor.

3. Present Position of Classical Relativity and Some of its Problems

49

The result is that one can write the equations of both electromagnetism and gravitation in a form in which only purely metric quantities
enter, as in Einsteins theory. It is a system of fourth order equations
which contain all the content of the two coupled second order equations
of electromagnetism and gravity. Further investigation of theories of this
kind deserve a score of, say, 25 points on our shueboard court.
In summarizing, let me remark that we have here, in the already
unied theory, electromagnetism without electromagnetism, just as we
have, in geons, mass without mass, and in connection with multiply
connected topologies, charge without charge. This then is just one mans
start at listing some of the problems with which we will be occupied in the
coming days.

Chapter 4
The Experimental Basis of Einsteins Theory
R. H. Dicke

It is unfortunate to note that the situation with respect to the experimental

checks of general relativity theory is not much better than it was a few years
after the theory was discovered - say in 1920. This is in striking contrast
to the situation with respect to quantum theory, where we have literally
thousands of experimental checks. Relativity seems almost to be a purely
mathematical formalism, bearing little relation to phenomena observed in
the laboratory. It is a great challenge to the experimental physicist to
try to improve this situation; to try to devise new experiments and rene
old ones to give new checks on the theory. We have been accustomed to
thinking that gravity can play no role in laboratory-scale experiments; that
the gradients are too small, and that all gravitational eects are equivalent
to a change of frame of reference. Recently I have been changing my views
about this. The assumption of the experimentalist that he can isolate his
apparatus from the rest of the universe is not necessarily so; and he must
remember also that the private world which he carries around within
himself is not necessarily identical with the external world he believes
in. Particularly in the case of relativity theory, where one thinks one knows
the results of certain experiments, it is all the more important to perform
these crucial experiments, the null experiments. For example, the Etvs
experiment has not been repeated, in spite of the tremendous improvement
in experimental techniques now available. An improvement by a factor of
1,000 in the accuracy of this experiment should be possible, and it is a
disgrace to experimental physics that this has not yet been done. It is a
measure of our strong belief in the foundations of relativity - the principle
of equivalence - that this has not been done; but an excellent example that
our beliefs are not always correct was provided for us by the events of last
week, when it was discovered that parity is not conserved in beta decay!
This experiment could have been done much sooner, if people had thought
it was worth doing.

4. The Experimental Basis of Einsteins Theory

52

Professor Wheeler has already discussed the three famous checks of

general relativity; this is really very imsy evidence on which to hang a
theory. To go a little further back: the Etvs experiment is in some ways
very powerful; it is really remarkable that such wide ranging conclusions
should be drawn from a single experiment. It measures the ratio of mass to
weight for an object, and sees whether this varies with the atomic weight
of the element involved, the binding energy of the atoms, and things of this
kind. One nds that, to about one part in 108 ; the ratio is independent
of the constitution of the matter. It does not prove exact equality of
mass and weight, contrary to the assertion in one well-known text on
relativity; it proves it to one part in 108 , which is quite a dierent thing!
To considerable accuracy, it also shows that electromagnetic energy has
the same mass-weight ratio as the intrinsic mass of nucleons, since in
heavy nuclei the electromagnetic contribution is an appreciable fraction of
the total binding energy. On the other hand, gravitational binding energy
is generally negligible and one cannot infer its constancy from the Etvs
experiment.
I would like now to refer to the table below, which may indicate where
some of the troubles lie.

PH Y SICA L A N D A STROPH Y SICA L CON STA N TS

(Units of , c, m)
1020

100

1040

Reciprocal weak
Reciprocal
Gravitational
Coupling Const.2
2
decay, decay  Coupling Const.

decay, etc.
Electrical force
Ratio

etc., etc.
2
Gravitational

1080

Masses Elementary
Particles

e2

e
c

Age of universe
Other Strong
Coupling Const.+2

Number of
Particles in
Universe to
Hubble radius

Hubble radius
of universe

Figure 4.1: Physical and Astrophysical Constants (Units of h , c, m)

4. The Experimental Basis of Einsteins Theory

53

These are the famous dimensionless numbers - and of course everyone

will say, Hes fallen into that trap! Ive written in dimensionless form
the important atomic and astrophysical constants, and arranged them according to order of magnitudes; we see that they group themselves in the
remarkable way indicated. The masses of elementary particles, the ne
structure constant, and other strong coupling constants are all of order
unity (using the term order of magnitude rather loosely!). Then the
weak coupling constants have reciprocals squared all of order 1020 . At
1040 we have the gravitational coupling constant; everything so far has
been an atomic constant. But also at 1040 we have the constants associated with the universe as a whole; the age and the Hubble radius; while
at 1080 we have the number of particles out to the Hubble radius.
What is queer about these? The atomic numbers are queer because if
we think nature is orderly and not capricious, then we would expect some
day to have a theory from which these numbers would come out, but to
expect a number of the order of 1040 to turn up as the root of our equation
is not reasonable. The other thing we notice is the pattern: 1040 being
1020 squared, and 1080 being 1040 squared; and nally the strange equality
of the universal constants - age of the universe, Hubble radius - to the
gravitational coupling constant, which is atomic in nature.
What explanations exist for these regularities? First, and what ninety
percent of physicists probably believe, is that it is all accidental; approximations have been made anyway, irregularities smoothed out, and there
is really nothing to explain; nature is capricious. Second, we have Eddingtons view, which I may describe by saying that if we make the mathematics complicated enough, we can expect to make things t. Third, there is
the view of Dirac and others, that this pattern indicates some connections
not understood as yet. On this view, there is really only one accidental
number, namely, the age of the universe; all the others derive from it.
The last of these appeals to me; but we see immediately that this
explanation gets into trouble with relativity theory, because it would imply that the gravitational coupling constant varies with time. Hence it
might also well vary with position; hence gravitational energy might contribute to weight in a dierent way from other energy, and the principle
of equivalence might be violated, or at least be only approximately true.
However, it is just at this point that the Etvs experiment is not accurate
enough to say anything; it says the strong interactions are all right (as
regards the principle of equivalence), but it is the weak interactions we
are questioning. I would like therefore to run through the indirect evidence
we have. There would be many implications of a variation with time of

54

4. The Experimental Basis of Einsteins Theory

the gravitational constant, and we must see what the evidence is regarding
them.
Assuming that the gravitational binding energy of a body contributes
anomalously to its weight (e.g., does not contribute or contributes too
much), a large body would have a gravitational acceleration dierent from
that of a small one. A rst possible eect is the slight dierence between
the eective weight of an object when it is on the side of the earth toward
the sun and when it is on the side away from the sun. This would arise
from the slightly dierent acceleration toward the sun of a large object
(the earth) and the small object. If we estimate his eect, it turns out to
be of the order of one part in 1013 on g, which I think there is no hope
of detecting, since tidal eects are of the order of two parts in 108 . The
mechanism of the distortion of the earth by tidal forces would have to be
completely understood in order to get at such a small eect.
Another way of getting at this same eect would be to look at the
period and orbit radius of Jupiter, and compare with the earth, to see if
there is any anomaly. Here the eect would be of the order of one part
in 108 , which is on the verge of being measurable. Possibly by taking
averages over a long period of time one can get at it; I havent talked with
astronomers and am not sure.
Another interesting question is this: are there eects associated with
motion of the earth relative to the rest of the matter of the universe? All
the classical ether drift experiments were electromagnetic; what about
gravitational interactions? Could their strength depend on our velocity
relative to the co-moving coordinate system? We can say something about
this, assuming the eect to be of the order of 2 , where is the ratio of
earth velocity to light velocity. We know the velocity of the sun relative
to the local galactic group; but we know nothing of the velocity of the
galactic group relative to the rest of the universe, except that it is not
likely to be much greater than 100 km/ sec. Supposing it unlikely that
the motion of the local group would be such as to just cancel the motion
of the sun relative to it, we can say that the velocity of the sun relative
to the rest of the universe is perhaps of the order of 100 km/sec. Then
the annual variation in 2 , owing to the fact that the earths velocity at
one time of the year adds to and at another time subtracts from the suns
velocity, amounts to about one part in 107 . This could conceivably give
rise to an annual variation in g which could be detected, for example, by
a pendulum clock. Now the best pendulum clocks are not quite up to this;
but an improvement of a factor of ten would make this eect detectable,
if it exists.

4. The Experimental Basis of Einsteins Theory

55

There is, however, an indirect way of getting at this eect: any annual variation of the gravitational interaction would give rise to an annual
variation of the earths radius, and hence of the earths rotation rate. Now
such an annual change is in fact observed. The earth runs slightly slow in
the spring. The eect is, within a factor two, roughly what you might expect from assuming 100 km/sec. and putting in the known compressibility
of the earth. However, it is possible to explain it also purely on the basis
of eects connected with the earth itself, such as variations of air currents
with seasons. Also the irregular character of the variation indicates at
least some contribution from such factors.
GOLD: It varies from one year to the next a bit, so it is certainly not to
be attributed to relativity.
DICKE: Its not too clear. Certainly part of it varies, but what the
variation amounts to is not too obvious because its measurement depends
on crystal clocks and they havent been too good until recently so we cant
go back far in time.
Now there is some other evidence on the question of a long period
change in the gravitational interaction as we go back into the past. If
gravity was stronger in the past, the sun would have been hotter, and we
ask whether we can then account for the formation of rocks and creation
of life. The accompanying gure (4.2) shows what we can say about the
temperature of the earth.

Figure 4.2
This gure gives the temperature of the earth as a function of time as we
go into the past, 0 on the scale representing the present. I have assumed

56

4. The Experimental Basis of Einsteins Theory

that as the temperature rises, the amount of water vapor present in the
atmosphere increases, and the sky becomes completely overcast. Then if
we use the known albedo of clouds (.8), and the black body radiation in
the infra-red, we get the curve shown for the temperature, assuming the
age of the universe to be 6.5 billion years. The temperature rises slowly
to about 50 C one billion years ago. Evidence for life as we know it
exists back to about 1.0 billion years. Life could have been present back
1.7 billion years, when the temperature would have been about 100 C.
According to biologists, algae from hot springs are known capable of living
at such temperatures, so life could have existed then. There is evidence
for a fairly sharp cut-o on the existence of sedimentary rocks at about 2.7
billion years, even though the solar system is about 4 billion years old. This
agrees with our curve, since at the temperature corresponding to that age,
about half the water would be in the atmosphere, and the pressure would
be so high that its boiling point would be about 300 C. There would
still be enough liquid water to form sedimentary deposits. However, at
slightly higher temperatures, the critical temperature of water would be
exceeded, only vapor would exist, and sedimentary rocks would not have
been formed.
I would say that the evidence does not rule out the possibility of the
suns having been hotter in the past.
There is some additional indirect evidence arising from the problem
of the formation of the moon. The moon has a density so low that there
are only two possible explanations: rst, dierent composition from the
earth; second, a phase change in the earth that leads to a very dense core.
The second is, however, hard to believe, because if the core is assumed
to be liquid iron, the total amount of iron in the earth is in agreement
with what we believe to be the abundance of iron in the solar system as
a whole. If we say the composition is dierent, we notice that it seems to
be about the same as that of the earths mantle. This in turn suggests
Darwins old explanation, that the moon ew out of the earth. The earth
was formed rst and was rotating with a period of about four hours, when
as a result of tidal interaction with the sun, a large tidal wave was set up
which split the earth into two parts; then as a result of tidal interaction
between moon and earth, the moon gradually moved out to its present
position. The latter part of this account is probably right, as we know
from the evidence on slowing down of the rotation rate as deduced from
comparison of Babylonian eclipse records with modern ones. However, if
the earth was rigid then as it is now, the natural frequency of oscillation
would have been too high for the rst part of the account to be correct.

4. The Experimental Basis of Einsteins Theory

57

With a liquid earth, the period is more nearly right, but Jereys objection,
that turbulent dampening would prevent the buildup of a wave to the
point of producing ssion, arises. However, Professor Wheeler has pointed
out recently that this objection may be met by including the eect of
the magnetic eld of the earth in damping the turbulence. Now the only
diculty is keeping the earth liquid, since the rate of radiation from a liquid
earth is so great that it would freeze up in a hundred years - nowhere near
a long enough time to set up the oscillation required. One thing which
could keep it liquid would be a hotter sun!
There is another bit of evidence on this: the moon is distorted in such
a way that it has been called a frozen tide; it corresponds somewhat to
the shape it would have had if it had frozen when at about a quarter of
its present distance from the earth.
GOLD: This is not correct; the present shape is not at all what it would
have been at any closer distance.
DICKE: This is true, but there is the possibility that the moon was somewhat plastic at the time of freezing so that the biggest distortions would
have subsided somewhat to give you dimensions compatible with what is
now seen.
GOLD: If the moon were formed by a lot of lumps falling together, it
would have an eect on how strong the moon is, and what disturbance it
could bear; then it could have three unequal axes, as it does.
DICKE: Yes; this is the other explanation for the moons shape, that big
meteorites piled into it and made it lopsided.
Another piece of indirect evidence on this is connected with the problem of heat ow out of the earth. There is evidence for the earths core
being in convective equilibrium; and heat owing out of the earths core
seems to be the only reasonable mechanism at present for a convective
core. The question is how that heat gets out. It may be that there is
radioactive material at the center which is the source of the heat owing
out; but potassium would be expected to be the biggest source of heat,
and it is so active chemically that we expect to nd it in the mantle only.
If, however, gravity gets weaker with time, there would be a shift along
the melting point curve of the mantle which would lead to a slow lowering
of the temperature of the core, and heat owing at such a rate as to keep
the mantle near the melting point. If we put in a reasonable melting point
curve for the mantle, we nd this contribution to the heat owing out of
the earth is of the order of one-third of the total heat. So the evidence

58

4. The Experimental Basis of Einsteins Theory

is not incompatible with the idea that gravity could be weaker with time,
even though radioactive materials are of sucient importance so that one
would not attempt to account for the heat ow out of the earth solely on
this basis.
There is some small bit of evidence that there may be something
wrong with beta decay: the beta coupling constant may be varying with
time. This comes from the evidence on Rubidium dating of rocks. The
geologists on the basis of their dating of rocks assign a half life for Rb 87
decay of about 5 1010 years. There has been quite a series of laboratory
measurements giving values of about 6 1010 years; on the other hand one
particular group has consistently got a value lower than ve. So this is up
in the air; but I think that before long we will have a denitive laboratory
value for the half-life.
Finally, l may just mention last weeks discovery that parity is not
conserved in beta decay. What the explanation for this is no one knows,
but it could conceivably indicate some interaction with the universe as a
whole.
DE WITT called for discussion.
BERGMANN: What is the status of the experiments which it is rumored
are being done at Princeton?
DICKE: There are two experiments being started now. One is an improved
measurement of g to detect possible annual variations. This is coming
nicely, and I think we can improve earlier work by a factor of ten. This is
done by using a very short pendulum, without knife edges, just suspended
by a quartz ber, oscillating at a high rate of around 30 cycles/sec. instead
of the long slow pendulum. The other experiment is a repetition of the
Etvs experiment. We put the whole system in a vacuum to get rid
of Brownian motion disturbances; we use better geometry than Etvs
used; and instead of looking for deections, the apparatus would be in an
automatic feed-back loop such that the position is held xed by feeding in
external torque to balance the gravitational torque. This leads to rapid
damping, and allows you to divide time up so that you dont need to
average over long time intervals, but can look at each separate interval of
time. This is being instrumented; we are worrying about such questions
as temperature control of the room right now, because wed like stability
of the temperature to a thousandth of a degree, which is a bit dicult for
the whole room.

4. The Experimental Basis of Einsteins Theory

59

BARGMANN: About three years ago Clemence discussed the comparison

between atomic and gravitational time.
DICKE: We have been working on an atomic clock, with which we will
be able to measure variations in the moons rotation rate. Astronomical
observations are accurate enough so that, with a good atomic clock, it
should be possible in three years time to detect variations in g of the
size of the eects we have been considering. We are working on a rubidium
clock, which we hope may be good to one part in 1010 .
BONDI: One mildly disturbing fact is this: the direction to the center of
the galaxy also lies in the plane of the ecliptic. Why should this be so?
I see no relation to the eects you have been discussing, but this might
be included in the list of slightly curious and unexplained facts about the
universe.
ANDERSON: There has been some discussion of the possibility that elementary particles have no gravitational eld - or that there exist both
active and passive gravitational mass.
GOLD: I would strongly advise the Etvs experiment with a lump of
anti-matter!
DICKE: One should also try seeing whether positronium falls in a gravitational eld - but how can one do it?
GOLD: I think a more relevant experiment is to do it with anti-protons or
anti-neutrons rather than positronium. Wilson at Cornell thinks it could
be done - for a few million dollars.
DICKE: Is there some reason why it is better to do it with anti-protons
rather than positrons?
GOLD: A scheme is possible according to which - not that I believe it anti-gravity would be possessed only by particles of a specic gravitational
charge; then it would not be possible to have more than one type of such
charge, and you would have to assign it to the nucleon.
WITTEN: We are starting at our laboratory some experiments to measure precisely the time dilation eect predicted by the special theory of
relativity. Two sets of observations exist regarding this prediction. One
is the famous experiment of Ives and Stilwell who observed the eect by
measuring the shifts of spectral lines omitted by moving atoms; the other
is the various measurements on the lifetime of the -meson in dierent

4. The Experimental Basis of Einsteins Theory

60

frames of reference. Neither method has so far been carried to its highest
attainable accuracy.
We are going to do the experiment using basically the techniques
of Ives and Stilwell with improvements. We shall observe lines emitted
or absorbed by hydrogen, helium, or lithium atoms or ions. The expected
accuracy at velocities comparable to those used by Ives and Stilwell should
be a factor of about 100 greater than theirs. By going to higher velocities
the relative accuracy should be greater. We hope to measure the angular
dependence of the eect; it has been measured so far only in one direction.
lves and Stilwell1 have observed in their experiment a tendency towards
disagreement with theory at high velocities and have suggested a possible
source of experimental error that leads to this tendency. This point should
be investigated. We shall eventually extend the measurements to v/c large
enough to detect eects of the fourth order (v/c)4 . We hope to develop
techniques suciently well to do a precise measurement of the eect at
highly relativistic velocities. Another goal is to make the time dilation
measurement for ions moving in a magnetic eld and being accelerated.
The purpose here is to see if there is any shift in the spectral line of a
kinematic origin due solely to the acceleration of the clock with respect to
the observer.

1 J.

Opt. Soc. Am. 31, 369 (1941)

Session II Unquantized General Relativity, Continued

Chairman: P. G. Bergmann

63

BERGMANN, who opened the session, proposed that the following

topics should be discussed together in this session: the integration of the
partial dierential equations of general relativity, the properties of their
solutions, the invariants of the theory, and various topological problems.
LICHNEROWICZ was the rst speaker. For details, including those of
notation and terminology, the reader is referred to his book, Thories
Relativistes de la Gravitation et de llectromagntisme, Paris, 1955.

Chapter 5
On the Integration of the Einstein Equations
Andr Lichnerowicz

This is a survey of the main results on the integration of the Einstein

equations and the problems which remain open.
5.1

The Space-Time Manifold

In any relativistic theory of the gravitational eld, the main element is a

dierentiable manifold of four dimensions, the space-time V4 . To be precise:
I assume that this dierentiable structure is C2 , piecewise C4 . This means
that in the intersection of the domains of two admissible systems of local
coordinates the local coordinates of a point for one system are functions of
class C2 , i.e., possess continuous derivatives up to second order, with nonvanishing Jacobian, of the coordinates of the point in the other system.
Third and fourth derivatives also exist, but are only piecewise continuous.
On V4 we have a hyperbolic normal Riemannian metric
ds2 = g dx dx

(5.1)

which is everywhere C1 , piecewise C3 . This metric is said to be regular on

V4 . Note that the manifolds which admit hyperbolic metrics are precisely
those on which there exist vector elds without zeros. Such a manifold
admits global systems of time-like curves, but generally does not admit
global systems of space-like hypersurfaces.
We suppose that no further specication of the dierentiable structure
of the manifold or of the metric has any physical meaning.
For the study of global problems assumptions belonging to the dierential geometry in the large are often needed. Frequently V4 is identied
with a topological product V3 R, R being the real numbers, where straight
lines are time-like curves and thus trajectories of a unitary vector eld u.
The space sections dened by V3 are then generally either closed manifolds
or complete Riemannian manifolds for the metric dened on them by

5. On the Integration of the Einstein Equations

66

ds2 = (g u u )dx dx

(5.2)

moreover, in this case V4 admits a Minkowskian asymptotic behavior at

innity.
The problem whether these assumptions are necessary or sucient is
open.
5.2 The Cauchy Problem for the Gravitational Field
In general relativity the metric ds2 satises the Einstein equations:
1
S R g R = T ( , = 0, 1, 2, 3)
(5.3)
2
where the energy-momentum tensor T is piecewise continuous. It denes
the sources of the eld. In the regions where T = 0 we have the exterior
case.
Let be a local hypersurface of V4 which is not tangent to the cones
ds2 = 0. If x0 = 0 denes locally, we have g00 = 0. The Einstein equations
can then be written as1
1
Ri j g00 00 gi j + Fi j = (Ti j gi j T ) (i, j = 1, 2, 3)
2

(5.4)

S0 G = T

(5.5)

where Fi j and G are known functions on if we know the values of the

potentials g and their rst derivatives (Cauchy data).
Eq. (5.5) links the Cauchy data to the sources. Eq. (5.4) gives the
values on of the six second derivatives 00 gi j which I have called the
signicant derivatives. The four derivatives remain unknown and may be
discontinuous at . But, according to our dierentiable structure, these
discontinuities have no physical or geometrical meaning. We grasp here the
connection between the covariance of the formalism and the assumptions
on the structure.
From Eq.(5.4) one sees that the signicant derivatives for can have
discontinuities only if is tangent to the fundamental cones, or if the
energy-momentum tensor is discontinuous at . Gravitational waves are
dened as the hypersurfaces f (x ) = 0, tangent to the fundamental cone,
i.e., such that
1 We

use the notation =

, =

2
x x

5. On the Integration of the Einstein Equations

67

1 f g f f = 0.

(5.6)

(WHEELER pointed out, as a matter of terminology that what is called

gravitational wave by Lichnerowicz is a purely geometrical concept, like
the eikonal in optics, and not the same as the physical gravitational
waves. The question whether gravitational radiation does or does not
exist was aired thoroughly in later sessions.) Gravitational rays are dened as the characteristics of 1 f = 0, or as the singular geodesics of the
metric.
In the hydrodynamic interior case the exceptional hypersurfaces are
(1) the gravitational waves, (2) the manifolds generated by streamlines,
and (3) the hydrodynamic waves.

5.3

Local Solutions for the Exterior Case

In the exterior case, T = 0, the system of Einstein equations has the

involution property: If a metric satises Eq. (5.5) and, on only, the
equations S0 = 0, then it satises these equations also outside of . This is
a trivial consequence of the conservation identities. Thus, for a space-like
hypersurface the Cauchy problem leads to two dierent problems:

1. The search for Cauchy data satisfying the system S f = 0 on ( f =

0). This is the problem of the initial values.
2. The evolutionary problem of integrating Eq. (5.4) subject to these
Cauchy data.
Assuming only dierentiability, Mme. Fours has recently solved this second problem locally by considering a dicult system of partial dierential
equations. She uses isothermal coordinates (for which x = 0) and thus
proves the local existence and physical uniqueness theorem for the solution of the Cauchy problem of the Einstein equations. The characteristic
conoid generated by the singular geodesics which issue from a point x plays
the essential part, and it is clear that the values of the solution in x depend only on the values of the Cauchy data in the part of interior to
the characteristic conoid. We obtain thus all the results of a classical wave
propagation theory.

5. On the Integration of the Einstein Equations

68

5.4 The Problem of the Initial Values

This, most interesting, problem is simple if is a minimal hypersurface.
If
ds2 = V 2 (dx0 )2 + gi j dxi dx j

(i, j = 1, 2, 3)

(5.7)

we take
1
i j = 0 gi j , K = ii , H 2 = i j i j
2
and we obtain for the exterior case
K = 0, j i j = 0, R + H 2 = 0

(5.8)

(5.9)

where is the operator of covariant dierentiation, and R the scalar curvature of . If we set
gi j = exp(2 ) gi j

(5.10)

where we assume the metric ds2 = gi j dxi dx j to be known on , then it is

possible to introduce the functions

2 = e , i j = e5 i j , L2 = gik gjl i j kl .

(5.11)

For these functions a very simple system of equations holds:

gi j i j = 0

j i j = 0

(5.12)

L2
.
7

(5.13)

and the elliptic equation:

8 + R =

The Dirichlet problem for Eq.(5.5) admits solutions if the domain of is

suciently small or if R 0 and if L2 is suciently small. This study can
be extended to some hydrodynamic interior cases, and it is thus possible to
construct examples of Cauchy data corresponding to the motion of n bodies. For the two-body problem, Newtons law is obtained approximately.
Recently Mme. Fours has extended my method to the general case.
It would be important to obtain new global theorems for this problem,
currently perhaps the most important problem of the theory.

5. On the Integration of the Einstein Equations

5.5

69

The Cauchy Problem for the Asymmetric Theory

The Cauchy problem of the asymmetric theory was studied in collaboration

with Mme. Maurer.
On the dierentiable manifold V4 we have:
1. An asymmetric tensor eld g of class C1 , piecewise C3 , with a
non-vanishing determinant g, the associated quadratic form dened
by h = g( ) being hyperbolic normal.
2. An ane connection of class C0 , piecewise C2 ; S is the torsion
vector of the connection.
If we substitute into this connection the connection L without torsion vector, which admits the same parallelism, the eld equations deduced from
the classical variational principle for the rst connection give two partial
systems. According to Mme. Tonnelat and Hlavat, the rst system gives
the connection L from g and the rst derivatives of the tensor. The eld
is now dened by g , S which satisfy the equations

2
(5.14)
R ( S S ) = 0, (g[ ] |g|) = 0
3
where R is the Ricci tensor of L. In addition we have a normalization
condition for S :

(g( ) )S

|g|) = 0.

(5.15)

The Cauchy problem can now be investigated. The system obtained still
possesses the involution property. The eld waves are hypersurfaces tangent to one of the following two cones:
(a) If l = g( ) we have the cone
(C1 ) l dx dx = 0

(5.16)

where l is dual to l .
(b) If =

2h
g h

l (h = det h ) we have the cone

(C2 ) dx dx = 0

where is dual to .

(5.17)

5. On the Integration of the Einstein Equations

70

If the skewsymmetric part of g is small compared to the symmetric part,

then C2 contains C1 . h itself has no wave properties in the unied eld
theory.
The evolutionary problem of the asymmetric theory remains unsolved.
5.6

Global Solutions and Universes

I now return to general relativity. The main question of the theory is the
following: When is a gravitation problem eectively solved?
A model of the universe - or shortly, a universe - is a V4 with a
regular metric satisfying the Einstein equations and certain asymptotic
conditions. When T is discontinuous through a hypersurface ( f = 0)
we assume that ds2 is always C1 , piecewise C3 in the neighborhood of
. The joining of the dierent interior material elds with the same eld
causes the interdependence of the motions, and the classical equations of
motion are due to the continuity through of the four quantities

V = S f .

(5.18)

In this view the main problem is to construct and study universes. This
being a hyperbolic non-linear global problem, it is very dicult to do this.
Clearly global solutions of the problem of the initial values would be very
helpful here.
Another approach might be the study of some elementary global solutions of the Einstein equations. It appears that such solutions are connected with the solutions with singularities introduced by various authors.
5.7

Global Problems

This view of the universe leads us to the following question: Is it possible

to introduce in a universe new energy distributions, the interior elds of
which are consistent with the exterior eld of the universe? Furthermore,
if a universe model is dened by a purely exterior eld, regular everywhere,
is the universe empty and thus locally at?
The regularity problem is the study of the exterior elds regular everywhere under some general topological or geometrical assumptions. Good
results are known only in the stationary case. We assume V4 = V3 R,
and we use the explicit assumptions made at the beginning of this survey.
Then we have the following results:
1. An exterior stationary eld which is regular everywhere is locally at

5. On the Integration of the Einstein Equations

71

(a) if V3 is compact, or
(b) if V3 is complete and has a Minkowskian asymptotic behavior.
2. In a stationary universe the exterior eld extended through continuity of the second derivative into the interior of the bodies is singular
in the interior.
3. A stationary universe which admits a domain surrounding innity
and which in this domain has a Minkowskian asymptotic behavior
and for which the streamlines are time lines, is static. There exist
space sections orthogonal to the time lines. This is of interest in the
Schwarzschild theory.
Much less is known in the non-stationary case, which is of course more
interesting. Concerning the regularity problem, some counterexamples
have been constructed, especially by Racine, Taub and Bonner. But for
these examples either the existence of the solutions is certain only in a nite
interval of time, or the constructed solutions do not behave Minkowskian
asymptotically. The general regularity problem is still an open one.
However, the results on the problem of the initial values give new
theorems under the following assumptions:
(a) It is possible to diagonalize the matrix i j which denes the second
fundamental quadratic form of and there exist foliations of by
the corresponding system of curves, that is to say by curvature lines.
(b) ds2 has a at asymptotic behavior.
(c) The tensor i j has a summable square on .
If these assumptions are satised on (x0 = 0) and also on the sections
corresponding to arbitrarily small x0 , the universe is static and thus locally
at.
***
MISNER rst stressed the physical ideas in which he and Wheeler were interested and which had led them to inquire into the details of the EinsteinMaxwell equations, particularly whether or not they are singular.
This comes about because, according to work of Rainich,2 the EinsteinMaxwell equations can be interpreted as an already unied eld theory.
Explicit mention of the electromagnetic eld is not necessary even while
2 G.

Y. Rainich. Trans. Am. Math. Soc. 27 106 (1925).

5. On the Integration of the Einstein Equations

72

working with a system equivalent in all cases, except the null electromagnetic eld, to the Einstein-Maxwell set of equations. One aim of unied
eld theory has always been the notion that elds are more fundamental
than particles, and that it should be possible to construct all particles
from the purely geometrical concept of the eld. To allow singularities
in the elds to represent the particles would be a delusion, since then
the stress tensor is not merely the electromagnetic one but includes mass
terms, even though idealized to delta functions. Since we wish to be careful about singularities the results of Mme. Fours and Lichnerowicz3 have
been important to us. They assure us that if we specify certain initial conditions we shall have non-singular solutions at least for a short time. We
are interested in nding solutions to these initial value equations and seeing what they lead to, whether they indicate any possibility of constructing
particles from these electromagnetic and gravitational elds. Notice that
even though one speaks of the electromagnetic eld here, one does not
really have to, since one can use the ideas of Rainich to give a complete
description of the electromagnetic eld in terms of metric quantities alone.
The inuence of the electromagnetic stress energy tensor upon the gravitational eld is suciently specic so that from the curvature quantities
one can work backwards to the electromagnetic eld and have a purely
metric description of both electromagnetism and gravitation.
We have studied a certain class of solutions to the problem of initial
values. These solutions are global but by no means the most general. They
serve as examples. These solutions are always continuous and instantaneously static. We nd that there exist exact solutions of the EinsteinMaxwell equations for which the elds on a certain surface t = 0 are given
by:

ds2 = dt 2 + ( 2 2 )2 (dx2 + dy2 + dz2 )

Fi0

= 2( 2 2 )1 (

Fi j

=0

(5.21)

g
=0
t

(5.22)

where , satisfy
3 A.

i)
xi
x

(5.19)

Lichnerowicz, loc. cit., p. 50.

(5.20)

5. On the Integration of the Einstein Equations

2 = 2 = 0,

2 =

73

2
2
2
+ 2 + 2.
2
x
y
z

(5.23)

This metric together with the electric eld Fi0 gives a solution of the initial value equations. In the neighborhood of the surface t = 0 we have
a solution of the time-dependent equation but do not know what it is.
Perhaps one should use a high speed computing machine. The regularity
conditions of Lichnerowicz, that the three-dimensional manifold should be
either compact or complete but asymptotically at, lead to elds which
are completely free of singularities. The functions and can only have
the form:

= 1+
a

a
a

a
a

(5.24)

where

a = |

a|

a > |a |

x2 + y2 + z2 .

(5.25)

These functions appear to contain singularities, but they really do not.

This brings us into the study of the topology of this situation. The point
at which the metric seems to become singular can be thrown away. This
is consistent with the requirement of completeness, but would not be consistent if the point which is thrown away were one at which the metric did
not become singular. A picture can be drawn to indicate what the space

so obtained looks like. Near the point

a the space, which is at further
away, curves and ares out and goes over into another roughly at portion
like this:

Figure 5.1

74

5. On the Integration of the Einstein Equations

We have here as many regions where the space is asymptotically at as we

have particles. The next problem is to put these portions together. One
should then try to construct the initial values which would give rise to the
wormhole picture of Wheeler (see Session 1). All these things are possible;
they just have to be worked out.

Figure 5.2: Wormhole

One additional advantage of the solutions exhibited here over introducing point singularities into the theory is that by using Lichnerowiczs
regularity conditions you eliminate automatically all negative masses, mass
dipoles and higher multipoles. It also, unfortunately, implies that for any
particle or cluster of particles described by these deformities the total
charge and mass have a ratio e/m 1, while for an electron in these units,
where G = c = 1, e/m = 2.04 1021 . The reason for this too large mass, for
a given electron charge, is that you dont have enough of a cut-o for the
electromagnetic mass. Perhaps one might hope that quantum mechanics
might help here. In any event, we have here strong indications that the
Einstein-Maxwell theory is a good model for a unied eld theory. It can
tell us what a unied eld theory may have to say about physics.
In the discussion, PIRANI asked for the denition of completeness.
MISNER said that a manifold is complete if all geodesics can be continued
to innite length.
BERGMANN compared the present topology with the symmetric one
of Einstein and Rosen.4
DE WITT noted the similarity between Misners model and the Schwarzschild solution for a charged mass particle. MISNER agreed and said that
he had been led to his solutions by the observation that the space part of
the Schwarzschild metric can be written in the form
4 A.

Einstein and N. Rosen. Phys. Rev. 48 (73) (1935).

5. On the Integration of the Einstein Equations

[(1 +

m 2 e2 2 2
) 2 ] (dx + dy2 + dz2 ).
2r
4r

Figure 5.3: Einstein-Rosen Topology

MlSNER then added some remarks on global solutions.

75

(5.26)

Chapter 6
Remarks on Global Solutions
C. W. Misner

Suppose we consider the following set of equations:

j
;j

E;ii = 0

(div E = 0)

(6.1)

H;ii = 0

(div H = 0)

(6.2)

= g i jk E j H k Si

(Poynting vector)

1 2
1
1
i j i j + R(3) = (E 2 + H 2 )
2
4
2

(6.3)
(6.4)

where = ii and i j = 1/4(i j gi j )

gi j
.
t
We study these equations in order of diculty. Eq.(6.1) has as its most
general solution:
i j =

1 ij
i
Ei =
( gA ) + ECoulomb
g xj

(6.5)

i
where ECoulomb
has vanishing divergence and also vanishing curl:

E i, j E j,i = 0

(6.6)

but

gi jk E i dx j dxk = 0 for some closed surface.

(6.7)
8
This is only possible if wormholes are present. There are as many independent solutions as there are wormholes, and one can specify arbitrarily
Q

6. Remarks on Global Solutions

78

the charge of each. Eq.(6.3) can now be regarded as an inhomogeneous

linear equation. A solution of the inhomogeneous equation exists on a
compact manifold if and only if

Ki Si gd 3 x = 0

(6.8)

for every Killing vector Ki (generating a group of motions) satisfying

Ki; j + K j;i = 0.

(6.9)

Whether this has any signicant applications I do not know. About Eq.
(6.4) I can say nothing.
l conjecture that the homogeneous equation could be solved by the
use of potentials, provided we use a rather broad idea of a potential. By
this I mean a solution which somehow involves the use of an arbitrary
function. To illustrate the idea I give a particular class of solutions to the
equation ;ijj = 0, namely:
1
] f (R(3) ) + ij f (R(3) )
(6.10)
2
where f is an arbitrary function of the scalar curvature on the surface, and
f is the derivative of f . i is the covariant derivative.
Besides the problems of topology in the small associated with ares
and wormholes, or with the symmetric Einstein-Rosen picture, there are
the problems of topology in the large. These are familiar problems: Is
the universe closed and spherical in a reasonable approximation, or is it
open? Other topological problems are the problems of classifying threedimensional compact manifolds. One can at present write out a list containing all three-dimensional compact manifolds, but one has no way of
telling whether two are the same or dierent by a nite algorithm, since the
list may contain redundant cases. Most mathematicians seem to believe
that the solution to this problem is not in sight.
Another point is this: Once you take the course indicated here and
decide to make a unied eld theory to avoid singularities and you bring
in the possibility of non-Euclidean topology, then you would like to know
the limitations on the number of topologies that are likely to appear. The
problem is not one of nding ingenious enough topologies to do what you
want them to do, but rather to nd a reason why a host of topologies will
never appear. One must examine the 4-dimensional manifold. Professor
Lichnerowicz has mentioned that for most manifolds which are not of the
type V3 R there will not exist reasonable space-like surfaces. However,
(3) j

ij = [i j ij k k Ri

6. Remarks on Global Solutions

79

the limitation V4 = V3 R might be too restrictive. In other words: Can the

number of particles in a universe with wormholes change? For instance,
one might ask if a wormhole can disappear in time. Can it happen that
starting from a wormhole topology at t = 0 you arrive at a Euclidean space
at t = 1?

Figure 6.1
From the work of Thom1 one can show that there are four-dimensional
manifolds which have these two surfaces as boundaries. The question
whether a continuous metric of Lorentz signature can be put on such manifolds can, I believe, be answered. In such a metric can you nd solutions
to any hyperbolic equations, or better, Maxwells equations? Anyone trying to study this would be much aided by certain qualitative classications
of metrics. I conjecture that if you divide the metrics which it is possible to put on a given manifold whose two boundaries are space-like, into
homotopy equivalence classes, then ordinarily there will not be more than
one of these classes which allows any solutions to a hyperbolic equation;
i.e., most homotopy classes are inconsistent with hyperbolic dierential
equations.
MISNER concluded with a few remarks about local invariants, such as
those given by Ghniau2 and others: Instead of the string of components
of the Ricci tensor, one looks at all the invariants which can be constructed
from these. One possible use of these might be as variables in discussing
the quantization of general relativity, since quantization seems to handle
1 Comment Math. Helv. 28, 17-86 (1954), Theorem IV, 13. Also see B. A. Rokhlin,
Doklady Akad. Nauk. SSSR 81, 355 (1951).
2 J. Ghniau and R. Debever, Bull. Cl. des Sci., Acad. roy. Belg., 5e Sr., 42,
(1956).

80

6. Remarks on Global Solutions

invariant quantities better than tensor components. Also it would be interesting to see Gedankenexperiment to measure each of these curvature
invariants. This would be a beginning for making simple quantum mechanical arguments about measurability in general relativity. Perhaps Pirani
could give a characterization of pure gravitational radiation in terms of
these invariants?
BERGMANN pointed out that in the study of the equations of general
relativity today one seems to have a choice of obtaining either general
local properties or special global theorems.
MME. FOURES then gave her solution of the problem of the initial values
using Cartans exterior dierential calculus:

Chapter 7
Solving The Initial Value Problem Using Cartan Calculus
Y. Fours

I write for the space-time manifold

ds2 =

(7.1)

=0

where I take as the basis in my four-dimensional manifold one vector orthogonal to the initial hypersurface, so that

0 = V dx0

(7.2)

i = ai j dx + i dx
j

(i, j = 1, 2, 3).

(7.3)

Then the dierential relations for the eld can be written in the form

S0k = h Phk k P

(7.4)

S00 = R + H P , P = Phh , H = Phk Phk

2

(7.5)

where h is the covariant derivative, and Phk is what Lichnerowicz in the

orthogonal calls hk . (For further notation see Section 5.4 of the talk by
Lichnerowicz earlier in this session.) If the coordinates were orthogonal
we would have i = 0, and hence
Phk =

1
0 ghk .
2v

(7.6)

The coecients Phk are the coecients of the second fundamental quadratic
form of the hypersurface .

82

7. Solving The Initial Value Problem Using Cartan Calculus

We are dealing with the purely gravitational case. To solve the dierential equations I choose on the hypersurface three particular vectors, the
eigenvector of the tensor Phk , that is to say we will have
Phk = 0

for

h = k.

(7.7)

P11 , P22 , P33 are unknowns. The metric of the hypersurface is

ds2 = 2i .

(7.8)

If I set u1 = P22 + P33 , et cycl., and if I use Lichnerowiczs idea to take the
metric on the hypersurface conformal to a given metric, denoted by an
asterisk:
ds2 = e2 ds2

(7.9)

then I obtain the following system of equations:

1 u1 1 (u2 + u3 2u1 ) j1kk

u1 = 0 etc.

42

2(i )2 + Le2

+R = 0

(7.10)
(7.11)

where 2 is the Beltrami operator of second order, and where

L = H 2 P2 , P = P11 + P22 + P33 , H 2 = Phk Phk .

(7.12)

It is possible to solve this system of equations which are linear with respect to the highest derivatives. For instance, I can solve them by giving
the values of the unknown u in my three-dimensional space on a twodimensional variety S, and I can solve then the Cauchy problem for the
system which contains . A general iteration method can be used to solve
the set of equations by writing them in the form of integral equations using
Greens functions. If I give the value of on the boundary, and the values
of the u on S, then I nd the general solution of these equations. This is a
rigorous mathematical treatment of the problem of initial values.
WHEELER pointed out the analogy with the electromagnetic case where

the equation div E = 0 yields E in all space if it is known over a twodimensional surface. However, one would still like to have a potential
which gives the eld automatically without having to do any integration.

7. Solving The Initial Value Problem Using Cartan Calculus

83

MME. FOURES, answering a question of MISNER, pointed out that the

statement that the metric is conformal to a Euclidean one is an assertion,
not an assumption, and that her method gives general solutions for an
open space.
DE WITT asked if the Cauchy problem is now understood suciently
to be put on an electronic computer for actual calculation. Do we now
know enough about constraints and initial conditions to do this at least
for certain symmetrical cases?
MISNER replied that you could take some of the particular solutions which
he has given, and which start out as instantaneously static elds, or the
interesting example of two particles which start to rotate about each other,
and you could perform actual calculations. You start out with a Euclidean
metric and would use the method of Mme. Fours proof.1 On the surface
z = 0 you put

e2 = 1+

m m
+
r1 r2

(7.13)

r1 and r2 being distances measured from two points in space. Then you
assign initial conditions for the eigenvalues u. You want to specify u in
such a way that the velocities of the two particles are initially like that:

Then you try to use the same method which was used for the general proofs
to nd particular solutions not just on z = 0 but through all space. These
can be interpreted as two particles which are non-singular, or they can be
thought of as the kind of 1/r type singularity of which one ordinarily thinks
in gravitational theory. These partial dierential equations, although very
dicult, can then in principle be put on a computer. MISNER thinks
that one can now give initial conditions so that one would expect to get
gravitational radiation, and computers could be used for this.
DE WITT pointed out some diculties encountered in high speed computational techniques. Singularities are of course dicult to handle. Secondly, any non-linear hydrodynamic calculations are always done in socalled Lagrangian coordinates, so that the mesh points move with the
material instead of being xed in space. Similar problems would arise in
applying computers to gravitational radiation since you dont want the
radiation to move quickly out of the range of your computer.
1 J.

Rat. Mech. Anal. 2, 951 (1956).

84

7. Solving The Initial Value Problem Using Cartan Calculus

BERGMANN remarked on the general discussion of the afternoon that six

years ago it was discovered that one can nd a closed form for the Hamiltonian of the Einstein equations, and it was thought then that all troubles
with the Cauchy problem for gravitational theory were over. This is not
so, for the simple reason that there are eight constraints for the twenty
canonical variables. Of the remaining twelve variables eight have nothing
to do with the problem, they correspond to an arbitrariness in the choice
of coordinates, so that one is left with four canonical variables. Can you
introduce new variables such that these variables could be set up arbitrarily? This is essentially a search for potentials in the general sense in which
that term has been used here. BERGMANN is personally convinced that
the search for locally dened potentials is probably hopeless, because they
probably do not exist. This is only a belief, of course.
ANDERSON commented on the problem of two-surface boundary conditions. He suggested that since it is possible to put the gravitational
Lagrangian into a form in which the time derivatives of g0 do not appear,
just as the time derivative of the scalar potential does not appear in the
electromagnetic Lagrangian, it may be possible to break up gi j in a fashion
analogous to the separation of the vector potential in the electromagnetic
case into transverse and longitudinal parts. Only one of these parts appears
in those equations of motion which are free from second time derivatives.
One could then initially and nally specify the part which does not appear
in these equations, and obtain a complete solution. There may be one
catch to this. There is reason to believe that one of the eld equations,
which do not depend on second time derivatives, is really the Hamiltonian
of the theory. If that is the case one would not be allowed to specify
independent initial and nal values for the quantities just discussed.
Discussion of these points, which are closely related with the problem
of quantization, was postponed until later.
WHEELER returned to the central problem of the afternoon, the initial
value problem. He asked whether, leaving technical details aside, there is
a point of view which would permit us to nd our way around some of
these subtleties.
I would like to raise in this connection the question of the use of the
variational principle itself. The variational principle in general relativity
has a character rather dierent from that in electromagnetic theory. In the
latter we have the square of the eld variables, whereas in the gravitational
case we have the curvature, i.e., the second derivatives of the potentials
entering. This has always puzzled us: Why is the variational formulation

7. Solving The Initial Value Problem Using Cartan Calculus

85

of the gravitational eld so dierent? I would like to emphasize that the

convenience which one so often has sought in transforming by partial integration the second derivatives into rst derivatives has concealed what the
equations are trying to tell us. I would like to illustrate this by a trivial
example. Take the free particle of ordinary mechanics. The variational
principle is:

2
1 2
x dt = 0.

(7.14)

How do we tell what to x at the endpoints of the time interval? We do

this by performing an integration by parts:
= x x|21

2
1

x xdt.

(7.15)

From this we draw two separate conclusions:

1. x = 0.
2. x = 0 at the two endpoints of the motion, i.e., we must specify
the coordinate x at the times 1 and 2. This is a time-symmetric
specication. One is so accustomed to this pattern of thinking that
when general relativity presents us with a dierent pattern we are
not prepared for that. Therefore, let me introduce a problem which
does have the pattern characteristic of general relativity, and let us
see how we would face up to it. Consider this:

(7.16)

xxdt
= 0.
1

Traditionally one abhors these second derivatives in the problem, so

one quickly transforms it into the previous problem. I propose to
deal rather directly with the new problem like this:
= (x x x x)|21 + 2

x xdt.

(7.17)

The conclusions would now be:

1. x = 0.
( )
2. xx = 0, i.e., we are to specify the value of
of the motion.

x
x

at the two endpoints

7. Solving The Initial Value Problem Using Cartan Calculus

86

In other words, we are led here to a dierent kind of specication of the

problem. I merely want to point out that the variational problem itself
has the property if we dont monkey with it, to tell us what it is that we
should deal with.
In the case of electromagnetism we can also let the variational principle tell us what the quantity is that we are concerned with. Note the
equation:2

2
1

(E H ) d = 0 =
2

A
dS
|21 volume integral
E

(7.18)

The volume integral gives us the usual eld equations. The boundary
term tells us that we should specify Atransverse at the limits. If you specify
A itself, rather than its transverse part, you would be taking too seriously
the demand that the boundary term must vanish. Clearly you dont have
to be that hard on A. Analogous considerations can and have been applied
to the general relativity by Misner and Fletcher.
MISNER outlined his present, tentative results on this question. He noted
that one has not yet been able to specify the gravitational variables which
are analogous to Atransverse in the sense mentioned by Wheeler. With
arguments based on the normalization in Feynman integration one can
single out the invariant variational principle:

R gd = 0.

(7.19)

From this one gets the surface term of form:

1
[ i j i j + i j i j ] g d 3 x
2
gi j jk = ik

i j =

gi j
t

if Gaussian coordinates are used.

Otherwise the formula for ij is long.

The suggestion is that if you look at the surface term carefully you will
nd what the correct variables to use are, and what you can specify on the
initial and nal surface.
2 We

are using the notation of Landau and Lifshitz, Classical Theory of Fields.

7. Solving The Initial Value Problem Using Cartan Calculus

87

MISNER summarized the discussion of this session: First we assume

that you have a computing machine better than anything we have now,
and many programmers and a lot of money, and you want to look at a
nice pretty solution of the Einstein equations. The computer wants to
g
know from you what are the values of g and
t at some initial surface,
say at t = 0. Now, if you dont watch out when you specify these initial
conditions, then either the programmer will shoot himself or the machine
will blow up. In order to avoid this calamity you must make sure that the
initial conditions which you prescribe are in accord with certain dierential
equations in their dependence on x, y, z at the initial time. These are what
are called the constraints. They are the equations analogous to but much

more complicated than div E = 0. They are the equations to which we have
been nding particular solutions; and on the other hand, Mme. Fours has
shown the existence of more general kinds of solutions. Mme. Fours has
told us that to get these initial conditions you must specify something else
on a two-dimensional surface and hand over that problem, the problem of
the initial values, to a smaller computer rst, before you start on what
Lichnerowicz called the evolutionary problem. The small computer would
prepare the initial conditions for the big one. Then the theory, while not
guaranteeing solutions for the whole future, says that it will be some nite
time before anything blows up.
LICHNEROWICZ emphasized that one seeks to answer the question whether
V4 is compact and whether V4 is orientable.

Chapter 8
Some Remarks on Cosmological Models
R. W. Bass and L. Witten

It is unusual to hypothesize that the four-dimensional space-time universe

of general relativity is compact (i.e., nite). But in such a case several
interesting conclusions can be drawn. In the rst place, if the mass distribution is assumed to be continuous, so that the metric tensor has no
singularities, then the Euler-Poincar characteristic of the universe must
be zero [6]. This implies, for example, that the universe cannot be a fourdimensional sphere. It also implies that a nite universe cannot be simplyconnected, in the sense that the rst Betti number cannot vanish. This is
reminiscent of Professor J.A. Wheelers non-simply connected models.)
In the second place, it seems to be generally known that in a nite
cosmology there must exist a closed curve in space-time whose tangent
vector at every point is time-like. Professor L. Markus has indicated
a proof to us. Let V4 denote the 4-manifold of the universe. Now, on V4
construct a continuous, nowhere vanishing eld of time-like vectors ([6], pp.
6-7; cf.[7], p. 207). By Birkhos fundamental theorem on the existence
of recurrent orbits in compact dynamical systems [3], there must exist
either an orbit of the type sought or else an almost-closed time-like orbit
which can serve for the construction of such a closed orbit by an obvious
procedure.
A more standard hypothesis, however, is that the universe V4 is not
compact, but is the topological product of the innite real line (a time axis)
with a 3-manifold V3 . The manifold V3 is often assumed to be compact,
and any local (hence experimentally veriable) condition which implies
compactness is of much interest. For example, if V3 has constant curvature
K then V3 is compact if, and only if, K is positive ([4], pp. 84 and 203), and
in this case is a 3-sphere if its rst Betti number vanishes, and in general
admits the 3-sphere as a covering space.
We wish to point out a new method for studying the topology of
manifolds such as V3 and V4 . This method consists of the construction of
a continuous, nowhere vanishing, irrotational vector eld on the manifold

90

8. Some Remarks on Cosmological Models

under consideration. Once such vector eld has been constructed, we can
assert that either the manifold is non-compact (i.e., open or innite), or
that it cannot be simply-connected.
We shall prove a slight generalization of this theorem; but rst, let us
note that a similar, but more restrictive and less easily applicable condition
is a trivial consequence of Hodges well-known theorem that the number
of linearly independent harmonic vector elds on a compact Riemannian
manifold is equal to its rst Betti number. For if after constructing on our
manifold an irrotational vector eld (which is non-trivial but may vanish
at more than one point), we then ascertain that it is also solenoidal (i.e.,
of vanishing divergence), then the vector eld must be harmonic ([9], p.
56).
Theorem 1 (Hodge): Let Vn be an n-dimensional Riemannian manifold
(with positive denite metric tensor), and let F denote a non-trivial
class C2 vector eld dened on Vn . Suppose that the curl and the divergence of F both vanish identically; or equivalently, suppose that
the eld F satises the generalized Laplace equation for harmonic
vector elds. Then, if Vn is compact, its rst Betti number is not
zero.
Corollary (Bochner-Myers): If Vn is orientable and has positive denite
Ricci curvature throughout, then its rst Betti number vanishes.
([9], p. 37).
Recall that the curl tensor of a vector eld is independent of the metric
tensor, and so is a non-metric notion. Accordingly, the following theorem
applies equally well to V4 with its indenite hyperbolic metric as to V4 with
its positive denite Riemannian metric.
Theorem 2: Let Vn be an n-dimensional dierentiable manifold, and let F
be a continuous, class C1 vector eld dened on Vn . Suppose that
F vanishes at most once and that its curl vanishes identically on
Vn . Then either Vn is non-compact, or Vn is compact and its rst
Betti number does not vanish. In either case, of course, if F actually
vanishes nowhere, the Euler-Poincar characteristic of Vn is zero.
For non-vanishing F this theorem is a consequence of a more general theorem [1] which applies, for example, to manifolds with boundary. In fact,
by a generalization to arbitrary ows of a theorem proved by Lichnerowicz
for a very special class of ows ([5], p. 79), we can prove [2] that Vn is
homeomorphic to the product of the real line with an (n 1)-dimensional

8. Some Remarks on Cosmological Models

91

space which is a connected subset of a Vn1 . But in the present case, because we are dealing with a manifold, there is a much simpler proof. We
wish to thank Professor Kervaire for pointing out to us this simpler proof
during the Conference on the Role of Gravitation in Physics. The proof
runs as follows. If Vn is simply connected, then the generalized Stokes
Theorem assures us that there exists on Vn a single-valued scalar potential
function of which F is the gradient eld. (See the survey of vector analysis
in [8].) But if Vn is compact, this potential function must assume both
its maximum and minimum values on M, and at these extreme points the
gradient must vanish. This contradicts the hypothesis that F has at most
one zero on Vn , and so proves the theorem.
It is possible that Theorems 1 and 2 have applications to the study of
specic cosmological models. In fact, there are many ways of constructing
on V3 , or on V4 continuous vector elds which are unique once the indenite
metric (or set of gravitational potentials) for V4 has been specied.
Professor J.A. Wheeler has pointed out to us an application of Theorem 2 to V4 .
Theorem 3: Consider the combined Einstein-Maxwell eld theory on V4 .
If the vector eld
ui = i jkl Rmj;l

Rmk pq
R
R pq

is dened everywhere and of class C1 on V4 , and if ui does not vanish

more than once, then the universe V4 cannot be compact.
The vector eld ui , which was dened by Dr. C. W. Misner, is essentially
the gradient of the ratio (in a certain coordinate system) of the electric to
the magnetic eld strength. Dr. Misner has shown that Maxwells equations imply that ui; j u j;i 0. Hence, Theorem 3 follows from Theorem
2.
References
[1] Bass, R. W. A Criterion for the Existence of Unstable Sets. A RIAS
preprint, December 26, 1956. Copies available on request.
[2] . Topological Dynamics and Non-linear Dierential Equations. A RIAS preprint, to appear.
[3] Birkho, G. D. Dynamical Systems. New York, 1927.

92

8. Some Remarks on Cosmological Models

[4] Eisenhart, L. P. Riemannian Geometry. Princeton University Press,

1949.
[5] Lichnerowicz, Thories Rlativistes de la Gravitation et de lElectromagntisme. Masson et Cie., Paris, 1955.
[6] Markus, L. Line Element Fields and Lorentz Structures on Dierentiable Manifolds. Annals of Mathematics 62 (1955), 411-417.
[7] Steenrod, N. The Topology of Fibre Bundles. Princeton University
Press, 1951.
[8] Weyl, H. The Method of Orthogonal Projections in Potential Theory.
Duke Mathematical Journal 7 (1940), 411-444.
[9] Yano, K. and S. Bochner. Curvature and Betti Numbers. Annals of
Mathematics Studies No. 32, Princeton, 1953.

Session III Unquantized General Relativity, Continued

Chairman: H. Bondi

95

Introductory remarks to the session on gravitational radiation were

made by BONDI.
There are two quite dierent attitudes one can take toward the general
theory of relativity. First, one can regard it principally as a theory of
gravitation. Then one knows that it must be an open theory into which
knowledge gained in other elds can be tted. Secondly, one can take the
attitude that relativity is more than other theories in that it says more
than just something about gravitation; and then one supposes that there
are somehow some wonderful clues hidden in it. The latter attitude, which
is what the speakers of yesterday put forward, is one which Bondi cannot
accept.
The theory of relativity is based on two principles: (1) the principle
of equivalence, and (2) the general principle of relativity. The principle
of equivalence, we all agree, is profound. The principle of general relativity, on the other hand, actually says nothing physical at all; it is purely a
mathematical challenge, which has been used successfully in gravitational
theory as a heuristic principle. On the basis of the rst ideology, BONDI
proceeded to discuss some of the principal questions in gravitational radiation.
The analogy between electromagnetic and gravitational waves has often been made, but doesnt go very far, holding only to the very questionable extent to which the equations are similar. The cardinal feature of
electromagnetic radiation is that when radiation is produced the radiator
loses an amount of energy which is independent of the location of the absorbers. With gravitational radiation, on the other hand, we still do not
know whether a gravitational radiator transmits energy whether there is
a near receiver or not.
Gravitational radiation, by denition, must transmit information; and
this information must be something new. The picture of a gravitational
transmitter Bondi has is a nite region of space, inside of which something
is going on and outside of which space is empty. An example of a gravitational transmitter is a person sitting very quietly holding two dumbbells,
who suddenly, unpredictably, starts taking exercise with them. What we
want to know is what is the eect of his motion? Does it transmit information to other regions of space of what the person taking exercise is doing,
and does it transmit energy? In connection with these questions, BONDI
reports on some work of L. MARDER carried out at Kings College. The
following is a summary of Marders results:

Chapter 9
Gravitational Waves
L. Marder, Presented by H. Bondi

9.1

Static Cylinder

The general external solution may be reduced to

ds2 = r2c dt 2 r2(1c) d 2 A2 r2c(1c) (dr2 + dz2 ).

(9.1)

This is equivalent to Wilsons result (1920), who nds c

= 2M by considerations of geodesic motion.
Consider solutions inside the cylinder (radius r = a) corresponding to
metrics of the type (not the most general)
ds2 = e2 dt 2 r2 e2 d 2 e2 2 (dr2 + dz2 ),

(functions r only)

(9.2)

of which (9.1) is a particular case.

Suppose we take = Xr, = Y rm , where X and Y are constants
determined by boundary conditions. Field equations give
C2
u2 (1 um3 )
8 a2 A2
2
C
p2 =
u2 (1 + mum3 )
2
8 a A2
C

=
2 a2 A2

p1 = p3
=

u = r/a
x1 = r,

x2 = ,

x3 = z,

x4 = t .

C is small (<< 1010 , say) since C

= 2M. Pressures are therefore small
compared with density which is positive and approximately constant. p2
is larger than other pressures and balances gravitational attraction. Integrating gives M = C2 + O(C2 ) where O(C2 ) depends on distribution.
Considerations of periodicity of show A = 1 + O(C2 ). Other solutions
for , give alternative but less interesting solutions. Geodesic motion

9. Gravitational Waves

98

shows that free particles outside the cylinder have, in general, an acceleration in the z-direction.
9.2 Periodic Waves
Take metric of form
ds2 = e2 2 (dt 2 dr2 ) e2 r2 d 2 e2 +2 dz2 ,

(9.3)

with , , functions r, t (not to be confused with previous , ). Rosen

solves the case = 0 [eld equations rr + (1/r)r tt = 0, t = 2rr t ,
r = r(r2 + t2 )] in free space, and takes as periodic wave solution (denoted
by , 8 on detailed paper).

= AJ0 (wr)cos wt + AN0 (wr)sin wt (do not confuse A with previous)

and a corresponding which has a factor A2 and contains an aperiodic
2
term, 2
A wt. We have superimposed the static exterior solution (transformed) on to Rosenssolution and extended to within the cylinder obtaining a system which is initially physical, for A of order C2 . The aperiodic
term makes the system unphysical by changing the sign of the density at
r = O. This is the rst unphysical behavior of the system. The mass of the
2
cylinder/unit length is initially M
= C2 2A wt, showing that M vanishes at
t = A2Cw . The density changes sign at r = 0 at t = 3AC2 w .

Figure 9.1

9. Gravitational Waves

9.3

99

Pulse Waves

A solution of eld equations relating to (9.3) is

1
2

tr

f ( )d
[(t )2 r2 ]1/2

where f (t) represents the source of the pulse.

If (1) f (t) is of nite duration
or (2) f (t) is of innite duration with a nite number of sign
changes,

and
f (t)dt is convergent

or (3) f (t) is of innite duration with an innite number of

sign changes,

and

| f (t)|dt and

| f (t)|dt are both convergent

then , r , t 0, t , for xed r, showing, by boundary conditions,

etc., that when the pulse has travelled to innity the mass of the cylinder
reverts to its value prior to the pulse. Considerations of a particular form
of pulse show that the mass does change during early motion of the pulse.
It is clear that and derivatives are not of such a form as to allow the mass
to return to its original value whatever the form of f (t), as such functions
as f (t) = cos wt do not correspond to true pulses, but to travelling waves
such as those considered above, which clearly radiate mass.
***
DE WITT asked, If the mass eventually returns to its original state, are
there any waves left out in space?
BONDI replied that the waves travel out toward innity. They carry no
energy with them. If one integrates the energy momentum tensor through
the interior during the time the motion is going on, then while the wave is
being sent out the mass is decreasing, but as the wave dies down the mass
returns to its original value.
In answer to a question as to why it was necessary for the process
of emission of gravitational waves to be so unpredictable, BONDI replied
that then and only then is information being transmitted.

9. Gravitational Waves

100

DE WITT added, In other words, if I know at an initial time that I am

going to give you a yes answer, then the eld already contains it.
BELINFANTE asked if there were any incoming waves. BONDI replied
that there were no incoming waves in this example at all. He then continued with the analogy between electromagnetic and gravitational eld: To
my mind the electromagnetic eld is like money spent. I do not get it back
unless someone is very charitable. The gravitational eld is more like my
breathlessness when I do my exercise. When I stop, I regain my breath.
If I do not stop (as in the periodic case) I will collapse. In the nite case,
which to my mind is the more physical one, no irreversible change has
taken place.
BELINFANTE asked if one would nd, although one starts with no elds,
as in this case, that at a certain time the energy ceases to go outward and
starts to go inward. Bondi replied that he has not so far examined this,
so he does not have the answer to the questions.
WHEELER remarked, How one could think that a cylindrically symmetric system could radiate is a surprise to me. There seems to be a
far-reaching analogy between this case and the problem of emission of electromagnetic radiation from a zero-zero transmission in an atom or nucleus.
The charge can oscillate spherically symmetrically, but the system doesnt
radiate. However, if we have an electron in the neighborhood, internal
conversion can take place, with still no electromagnetic radiation emitted. This would correspond to the uptake of energy of the gravitational
disturbance created by the cylindrical symmetric exercise of yours.

Figure 9.2

9. Gravitational Waves

101

BONDI replied that he has had suspicions on that side also. To put it
crudely, what stops the emission of electromagnetic radiation in the atom
is the law of conservation of charge, and what stops gravitational radiation
from taking place is the conservation of mass and of momentum. But he
does not think there is necessarily anything against radiation of cylindrical
symmetry. However, he hopes to be able to demonstrate some day that if
one has a cylinder surrounded by a shell of matter one can transfer energy
from the cylinder to the shell by means of cylindrically symmetric motions.
This, he agrees, is required to complete the problem.
BONDI then reported on some of his own research. He takes a nite threedimensional region in which there is mass, and considers the gravitational
eld at large distances from the mass. In this region the metric has the
approximate form
)
(
)
(
t r
g = +
r1 +
r2 +
1
The coecients are functions of the variables in the brackets. He investigates what one gets if one proceeds to the approximation which includes
the r1 term. His strong impression is that this coecient is independent
of time. If we again consider the transmitter, only this time a much more
general one (the system is three-dimensional) - but suppose the system to
be spherically symmetrical both before and after emission - the m of the
two Schwarzschild solutions is then the same: hence no energy will have
been lost.
The next speaker was WEBER, who made some remarks on cylindrical waves, using the pseudo-tensor formalism. He considered the EinsteinRosen metric, and its relevant solution. Because of the linear nature of
the equation one can construct a pulse which, for example, can implode
from large distances. This cylindrical pulse will have a metric which is well
behaved at large distances and this implies immediately that the energy
per unit length of this wave will be zero. This horse sense argument is
bolstered by an exact calculation of the pseudo-tensor. If one calculates
t00 and t10 , he nds that they vanish everywhere. This has the consequence
that energy cannot be transferred around as long as one has this type of
symmetry and the above metric.
BONDI: Where did you feed in the condition that the solution is well
behaved at innity?
WEBER replied that he did not make use of this at all, but if it is well
behaved at innity, then nothing can happen. He continued: If one has a

102

9. Gravitational Waves

wave with energy per unit length which is zero, a particle put in the system
will move in some fashion. Since this seems nonsensical, one thinks of some
alternative. A possible one is that the total energy of the wave is not zero.
He has shown that the energy of the entire disturbance is zero. He has
also obtained an approximate solution of the Hamilton-Jacobi equation
of a particle which interacts with this wave. The approximate solution
says that if the particle is initially at rest, it will be at rest also after the
disturbance has passed over it. The wave interacts with the particle in
this respect like a conservative eld - it takes no energy from the wave. In
connection with the question of whether or not these solutions are trivial,
he has calculated all of the components of the Riemann tensor, and has
found that not all of them vanish.
BONDI remarked that it is vital, in this confusing subject, to make sure
that one can physically detect what one is talking about. Also a single
particle is a very poor absorber of any sort of energy.
WEBER replied that he thinks one could carry this sort of calculation out
for a couple of particles.
BERGMANN remarked that if we try to bring in decent conditions at
innity, we are licked before we start. Also if we have a spatially limited
area of disturbance, we cannot assume too low forms of symmetry because
we are limited by four conservation laws, which have rigorous signicance.
The next speaker was PIRANI. An attempt was made to formulate
a denition of gravitational radiation in an invariant way. The denition
was arrived at by making two assumptions: (1) gravitational radiation is
characterized by the Riemann tensor, and (2) radiation must be propagated along the null cone. From this point of view, on a wave front one
could expect to nd a discontinuity in the Riemann tensor. One takes
a space-time in which Lichnerowiczs conditions hold, and calculates the
permitted discontinuity across the wave front in the Riemann tensor. To
make a physical interpretation one introduces a vierbein at any space-time
event; its interpretation is that the time-like vector is the observers fourvelocity, and the three space-like vectors are the Cartesian coordinate axes
which he happens to be using at that time. In order to write down the permissible discontinuities, it is convenient to introduce the six-dimensional
formalism. One re-labels the physical components H of a skew tensor,
regarding it as a six dimensional vector. The curvature tensor R can
be similarly cast in the six dimensional formalism, appearing as a symmetric 6-tensor. If one imposes the empty space-time eld equations one
nds that its form must be

9. Gravitational Waves

103

[
[R( )( )] =

P
Q

Q
P

where P and Q are symmetric 3 3 arrays. The spurs of both P and Q

are zero. Lichnerowiczs continuity conditions permit the following array
of discontinuities (taking the discontinuity across the surface t x = 0):

P = , Q = ;

and are independent numbers. This form is obtained by taking a

particular choice of x and t axes. Here can be made to vanish by a
suitable choice of y and z axes. The above array, using the preceding
denition, characterizes a gravitational wave front (or shock wave). The
remarks from here on refer only to the pure gravitational eld, which is
analogous to the pure electromagnetic eld.
In the electromagnetic case, one can dene an invariant Poynting vector
P = ( v v )T v .
If the eld is not of the self conjugate type, then an observer following
the eld can make the Poynting vector vanish by acquiring a suitable 4velocity. If it is self-conjugate, then the eld contains pure radiation, and
the Poynting vector vanishes only if the observer acquires the velocity of
propagation of the wave.
Similarly, in the gravitational case one can dene a certain timelike
eigenvector, in terms of the Riemann tensor. This vector is interpreted as
the 4-velocity of an observer following the eld, and if for some eld this
vector collapses onto the null cone, one has radiation. The eigenvectors of
the gravitational eld are dened with the aid of Petrovs classication of
empty space-time Riemann tensors into three canonical types. These are:

9. Gravitational Waves

104

2
P=

1
I: P=

II :

III :

P =

1
Q=

= = 0

2
Q=

Q=

The s and s are independent scalar invariants of the Riemann tensor.

For example, in the Schwarzschild solution, which is of the rst type,
1
m
1 = 2 = 3 = 3 ,
2
r

r = 0.

The Einstein-Rosen metric gives a Riemann tensor of the second type.

Using these forms one may work out the eigen 6-vectors of the Riemann
tensor, and if one takes the intersections of the corresponding 2-planes in
pairs, one obtains 4-vectors, the Riemann principal vectors. The denition
of gravitational radiation is that if the Riemann tensor is of the second or
third type, one has radiation, and if it is of the rst type, one does not
have radiation. It can be shown that the dierence between the nonradiation type and one of the radiation types can be made to correspond
to the discontinuity in the Riemann tensor across a wave front permitted
by Lichnerowiczs conditions. (Details of this work are to be published in
the Physical Review, February 1957.)
Next, SCHILD presented some remarks on an apparent possibility to
get indirect information on gravitational radiation by looking for gravitational radiation reaction force. The idea is built on an analogy to
electromagnetic radiation reaction force. In the electromagnetic case, the
equations of motion of a particle are
mx ex

2 2
ext

= e (x x x x ).
F
3

9. Gravitational Waves

105

One proceeds by analogy and attempts to write the geodesic equations in

the gravitational case as
{
}
d 2 x
dx dx

m 2 +m
= m2 A

ds
ds ds
where, in analogy with the electromagnetic case
( 3
)
d x

A =
tangential terms .
ds3
A procedure like this does not work in principle, and this for the following
reasons: For an equation like this to make sense, one must go to a limiting
background eld, letting the mass tend to zero. That is, what one really
is considering is the limit of the eld containing mass as the mass goes to
zero:
g (x; m) g (x; 0) = g
0

ds20

= g dx dx .
0

This means that we are considering dierent Riemannian 4-spaces, with no

natural one-to-one correspondence between the points of any two spaces.
Thus we must demand invariance under independent transformations in
dierent spaces:
x = F (x1 , x2 , x3 , x4 ; m).
Since the equations are tensorial under
x = F (x1 , x2 , x3 , x4 )
it remains to demand (to order in m considered) invariance under
x = x + m f (x).
However, the equations are not invariant under such a transformation. It
appears from an investigation still in progress that any right hand side
m2 A can be wiped out by a suitable choice of such a transformation.

9. Gravitational Waves

106

Geometrically one may picture this as follows:

Figure 9.3
If one changes coordinate systems in Rm (or equivalently, a one-to-one
correspondence Rm R0 ) one can move Lm into coincidence with L0m .
PIRANI reported the following communication from ROSEN.

Chapter 10
Gravitational Field of an Axially Symmetric System
N. Rosen and H. Shamir, Presented by F. Pirani

In order to investigate the possibility of a physical system radiating gravitational waves, it seems desirable to choose a simple system, one with
axial symmetry. If the eld of such a system is described by means of
a spherical polar coordinate system, (x1 , x2 , x3 , x4 ) (r, , ,t), then by a
suitable choice of coordinates one can satisfy two conditions:

(a) The metric tensor g is independent of the angle .

(b) It is diagonal.

If one writes down the eld equations in the empty space surrounding the
system
1
G R g R = 0,
2
one obtains a set of 7 equations (since G vanishes identically if one index
is equal to 3) for the 4 diagonal components of g . Among these there
exist 3 identities (the Bianchi identities, except for the one with index 3).
The eld equations are non-linear and dicult to solve. It is proposed
to investigate them by the method of successive approximations. As a
beginning, the rst approximation can be calculated. Let us write the line
element in the form
ds2 = (1 + )dr2 r2 (1 + )d 2 r2 sin2 d 2 + (1 + )dt 2
where , , , and are regarded as small of the rst order. The linear approximation of the eld equations has the following form (indexes
denoting partial dierentiation):

108

10. Gravitational Field of an Axially Symmetric System

44 + 44 r12 (22 + 22 ) + cotr2 (2 22 2 ) 1r (1 + 1 + 21 )

+ r22 ( ) = 0,

11 11 + 44 + 44 cot
(2 + 2 ) + 1r (1 21 1 ) = 0,
r2

11 11 + 44 + 44 r12 (22 + 22 ) + 1r (1 21 1 ) = 0,

11 + 11 + r12 (22 + 22 ) + cotr2 (2 2 + 22 ) 1r (21 31 31 )

r22 ( ) = 0,

12 + 12 1r (2 + 2 ) cot (1 1 ) = 0,
14 + 14 1r (24 4 4 ) = 0,
24 + 24 cot (4 4 ) = 0.
From these equations it is possible to derive the wave equation for ,
2
1
cot
11 + 1 + 2 22 + 2 2 44 = 0,
r
r
r
and to express the other unknowns in terms of the solution for .
If we represent the radiating system by a quadrupole mass moment
, then the solution describing outgoing monochromatic waves can be
written in terms of the frequency and the amplitude of the moment
p0 (p = p0 cos t) as follows:

=
=

3
p0 [ j2 (x) sin t + j2 (x) cos t]P2 (cos ),
6
3
p0 {[(3k2 (x) + j2 (x)) sin t + (3k2 (x) + j2 (x)) cos t]P2 (cos )
12
(k2 (x) + j2 (x))sin t (k2 (x) + j2 (x))cos t},
3
p0 {[(k2 (x) j2 (x)) sin t + (k2 (x) j2 (x)) cos t]P2 (cos )
12
+ (k2 (x) + j2 (x))sin t + (k2 (x) + j2 (x))cos t},
3
p0 [(2q2 (x) + j2 (x))sin t + (2q2 (x) + j2 (x))cos t]P2 cos ].
6

10. Gravitational Field of an Axially Symmetric System

109

Here
x = r ,
( )1/2
jn (x) =
x1/2 Jn+1/2 (x),
2
1 x
kn (x) =
jn (u)du,
x
x

qn (x) = x

u2 jn (u)du.

It is planned to use this solution as the starting point for a more accurate calculation. The interesting question, of course, is whether the exact
equations have a solution going over into the above for suciently weak
elds.
***
J. N. GOLDBERG presented an approximation method for high velocities. The obvious one is a mass expansion of the gravitational eld. In
the rst approximation there is no acceleration of the particles, which are
represented by singularities. This situation diers from the corresponding
case in electromagnetic theory. The reason for this is that the electromagnetic eld is described by a vector potential whereas the gravitational eld
is tensor in character. In the E.I.H. method, the motion of the particles is
determined by certain consistency conditions in the form of surface integrals each of which contains two terms. One of these is the time derivative
of an expression which is linear in the rst derivatives of the potentials, the
other is quadratic in the rst derivatives of the potentials. Therefore, in
each order the surface integrals place restrictions on the solutions of lower
order. These restrictions may be relaxed by adding appropriate poles and
dipoles in the lower orders. The requirement that the sum of the dipoles
added must vanish leads to the equations of motion. However, since we do
not discriminate against the time in the mass expansion, the surface integral conditions must be satised in the same order as the corresponding
eld equations. This situation is very unsatisfactory. The surface integrals, however, are related to the Bianchi identities. Instead of the surface
integral conditions, one can require that the Bianchi identities be satised
everywhere, even across the singularities. In this manner one can obtain
higher order equations of motion from the lower order eld equations both
in the E.I.H. method and in the mass expansion.

110

10. Gravitational Field of an Axially Symmetric System

TONNELAT: The starting point of the non-symmetric unied eld theory

is usually the invariant density:
L = G R
in which R is the Ricci tensor built from an arbitrary ane connection
. The variations , G lead to the eld equations. However, if

one tries to apply the Einstein, Infeld, Homann method to these equations, one obtains merely the results of general relativity, and one does
not obtain the equations for a charged particle. This result stems from the
condition

G = 0
which is imposed by the theory.

To avoid this situation, one can start from an ane connection

with vanishing torque

L L
= 0.

Lagrange multipliers are needed in the variations of L because the 64

L are not independent and, moreover, the vector A related to the torque
is introduced. In this case, one obtains

G = 0.
Introducing the metric

a a = G
one obtains

= F J
with
1
S = R a R a .
2
Setting
R G ({ }a ) +V
denes a tensor V whose divergence gives a Lorentz force. This tensor
plays in the eld equations the role of a Maxwell tensor and leads, with

10. Gravitational Field of an Axially Symmetric System

111

the use of an extension of the Einstein, Infeld, Homann method, to a

Coulomb force.

Chapter 11
The Dynamics of a Lattice Universe
R. W. Lindquist

One considers a cosmological universe with the cosmological constant set

equal to zero and asks whether it is possible to construct a closed universe
which contains a nite number of mass singularities. In dealing with this
problem, one works with what is essentially the Wigner-Seitz approximation carried over bodily from solid state physics. One begins by taking a
space-like hypersurface or zone of inuence surrounding a particle. These
zones are assumed to be spheres and for the aggregate of particles each
sphere is tangent to all of its nearest neighbors. Next one makes the somewhat more drastic assumption that inside each zone of inuence one has a
pure Schwarzschild solution. These singularities can then be tted onto the
surface of a hypersphere as shown in the gure to the left. The requirement
that each zone touches its nearest neighbor at only one point requires, in
view of the very high symmetry of the problem, that one takes the number of particles in the problem to be equal to the number of vertices in
a regular hyperpolyhedron, i.e., N = [5, 8, 16, 24, 120, 600]. One must now
set up the tangency condition, and require that the Schwarzschild funnels
shall be tangent to the hypersphere.

Figure 11.1

11. The Dynamics of a Lattice Universe

114

This can be done in an invariant manner. One must also nd how to

dene the hypersurfaces to begin with. This can also be done. This, with
the tangency condition, gives the motion of the hypersphere. One gets
from this model an expanding and contracting universe, with a cycloidal
behavior.
BONDI commented that this is interesting from the point of view of the
Eddington universe, admitting that this is the case with the cosmological
constant not equal to zero. One can ask, if one allows for the granular
structure of mass, how many granules can one allow? Of course it is your
number 600.
LINDQUIST agreed and continued: In order to get an idea of how good
an approximation the Schwarzschild singularities are, one can compute
the maximum radius of the tangent hypersphere, for the various number
of particles.

K
2m

!"#$%&'()*(#+$",$

04.00
- 3

./ 01

.02

Figure 11.2
This is shown in the above graph, in which 0 is the maximum radius of
expansion. We have also computed the invariant distance between two
Schwarzschild singularities (at the radius of maximum expansion). This
has been done both rigorously - as an initial value problem - and with the
above approximation. One nds the values given below:

d/2m

Rigorous
.9994

Approx.
1.18

The session was concluded by WHEELER, who summarized the results presented by the various speakers.

11. The Dynamics of a Lattice Universe

115

First, from what PIRANI has said, we have gained some insight into
how we may dene what the measurability properties are locally of the
gravitational eld. The tensors and invariants he has described are at
the heart of the matter. Second, as concerns the radiation problem, we
would like to know what is the highest degree of symmetry one can have
in a problem, and still have interesting radiation. This leads one to the
question of whether, even in the cases where there is no symmetry, one
has reason to expect radiation. On this score, it would be well to recall
an important physical fact: that the gravitational eld of a point charge
has close analogies to the electric eld. One knows that there is a certain linear approximation to the eld equations similar in nature to the
electromagnetic equations, so that if a mass is accelerating, one nds it
produces radiation similar to the electromagnetic radiation of an accelerating charge. On this account, one expects gravitational radiation. Using
this analogy, Einstein was able to calculate the rate of radiation from a
double star. The issue which SCHILD brought up is one quite dierent
from the double star radiation, in that the system is time-nonsymmetric.
He invites us to consider the radiation produced in a gravitational scattering process. The acceleration of the particle as it zooms by the stationary
one will give radiation at large distances, but the recoil acceleration of
the particle which was initially at rest will produce a wave of nearly the
opposite phase of the rst, so that one expects the rate of radiation to
be proportional to something analogous to the moment of inertia of the
double star system. This means that the damping of this system depends
on some complicated geometric concept like mr2 . Therefore, if one hopes
to get a proper answer to the question of how to describe the frictional
force between the two masses, one must nd some quite general way of
describing invariantly that region of space in which the particle undergoes
its acceleration.

Figure 11.3

116

11. The Dynamics of a Lattice Universe

BONDI has reminded us that if one looks for radiation pressure on a

particle in gravitational waves, he must take into account the radiation
produced by the motion of the particle itself. The situation here is analogous to an electromagnetic wave passing over a particle. To the lowest
approximation, the particle only feels the electric eld and oscillates with
it. If one improves his approximation, he nds that the particle begins
to respond to the magnetic eld, and moves in a gure eight. Still there
is no radiation pressure. It is only when one includes the radiation that
the particle itself gives out that one gets radiation pressure. That is, it is
only when one allows for the radiation damping force that one nds the
particle moving forward. Similarly, in the case of gravitational radiation,
one faces similar problems.

Figure 11.4
As WEBER brought out, in the case of the cylindrical wave, a gravitational metric charge passing over a particle leaves the particle with its
initial energy after the disturbance has left. At rst sight, one might believe that there is no observable consequence of the action of the wave on
the particle. However, the electromagnetic analogy suggests that if one
were to go further, one might expect to nd radiation pressure.

11. The Dynamics of a Lattice Universe

117

One has also to consider the nature of the one-sidedness of gravitational radiation. Here one faces the problem of what is to be meant by
the dierence between retarded and advanced waves. If one employs the
absorption theory of radiation damping in treating the above problem, one
must employ the use of advanced and retarded waves. In at space the
concepts of advanced or retarded waves are easily understood. However,
with gravitational waves, space is curved, and this has the consequence
that it is dicult to distinguish between retarded and advanced waves.
This is due to the fact that a pulse sent out by a source gets defracted by
the curvature of space and secondary waves are thereby generated, which
are in turn scattered. In this way, one may ultimately get contributions to
an incoming wave, so that the distinction between retarded and advanced
waves is lost.

Session IV Invited Reports on Cosmology

Chairman: F. J. Belinfante

Chapter 12
Measurable Quantities that May Enable Questions of
Cosmology to be Answered
Thomas Gold

12.1

Introduction

A discussion of a number of cosmological problems was presented including

radio and optical astronomy as important observational tests of cosmological theory. The point of view adopted was that, while locally discovered
laws of physics may possibly suce to describe observations on the large
scale, it is conceivable that this may not be true on the very large scale
requiring extreme extrapolation. Cosmological models should therefore be
constructed that are simple and can be subjected to observational tests.
With respect to the distribution of matter, it is necessary to discuss not
only the geometrical and kinematical aspects but also physical characteristics such as the abundance of the elements, galaxies, and galactic clusters.
Models must be capable of allowing for such evolution. We cannot be sure
that we know all the laws of physics required for a real understanding of
the situation, and we should consider seriously any model which is simple
and presents the greatest variety of experimentally veriable conclusions.
12.2

Measurements and Observations

The observations which are most easily interpretable as of a cosmological

nature with respect to galaxies are the following quantities:
1. distance (brightness)
2. red shift
3. number of galaxies
4. distribution of and (on the celestial sphere)
5. color and other physical characteristics

122

12. Measurable Quantities and Questions of Cosmology

6. mean density of matter in the world

7. distribution and abundance of elements
In the above list, 1 to 4 would represent a distribution in phase space.
These would be expected to be predicted (as well as the other measurements) by a cosmological model.
12.3 Cosmological Models
A. The Explosion Model
The present state of the universe is the result of an explosion of
a previous dense state of matter at a time of the order Hubbles
constant years ago.
B. The Oscillatory Models
These are similar to the explosion model, and we are supposedly now
on an expansion phase of the oscillation which should be followed
by a contraction of the universe to a repetition of the cycle.
C. The Steady State Model
In this model the observational quantities on the average do not
change with time. It eliminates the eects of local evolution on
all statistical observations and avoids therefore a great diculty in
testing the model in contrast to other models. All observations on
distant masses are past history of those distant masses. On the
steady state model the average of such observations is to be the
same as for nearby masses. Unless we know how to calculate the
changes in distant galaxies with time as a function of a wide range of
detailed information, we would not know how to interpret the data.
The virtue of the steady state model is that it bypasses this problem
and that it could therefore be disproved by the observations as they
are all crucial.
12.4 Observational Data
A. Distance vs. Red Shift (C / )
The best summary has been given by Sandage on the work of Sandage
and Humason. According to the Cosmological Principle, at a suitable
moment in time all the localities, suitably chosen, look alike.

12. Measurable Quantities and Questions of Cosmology

123

The data is to be summarized by the relations:

C ( / ) = H D
where
C = speed of light,
H = Hubbles constant,
D = distance,
/ = fractional shift in wave length;
1 dR
= H0
R dt
denes the present observational magnitude of Hubbles constant;
1 d2R
= q0 H02
R dt 2
where q0 is a quantity to be discovered by the observations.
The observations to be made during the next few years should give
us the required kinematical information. The predictions of the different models in regard to q0 are not resolvable by present observations when one compares the red shift vs. bolometric magnitude.
It appears that the newly-developed photoelectric image multiplier
technique used at Mt. Palomar by Baum should extend the data
suciently to make possible the resolution between models in regard
to the red shift data.
In regard to the quantity q, it is to have the following cosmological
signicance:
q
q > 1/2
q = 1/2
0 q < 1/2
q = 1

Type of Universe
closed spherical universe
at
open (hyperbolic)
steady state

124

12. Measurable Quantities and Questions of Cosmology

B. Red Shift and Number of Galaxies

This correlation would be an especially attractive result. However,
insucient data exist because of the laborious eort involved in examining hundreds of spectrograms of distant galaxies to obtain number counts and spectral measurements. It would be helpful to have
this information obtained by mechanical means.
C. Color vs. Distance (Red Shift)
The absence of color eect as a function of distance (dened by red
shift) as now observed by Stebbins and Whitford can be interpreted
to mean there is no large space absorption and (at great distances)
there are no large evolutionary eects in times of the order of one
quarter of Hubbles constant. This would agree with the steady state
model.
D. Radio Counts of Galactic Number (Number vs. Brightness (distance))
All number counts are subject to a type of error which must be
avoided. In a homogeneous system the law of number (N) versus
intensity (I) should be
ln N 3/2 ln I
If the error in I is symmetric and equal at all intensities, it does
not aect the gradient of the relationship which is 3/2 in magnitude
but will shift the curve parallel to itself. When the error is greater
at small intensities, the eect is to change the gradient and is then
very strongly dependent on the extent of the tail of the error distribution (the small number of large errors). The eect is to steepen
the gradient, and can be calculated from the radio source data if an
independent measure of the error exists.
The radio counts are not able to produce a cosmological answer at
this time except if it were found unambiguously that the gradient
was greater than 3/2 or that it was found possible to observe independently the distance of sources rather than just their brightness,
for in the case that they follow the 3/2 law they can be assumed to
be too near to have any cosmological signicance.

12. Measurable Quantities and Questions of Cosmology

125

E. Mean Density of Matter in the Universe

Another type of radio observation of cosmological signicance is the
mean density of matter. This is involved in the relation (according
to some cosmological theories)
4 G 2 6
where
G = universal gravitational constant,

= mean density 1028 gm/cm3 ,

= Hubbles constant (as a time).
The radio observations on neutral hydrogen (21 cm wavelength) in
distant galactic clusters indicate appreciable addition of neutral hydrogen so that much of the matter of the universe can no longer be
considered luminous. This is to the eect that will always tend to
increase as methods are found to detect matter. It would be nice to
know the amount of matter in intergalactic space. It is presumably
largely ionized because the recombination time is about 1010 years.
(Dust is exceedingly rare.) Enough dust to aect the recombination
time would make intergalactic space opaque to visible radiation.
F. Origin of the Elements and Abundance
The recent developments (the work of Fowler, Hoyle, Burbidge, and
Cameron on element generation in stars) are to be regarded as a
great triumph. According to this theory, elements are being generated at present in stars that are seen at present by calculable nuclear
processes. The massive stars have a shorter life and scatter themselves. They throw out their matter containing elements up to the
Fe-group. This matter is caught in other stars and in turn captures
neutrons (because of the larger cross-section) from light element reactions within the stars. In this way a small proportion of heavy
elements will be built up. It has been possible to give explanation
for some of the details in the nuclear abundance diagram - an extremely complicated picture. A particular triumph within this theoretical picture of element buildup in stars is the prediction by Hoyle
of a C12 nuclear energy level (which was veried to exist). A cosmological requirement thus served as a prediction of a nuclear physics
experiment.

126

12. Measurable Quantities and Questions of Cosmology

According to the steady state model, there is no need to imagine

a particular state of matter which was dierent in the past. As the
formation of the elements is proceeding continuously, the overwhelming abundance of hydrogen now implies that the bulk of material is
young enough not to have undergone these processes. In other models it is necessary to assume also that the material is mainly young,
or that it has been in a non-reactive state until recently. To suppose
that some process undid the nuclear combinations imagined as having taken place previously is very dicult and requires circumstances
so far from those we know that such speculation seems unprotable.

Discussion
BONDI remarked that q is not a purely geometric quantity. The rst three
values given (page 123) involve the use of the eld equations of general
relativity and the conservation of matter. The steady state model is at
in the large.
BELINFANTE inquired about the loss of number counts because of absorption of intergalactic matter.
BONDI replied that this could almost be ruled out because one could see
so very far and the photoelectric measurements of Baum indicate there
cannot be too much absorption out to the limits of observation.
DICKE observed that one could not see anything beyond that point and
raised the question of the signicance of number counts without going into
red shift.
GOLD stated that there exist no good number counts since 15 years ago,
and they do not distinguish between the models.
BONDI stated the astronomers say they cannot get such numbers easily
because of the great accuracy required in brightness measurements and
the steepness of the curve relating number and brightness.
GOLD remarked that the question of absorption is important but in a way
the observations indicate it does not exist in fact to a signicant extent.
BONDI inquired what fraction of the mass of galactic clusters is estimated
to be neutral hydrogen.
GOLD replied several times the stellar masses.

12. Measurable Quantities and Questions of Cosmology

127

DE WITT inquired about what happens to the energy-stress tensor which

is not conserved in the steady-state theory.
BONDI replied that there is no clear-cut mathematical analysis. The
assumption of the steady state theory is a physical hypothesis from which
it is possible to make predictions, veriable by observation, without going
into the tensor calculus. McCrea suggests there is a zero point tension in
the vacuum which on expansion produces outgoing energy.
DE WITT asked why protons should be chosen as the entities which are
created out of nothing. Since they are already complicated structures,
why not go all the way and use heavier nucleii - such as Fe - as well? Or
is the theory to be considered purely phenomenological?
GOLD said that if any more basic heavy particles were to be made they
would end up as protons and hence hydrogen gas. To assume the creation
of Fe would seem very articial and is in fact unnecessary.
BONDI pointed out that the evolutionary history of any galaxy required
the pre-existence of matter and in the steady state model this means cold
hydrogen gas is the simplest hypothesis and reasonable.
GOLD remarked that one is forced to the supposition of the coming into
being of the matter anyway, as it could not have been around for an indenite period without going into heavier elements.
DE WITT said that the creation of matter required the existence of some
unknown dynamical process as a precursor to the creation which forces the
production of protons rather than something else.
SCIAMA remarked that any future detailed formulation of the theory
would have to account for the production of protons.
BERGMANN emphasized that the principle of equivalence of the general
theory carried with it certain detailed consequences which includes the
conservation of the stress tensor which follows from the covariance and
action principles.
BONDI stated that what one has is something which is interpreted as
having components measurable in the form of stress.
BERGMANN replied that whatever it is which is conserved is stress by
denition. If expansion and creation occur simultaneously, stress can be
conserved only by having inux of matter along with eux so that the
concept of creation is superuous. The steady state hypothesis requires
true creation which is a violation of stress conservation. It is conceiv-

128

12. Measurable Quantities and Questions of Cosmology

able that the principle of equivalence and the action principle may require
modication from the other point of view.
PIRANI proposed that one envisage a Dirac type distribution of states
which contributed to the energy tensor and condensed into the protons at
the required rate.
BERGMANN said this could not lead to steady state.
PIRANI said it could.
GOLD pointed out that if creation is denied currently, it cannot be maintained for the past; and this issue cannot be avoided on any theory.
BERGMANN stated that he did not favor the explosion hypothesis either.
SCIAMA felt that the conservation equations as now known are interpreted too seriously. One could devise a more complicated theory with
conservation equations arising from identities. The extra variable in such
a theory could allow for creation.
WHEELER took the point of view that one should not give up accepted
ideas of wide applicability such as general relativity but should investigate
them completely. The two things left in question are the expansion and the
creation of elements. The rst does not appear to cause any diculty in
general relativity theory. Hydrogen can be created out of heavy elements
by the reaction of matter in stars. This process continued suciently far
will get one down to the absolute zero of temperature. There is a well
dened state of absolute zero and, in the beginning, for not too great
a mass, one has iron. If the mass is too great, the composition of the
interior is pushed into neutrons. A neutron core star undergoing explosions
will throw large quantities of neutron-rich matter into space which yields
hydrogen.
GOLD questioned whether such a process could yield 99% hydrogen and
helium.
WHEELER replied that no detailed calculation had been made but would
not say that it could not be explained within the present framework of
accepted ideas.
BERGMANN inquired about the thermodynamic analysis of the neutron
core model leading to the hydrogen clouds from the point of view of entropy.

12. Measurable Quantities and Questions of Cosmology

129

WHEELER said that initially it is a neutron situation at absolute zero

under high pressure.
BONDI commented on the fact that Einsteins proposal of the Cosmological Principle 40 years ago occurred at a time without knowledge of the
galaxies. The extension of the range of observation by radio and optical
means has shown that a homogeneous system obeying the Cosmological
Principle demands the distance vs. recession-velocity relation according
to McCrea and Milne. The body of evidence in support of the Cosmological Principle makes it an astounding prediction, more striking, in a sense,
than those of general relativity
WHEELER remarked that Einstein said also that symmetric motions say
nothing about laws of motion. What is simple in physics are the laws of
motion and not the motions themselves as, for example the situation in
hydrodynamics with turbulence. The Cosmological Principle is not in the
same category as other physical laws because it is not a fundamental law.
BONDI said its fundamental character may be debatable but not its success.
ERNST questioned if a steady-state theorist would drop the generalized
Cosmological Principle if observations did not t the steady-state theory.
BONDI replied, Like a hot brick.
ERNST continued with the statement that this would perforce leave only
the stress-tensor and asked where could one go from there.
BONDI said he would drop the subject.
GOLD remarked that the steady-state theory is certainly successful in the
sifting of observational data.

130

12. Measurable Quantities and Questions of Cosmology

Figure 12.1: Map showing the distribution of radio sources in galactic coordinates. The open circles represent the sources of large angular diameter, and in both cases the sizes indicate the ux
density of the sources

Chapter 13
Radio Astronomical Measurements of Interest to Cosmology
A. E. Lilley

With the discovery of radio radiation of galactic origin in 1932 by Karl

Jansky, this new science of radio astronomy has rapidly matured and already has produced measurements which are of interest to cosmologists.
This afternoon we shall take up several of these, and in particular discuss
those radio astronomical measurements which have or may produce data
of quantitative signicance for cosmologists; however, before examining
these measurements in detail, let us review briey the appearance of the
radio sky as evidenced by the radio data. We may conveniently divide the
radio radiation into two components: that originating from an apparently
continuously distributed source and that originating from discrete sources,
the so-called radio stars. The continuous radiation eld displays a distribution on the celestial sphere which concentrates itself toward the plane
of the Milky Way. It reaches maximum intensity in the Sagittarius region,
which contains the nucleus of our galaxy. The variation of intensity with
wavelength is such that the longer the wavelength the greater the observed
intensity. This radiation component therefore is of non-thermal origin. Superimposed on the continuous radiation eld are the radio stars. It is the
analyses of the radio star studies which have been made to date which we
shall consider rst as one example of radio astronomical measurement of
cosmological interest.
Extensive surveys have been made of radio stars, and these surveys
continue currently. The survey data produce positions and apparent intensities of the radio stars. Approximately 2,000 discrete sources have
been observed with current instruments. This number will undoubtedly
increase as instrumentation improves.
If we assume that the radio sources have a constant luminosity, or,
obey some regular luminosity function which does not vary with distance,
the distribution of radio sources throughout the observable universe may
be inferred from the survey data. We may write down how the number

13. Radio Astronomical Measurements of Interest to Cosmology

132

of observed radio sources per unit solid angle will increase with decreasing
apparent intensity if one is viewing a universe having an isotropic distribution. The number of observed sources having an intensity greater than
I is plotted against intensity and the resulting curve is compared with the
curve which would result from an isotropic distribution. On a log N versus
log I plot, an isotropic distribution would produce a straight line having a
slope of 3/2.

Sources of small angular diameter, (a)

2.0

Extended Sources b > 12,

Extended Sources

1.0
LOG N

b<12,

(b)
(c)

1.0

2.0

3.0

LOG I

Figure 13.1: Curves of log N against log I, where N is the number of sources
per unit solid angle having a ux density greater than I.

13. Radio Astronomical Measurements of Interest to Cosmology

133

Ryle and Scheuer of Cambridge analyzed the rst such survey data
which involved a statistically signicant sample of radio sources. The slide
shows the observed distribution of radio sources over the sky displayed in
galactic coordinates. The circular region in which no radio stars appear
is simply the region inaccessible to observation. The next slide which is,
as the rst, taken from the work of the Cambridge group, displays the
log N versus log I behavior of the data. You will note that as the data
proceed toward decreasing intensity, the actual plot shows that there is an
apparent accumulation of radio stars which is more rapid that one would
expect from an isotropic distribution. The interpretation placed upon the
observed curve by the Cambridge group suggests that there is an increasing
density of radio stars with increasing distance from the neighborhood of
our galaxy.
This will suggest departures from an isotropic and uniform universe
and the results, if valid, are not consistent with a steady-state universe.
However, the interpretation of this curve has been discussed by Bolton,
who has suggested that when one has observational errors which increase
with decreasing intensity, even an isotropic distribution can produce a
curve of the form shown in the slide.
In addition to the interpretative diculty pointed out by Bolton, observational conict now exists. A similar observational survey has been
conducted by Mills in Australia with a dierent type of antenna system;
and this system also possesses a greater sensitivity than the Cambridge
instrumentation. Where the surveys of the Australian and Cambridge
workers overlap, and where a detailed intercomparison can be made, the
agreement between the separate surveys has been disappointing. In addition, the log N versus log I analysis by Mills shows no signicant departure
from the 3/2 curve until the approximate sensitivity limit is reached
where the curve does display some increase; however, this faint limit increase is very suggestive of the eect pointed out by Bolton.
Thus, the analysis of radio star data in quest of their distribution
throughout space has resulted in observational conict. Since this topic
is of considerable interest to the next speaker, Dr. Gold, I will leave its
further discussion to him.
Let us now discuss several possible measurements which can be made
by employing microwave spectral lines which originate in the gases which
compose the interstellar medium. Although others are expected, only one
such line has been successfully detected and studied to date - the hyperne transition originating in the ground-state of atomic hydrogen. This
transition occurs in the microwave domain near a wave length of 21 cm.

134

13. Radio Astronomical Measurements of Interest to Cosmology

As the rst surveys of this radiation in our galaxy were nearing completion, consideration was given to the possible behavior of the spectral line
prole in directions which contain radio stars. Although the line predominantly appears in emission distributed around the galactic plane, the rst
observations in radio star directions revealed the line in absorption.
An analysis of the absorption eect shows the absorption studies to
be extremely high resolution investigations of the interstellar gas. Minimal
distances to radio stars and observations of small-scale turbulent structure
of the interstellar medium were early consequences of the absorption studies. The ability of the absorption eect to make extremely high resolution
studies of the interstellar medium will probably prove more valuable than
utilization of the data for measurement of radio star distances.
The absorption lines produced in the continuum of radio stars are
sensitive to the size of the antenna which views the gaseous assembly. By
employing larger antennas and looking for absorption lines in the spectra
of the radio stars, we ultimately hope to observe other gaseous components of the interstellar medium. A transition in deuterium at a frequency
of 327 mc and in the hydroxyl radical at a frequency of 1667 mc are examples of new lines which may ultimately be detected by larger radio
telescopes. With the development of a new type of microwave receiver of
vastly improved sensitivity, the so-called solid-state maser, we may condently expect detection of other gaseous components in the interstellar
medium. When such lines are detected, radio astronomy will provide a
few numbers in the tables of cosmic abundance.
Of considerable interest to cosmologists is the size of the observable
sample universe. Let us briey compare the size of the universe available to
radio astronomical measurements with the size available for optical examination. Two billion light years may be taken as a measure of the limiting
distance at which objects are detectable by the Hale 200 telescope. This
is a measure of the limiting distance without electronic aids. We may
compare the optical 2 billion light year gure with a hypothetical radio
case. Restricting our attention to the detection of radio ux (neglecting for
the moment red shift corrections) a 150 diameter parabolicradio telescope
equipped with a conventional microwave receiver could detect a radio star
of the Cygnus A type at a distance of 8 billion light years. If the 150 antenna were equipped with a solid-state maser, the maser would overcome
some of the red-shift ux reduction and fruitful measurements could be
made at distances signicantly beyond the range of 2 billion light years.
Another topic of interest to cosmologists and of interest to workers
employing the hydrogen line is the possible existence of hydrogen gas in

13. Radio Astronomical Measurements of Interest to Cosmology

135

the intergalactic medium. Unfortunately, the intergalactic gas is probably

ionized, and the time for recombination is expressed in billions of years.
However, it is of interest to ask what density could be detected in the
intergalactic medium if it were neutral form. We would search for an
absorption line in the spectrum of an extragalactic radio source. The
minimum detectable density is of the order

igm 3 1011

TS Tmin H
TA

where H is Hubbles constant, Tmin is the sensitivity limit of the microwave

radiometer, TS is the state temperature of the intergalactic gas and TA is
the antennatemperature of the extragalactic source. Taking as an example
the 150 antenna equipped with a solid state maser and using the Cygnus
A source as a test object, the corresponding minimum detectable density
is about 3 1033 TS gm/cm3 . This number refers only to the unionized
component (if any) in the intergalactic medium.
It also refers to average regions of intergalactic space which are well
removed from rich clusters of galaxies. The density of material distribution
in the connes of rich clusters of galaxies could not be regarded as indicative of the average mean density throughout intergalactic space. However,
the contribution to the average density by the clusters is most important,
and measures of the cluster masses are required.
Heeschen has succeeded in detecting emission from hydrogen gas in
the Coma cluster of galaxies. This is presumably gaseous emission from
the cluster medium. The emission results indicate that the gaseous mass
of the cluster medium is of the order 1/3 the total mass of the cluster.
Measurements of 21 cm emission from clusters of galaxies may revise and
strengthen our estimates of the mean density of material in the universe.
Finally, we shall take up the measurement of the red shifts of extragalactic objects by radio techniques. The identication by Baade and
Minkowski of the Cygnus A radio star as a pair of galaxies in collision
produce a strong out-pouring of radio energy. The radio intensity of the
galaxies in collision is enhanced by a factor of the order 108 compared to
the normal radio intensity of isolated galaxies. In such colliding galaxies,
it is possible that peripheral gases not yet involved in the collision contain
atomic hydrogen gas. This gas would absorb part of the radio continuum
originating in the collisional zone thereby producing an absorption line in
the continuous spectrum. The optical studies of Baade and Minkowski
showed that the Cygnus A system has a red shift indicating a recessional
velocity of approximately 17,000 km/sec. The corresponding shift of the

136

13. Radio Astronomical Measurements of Interest to Cosmology

hydrogen line would be a decrease of about 81 mc from the rest frequency

of 1420 mc. This line was sought and found by investigators at the Naval
Research Laboratory in Washington, who used a 50 antenna. Although
this rst measurement was crude, it revealed that the microwave Doppler
displacement was, within the limits imposed by experimental error, identical for optical and microwave determinations. This is what the result
must be if the universe is expanding.
With the increasing distances available for observation in radio astronomy, the possibility of making red shift measurements on Cygnus A
type systems well beyond the two billion light year range suggests itself.
Measurements such as this may ultimately prove of considerable value in
determining the exact shape of the red shift curve with distance which,
as you will hear in Dr. Golds talk, is a means for testing models of the
universe.
Although radio astronomy is in its rst stage of development, it is
already evident that this new endeavor can produce measurements of considerable interest to cosmology.

Session V Unquantized General Relativity, Concluded

Chairman: A. Lichnerowicz

139

WHEELER opened the session with a presentation of some conclusions

reached by members of his group. Komar has investigated the interpretation of Machs Principle and has found great diculties, which are ascribed
to the lack of understanding of the initial value problem. The stability of a
Schwarzschild singularity in empty space to small disturbances was investigated with T. Regge. It was concluded that the Schwarzschild singularity
is, to the extent investigated, stable against small disturbances away from
its normal spherical form. Thermal geons were investigated with E. Power.
A self-consistent problem was set up for the motion of photons of thermal
radiation in a spherically symmetric gravitational eld produced by these
photons. It was found that a region existed in which photons would be
trapped; there is a stable circular orbit in this region, and the other orbits
consist of oscillations about the circular orbit. The limiting orbit is a spiral
which is asymptotic to the outer boundary.
Ambiguity in a Gravitational Stress Energy Pseudotensor Interpreted in Terms of Arbitrariness in a Base Line Coordinate
System
W. R. Davis
In order to avoid the non-localizability to transform the stress energy pseudotensor, Kohler attempts to transform the stress energy pseudotensor in
the given coordinate system, introducing in addition to the gik another
set of quantities, gik , such that the articial metric described by the gik s
possesses a zero curvature. He then shows that the dierence between the
gravitational pseudotensor dened by the gik and gik itself transforms like
a tensor under coordinate changes. This result might seem to give a certain uniqueness to the denition of a gravitational stress energy tensor if
one could imagine that the quantities, gik , were unique. However, a simple
investigation shows that the gik s are highly arbitrary. In fact, for every
coordinate system, xi , that is asymptotically straight at innity one can
introduce a set of gik s which have the normal Lorentz values everywhere.
By this procedure the gik s are of course dened in any other coordinate
system. The ambiguity in the choice of the gik s in some specic coordinate
system is therefore as great as the ambiguity in the choice of the original
coordinate system which becomes asymptotically at - an ambiguity can
be regarded as a base line. Motions with respect to this base line can
be thought of as due to gravitational forces. These gravitational forces
dene a gravitational stress energy tensor. This point of view has the
following usefulness: One is in this way easily able to extend the discus-

140

sions of the equivalence principle from the domain of equivalence between

gravitation and acceleration at a point to the question of equivalence in
the large. At a point the magnitude and direction of the acceleration are
arbitrary. For discussing the equivalence principle in the large, the choice
of the base coordinate system is arbitrary. Once one has selected this base
coordinate system all else is uniquely determined.

Chapter 14
Measurement of Classical Gravitation Fields
Felix Pirani

Because of the principle of equivalence, one cannot ascribe a direct physical interpretation to the gravitational eld insofar as it is characterized

by Christoel symbols . One can, however, give an invariant interpretation to the variations of the gravitational eld. These variations are
described by the Riemann tensor; therefore, measurements of the relative
acceleration of neighboring free particles, which yield information about
the variation of the eld, will also yield information about the Riemann
tensor.
Now the relative motion of free particles is given by the equation of
geodesic deviation

+ R v v = 0 ( , , , = 1, 2, 3, 4)
2

(14.1)

Here is the innitesimal orthogonal displacement from the (geodesic)

worldline of a free particle to that of a neighboring similar particle.
v is the 4-velocity of the rst particle, and the proper time along .
If now one introduces an orthonormal frame on , v being the timelike
vector of the frame, and assumes that the frame is parallelly propagated
along (which insures that an observer using this frame will see things in
as Newtonian a way as possible) then the equation of geodesic deviation
(14.1) becomes

2 a
+ Ra0b0 b = 0 (a, b = 1, 2, 3, )
2

(14.2)

Here a are the physical components of the innitesimal displacement and

Ra0b0 some of the physical components of the Riemann tensor, referred to
the orthonormal frame.
By measurements of the relative accelerations of several dierent pairs
of particles, one may obtain full details about the Riemann tensor. One

142

14. Measurement of Classical Gravitation Fields

can thus very easily imagine an experiment for measuring the physical
components of the Riemann tensor.
Now the Newtonian equation corresponding to (14.2) is

2 a
2v
+
(14.3)
b = 0
2
xa xb
It is interesting that the empty-space eld equations in the Newtonian
and general relativity theories take the same form when one recognizes
2
the correspondence Ra0b0 xa vxb between equations (14.2) and (14.3),
for the respective empty-space equations may be written Ra0a0 = 0 and
2v
= 0. (Details of this work are in the course of publication in Acta
xa xb
Physica Polonica.)
BONDI: Can one construct in this way an absorber for gravitational energy by inserting a dd term, to learn what part of the Riemann tensor
would be the energy producing one, because it is that part that we want
to isolate to study gravitational waves?
PIRANI: I have not put in an absorption term, but I have put in a spring.
You can invent a system with such a term quite easily.
LICHNEROWICZ: Is it possible to study stability problems for ?
PIRANI: It is the same as the stability problem in classical mechanics,
but I havent tried to see for which kind of Riemann tensor it would blow
up.
Interaction of Neutrinos with the Gravitational Field
D. Brill
The wave equation of a neutrino in a centrally symmetric gravitational eld
was derived, using the formalism of Schrdinger and Bargmann. In order
to construct a neutrino geon, one must also nd the gravitational eld
produced by a statistical distribution of neutrinos among available states
of such a character that the resulting stress energy tensor is spherically
symmetric. The stress energy tensor was worked out from a variational
principle. The equations which must be solved self-consistently were also
worked out for the case of a spherically symmetric neutrino distribution.
BERGMANN: What is the present motivation for the geon research?
WHEELER: The motivation is simply to understand more about how one
deals with non-linear eld equations. The idea is not that the geon has the

14. Measurement of Classical Gravitation Fields

143

slightest to do with an elementary particle, nor with astronomical objects.

The interesting things that come out, in the case of the electromagnetic
geon, are: if they have any stability at all, and the nature of the radial wave
equation for the electromagnetic eld. In the neutrino case: rst, there is a
similarity to the electromagnetic case; second, the radial equation contains
a spin orbit coupling term which does not appear in the electromagnetic
case, which may help to understand the dynamics of neutrinos.
DE WITT: In this work the neutrinos are not quantized; they are not real
neutrinos, are they?
WHEELER: One puts into each neutrino state just one neutrino; this
includes all of the results of second quantization.
DE WITT: Might not the gravitational eld induce transitions into negative energy states?
WHEELER: This already comes up in the electromagnetic case; what
should one use for the stress energy tensor? One counts as gravitation
producing only the excitations above the vacuum state, in order to avoid
innities. Here the only neutrino states one considers are the positive
energy states; the negative energy states are all lled.
BELINFANTE: You consider a complete absence of antineutrinos; no holes
in negative energy states.
WHEELER. Yes, just exactly as one does in the Dirac theory of the
electron.
Linear and Toroidal Geons
F. J. Ernst
Tolman has shown that there is no gravitational attraction between parallel light rays and that the attraction increases with the angle between the
rays, until they are antiparallel. For this reason, a spherical electromagnetic geon is expected to be less stable than a toroidal one. However, the
mathematical diculties in this case are great, and thus as a preliminary
step we consider a linear geon, in which the major radius of the toroid
has become innite; we consider light rays travelling along a central axis
in both directions. We look for the solution of the coupled equations of
general relativity and electromagnetism with a 4-vector potential of a form
deduced by analogy with the modes of excitation of a circular wave guide.
The equations can then be written down and numerically integrated. One

144

14. Measurement of Classical Gravitation Fields

obtains a complete solution of the eld equation which contains no singularities. This solution contains an arbitrary constant through which it
can be related to the toroidal solution. When the linear geon is bent into
torus, this constant is proportional to the major radius.1
Unied, Non-Symmetrical Field Theory
M. Tonnelat
The unied and non-symmetrical eld theory starts with a principle which
extends the principle of the Born-Infeld electrodynamics. It consists in
reducing the sources of the electromagnetic and gravitational eld to the
eld itself.
In order to avoid a point singularity, it is necessary in the static case
to dene an electric eld which is nite at the origin. The formalism of
the unied and non-symmetrical eld theory immediately introduces the
characteristic expressions of Born-Infeld theory.
By writing the fundamental tensor as the sum of two parts, we can
relate it to the invariant Lagrangian that appears in the Born-Infeld electrodynamics and the two invariants of the Maxwell theory.
On the other hand, the Einstein theory introduces the contravariant tensor, which may also be written as a sum of symmetric and skew
symmetric parts, which shall be called inductions. The relations foreseen
partly by the Born-Infeld theory appear between the eld and the inductions after making appropriate denitions. The denitions of metric, eld
and inductions may be chosen in two ways. These denitions depend only
on the properties of determinants and on the choice of eld and of metric;
they do not depend on the Lagrangian of the theory.
We apply the equations of the eld to the calculations of a static and
spherically symmetric solution of the form chosen by Papapetrou. We can
then deduce the radial component of the electrostatic eld, p14 . Now introducing like Born-Infeld a eld b = e/r02 , we set E = bp14 . Then, when r
approaches 0, E approaches b, a nite value. The originality of this conclusion comes from the fact that the values of the eld and of the metric
are not given a priori, but result from the eld equations in the particular
case of the Schwarzschild solution. We are now led to dene a current
following the principles of the Born-Infeld theory. We nd the charge
density, after substitution of the values obtained in the Schwarzschild solution, and upon integrating this over all space, we nd the nite value e.
So from the very principles of the unitary and non-symmetric theory, the
1 For

full discussion, see F. J. Ernst, Phys. Rev. 105 (1957).

14. Measurement of Classical Gravitation Fields

145

distribution of charge is characterized by a free density which leads to

a nite expression when integrated over the whole space. It is remarkable
that this conclusion which results from the Born-Infeld electrodynamics,
is connected here with a choice of the fundamental tensor and with an
extension of the Schwarzschild solution.
FEYNMAN: How big is r0 for an electron?
TONNELAT: It is exactly like in the Born-Infeld theory; it depends on
the choice of b; r0 is a fundamental length. There are two constants of
integration which correspond to r0 and mass, but there is a further arbitrariness in the choice of b, so that the charge e is not determined by the
choice of r0 and m.
DAVIS: I understand that b can be regarded as the ratio of the ordinary
electromagnetic eld to the natural electromagnetic eld. It appears that
b, as in the Born-Infeld theory could depend on the sign of the charge. Is
this correct?
TONNELAT: Yes, b appears like the quotient of the physical eld and the
mathematical eld.

Chapter 15
Correspondence in the Generalized Theory of Gravitation
Behram Kursunoglu

It is hoped that a reconciliation of quantized gravitational elds, the matter eld, and electromagnetic eld might induce a further structure on
the elementary particles, and might result in softening the divergences,
or eliminating them altogether. It is well known that the gravitational
eld has a non-local character, so it probably would not be possible by
quantizing the gravitational eld to nd a 4-vector momentum giving a
particle property to gravity. One tries the approach of generalizing Einsteins theory, making it more complicated than it is, hoping to get some
more physics out of it. Since there is no physical basis in unied eld theory, as there is in general relativity, one proceeds in a more mathematical
fashion. Nature must be described by a complete tensor with symmetric
and antisymmetric elds. One then constructs all of the quantities recognized in general relativity and obtains, for example, Bianchi identities and
Maxwell identities. In order to compare the theory with general relativity,
we have a conservation law in the absence of charges and this introduces
a fundamental length r0 . The real motivation for this procedure is the
development of a correspondence argument in order to identify the formalism with a theory. The question arises what happens if we assign no
structure to the theory and r0 approaches 0. We obtain the equations of
general relativity with the electromagnetic eld as the source of gravitation
plus Maxwells equations for charge-free elds. The existence of charges is
linked up with a non-vanishing fundamental length just as the existence
of spin arises from a nite h.
Now, the metrics depends on length. We have calculated the propagation of light in the presence of very strong background elds; the result
is in agreement with those of Schrdinger from non-linear optics. One may
also change the sign of r0 , but we cannot nd a way of measuring negative
length. The concept of negative length may be linked up with negative
mass (or antimatter).

148

15. Correspondence in the Generalized Theory of Gravitation

DE WITT: Can you state any conclusions about matter and antimatter?
KURSUNOGLU: I cannot state any conclusions because I must solve
the equations and show that solutions involving a negative and a positive
length will give repulsion.
TONNELAT: It is not necessary to introduce the factor 1/r0 into the
Lagrangian itself in order to obtain E D, B H when r0 0, because it
is a conclusion of the Born-Infeld theory. However, it is necessary to have
this factor 1/r0 or something else, in order to obtain equations of motion.
KURSUNOGLU: Firstly, I dont believe in the Born-Infeld theory; secondly, equations of motion are a dicult concept in this theory because
one has to dene what a particle is rst. Usually the eld variables are
singular at the point of a particle; here what can I let move if I dont have
a particle yet?
SALECKER: Can you not quantize this theory to get the results for elementary forces?
KURSUNOGLU: One develops a unied theory in order not to do anything with quantum theory. If there are any quantum eects, they must
be contained in the non-linearity which is caused by the nite value that
we have introduced for the electromagnetic eld. Quantization is out of
the question.
LICHNEROWICZ: It seems to me that many physicists do not like this
type of unied eld theory. We have a very good theory of propagation,
but it is dicult to nd physical interpretations, because the theory is in
a sense too unied. We have a good interpretation for the metric or the
gravitational part, but it is dicult to obtain a good interpretation for
the electromagnetic part. The rst problem is not to obtain the equations
of motion; the rst problem is to obtain by a process of approximation a
good identication.
BERGMANN: I believe that we have the following principal problems:
Firstly, the identication, I agree with you. Secondly, the actual nding of
the solutions which are free of singularities. If one believes in the unied
eld theory approach, he must face the question: does the theory lead to
solutions which can be interpreted as particles? The equations of motion
are a subsidiary problem. If there are such solutions, are they stable?
WHEELER: Perhaps the rst question to be asked about unied eld
theory is why?. We have non-singular solutions in geons in the present
theory.

15. Correspondence in the Generalized Theory of Gravitation

149

BERGMANN: All of the geons involve stochastic methods. A rigorous

solution would be one in which no averaging operation were performed,
where you can tie down a single solution.
WHEELER: A geon has exactly the property of being only an approximate
solution; or rather, an accurate solution which is not fully stable with time
- it leaks energy. Thus it is not in agreement with ones preconceived idea
that there should be a particle-like solution that is fully stable; but arent
we being very brash if we say that the world isnt built that way? How do
we know that the leakage of energy is not a property of high mass particles
which leak energy and drop down to lower mass particles? Perhaps the
stability of the particles we know is due to some intrinsically quantum
character, which we cannot expect to show up before we have gone to the
quantum level.
BERGMANN: The original motivation of unied eld theory is get a
theory of elementary particles, which includes electrons and not only hyperfragments, and furthermore to obviate the need for quantization which
would result from the intrinsic non-linearity.
FEYNMANN: Historically, when the unied eld theory was rst tackled,
we had only gravitation, electrodynamics, and a few facts about quantization, so it was natural to try to write down equations which would only
unite the Maxwell equations with the gravity equations and leave out of
account the strange quantum eects, with the hope that the non-linearity
would produce this business. In the meantime, the rest of physics has
developed, but still no attempt starts out looking for the quantum eects.
There is no clue that a unied eld theory will give quantum eects.
ROSENFELD: We know that there are forces which are certainly not reducible to electromagnetic and gravitational forces; forces of nite range.
It seems that this unication program is anachronistic. I think that it is
an illusion to hope that the particle can be reduced to some distinct distribution, either singular or non-singular, for the particle must be regarded
as a eld itself in quantum theory.
LICHNEROWICZ: We have here a theory which has a denite mathematical meaning. In quantum theory, there are many good experiments, but
many parts of quantum theory are not good theories. There are various
diculties with mathematical consistency in a large part of the quantum
theory.

150

15. Correspondence in the Generalized Theory of Gravitation

BONDI: We should regard general relativity not as a supertheory, but as

the best theory of gravitation we have got, which describes a certain set
of phenomena better than any other theory we have.
KURSUNOGLU: In unied eld theory, one tries to make particles, sort
of mass, out of the electromagnetic eld. This is also the case in quantum
electrodynamics; the eld creates mass, the electron-positron pairs.
FEYNMAN: Lets ask ourselves the question: If we looked at things on
a large scale, and saw them gravitating, we should discuss the character
of the singularities; but, gentlemen, look at one of the singularities: were
sitting on it. Look at the degree of complexity that it has; the lights and
the colors, and the trees, and the particles that its made out of. Can you
believe that in the fact that they gravitate there lies the key to all this
confusion? Wouldnt it be more reasonable that gravity were a result of
something on the small scale?
ANDERSON: To describe elementary particles, we must adopt a spacetime description; you must say something about the metric. Relativity is
not only a theory of gravity, but also a theory of space structure.
WITTEN: There may be another reason to look for a unied eld theory,
as a step toward quantization. Now we quantize a lot of elds, and it
might be easier to quantize just one unied eld.
BONDI: One important point: gravitational forces are not self-compensating and that is why they predominate on the large scale. I think thats
a side that should be looked at more, in the large.

Chapter 16
Presentation of Work by T. Taniuchi
Ryoyu Utiyama

UTIYAMA: I am going to talk about an article of my colleague, Dr. T.

Taniuchi.
In quantizing the gravitational eld, or any type of non-linear elds,
particular attention should be paid to the following points.
In general a disturbance of non-linear elds in Minkowski-space propagates with a super-light velocity in some cases and with a sub-light velocity
in some other cases. Namely, the velocity of non-linear elds usually depends on the strength of elds. This means that it is not so clear as in the
case of linear elds whether any couple of eld-quantities on a space-like
surface is commutable or not. Another remarkable feature of non-linear
elds is the following. Suppose that some initial value is given which has no
discontinuity on the initial spacelike surface. In some cases, however, some
discontinuity of the eld strength (shock wave) may be expected to occur
after the time passed. On the contrary, even though the initial value has
some discontinuity, at a later time such a discontinuity will be expected
to be smeared out owing to the non-linearity. In general the occurrence of
shock waves depends not only on the type of non-linear elds but also on
the type of initial values.
Owing to these situations mentioned above, it seems necessary to
investigate the character of propagation of the gravitational eld before
making quantization of this eld.
However, since the gravitational eld is too complicated to deal with,
we considered the following two cases as our training, the Born-type eld
with the Lagrangian density [1]
L = l2

1 + l 2 (2,x 2,t )

l = constant with a dimension (length)2 ,

and the Landau-Khalatnikow-type eld [2, 3]

16. Presentation of Work by T. Taniuchi

152

1
L = {2,x 2,t }2 .
4
These two types of non-linear eld were adopted by Heisenberg and Landau respectively in order to explain the phenomena of multiple mesonproductions.
Our method of solving these equations is completely similar to that of
Courant and Friedrichs stated in their textbook [4]. We are also planning
to investigate the equation of gravitational eld in some simple case.
(1) Case of Born-type eld: The Lagrangian density is

L = l 2 1 + l 2 (2,x 2,t ).
u = ,x , v = ,t .

Put

Then the eld equation is

(v2 l 2 )ux + (ut vx )uv (u2 + l 2 )vt = 0,

ut + vx = 0.
In this case the two kinds of characteristics c+ -curves and c -curves
consist of two sets of parallel straight lines respectively and shock
waves do not occur.
Consider the following initial condition:
at t = 0
u(x)
v(x)

=0
=0
= const. =

for all x
for |x| > a
for |x| < a

The solution in this case is shown in Figure 16.1, where

OP = a/{1 ( /l)2 }1/2 .

16. Presentation of Work by T. Taniuchi

153

t
C+

u=v=0
C
C
C+

u=v=

l(l

l 2 2 )

u=v=0

u=0
v=

45

45

Figure 16.1: Solution of Born-type eld

The initial discontinuity at A propagates with the light velocity. The
feature of propagation is quite similar to that of linear elds, because
this particular non-linear eld becomesidentical with the linear eld
in case of an extremely weak eld.

16. Presentation of Work by T. Taniuchi

154

(2) Case of Landau-Khalatnikow-type eld: The Lagrangian density is

L = l2
Put

1 + l 2 (2,x 2,t ).
u = ,x , v = ,t .

v(x)

v(x)

(x)

(x)

A
Case (a)

Case (b)

Figure 16.2: Initial value for L-K-type eld

Let us consider the two kinds of initial values shown in Figure 16.2.
In the case of (a), namely (x) > 0 , the c+ -characteristics intersect
with each other and the shock wave occurs in spite of the continuous
initial value (see Figure 16.3). On the other hand, in the case of (b),
namely (x) < 0 , the c+ -characteristics diverge from each other,
that is, the rarefaction wave occurs (see Figure 16.4).
Let us also consider the same initial condition as that of Born-type
eld. The solution is illustrated in Figure 16.5. In this case the
discontinuity of v at A is smeared out, and u and v at any later time
t > 0 continuously vanish on the light-cone AB. Contrary to linear
elds, the disturbance of the eld soaks into the region CPC.
From these results we can see that the latter type of eld is more adequate
to the explanation of the phenomena of multiple meson-production.

16. Presentation of Work by T. Taniuchi

155

t
C+

C
C+

C+

C+

60

60

A
a

Figure 16.3: Characteristics for (a)

16. Presentation of Work by T. Taniuchi

156

t
C+

C+

C+
C+

60

Figure 16.4: Characteristics for (b)

60

16. Presentation of Work by T. Taniuchi

157

C+

C
C+
C

C+

C+
C

C+

C+

C
C+

C+

C
C+

C+ = C-

C- = C+
P

C+

C
C

u=v=0

u=v=0
60

45

45

Figure 16.5: Solution of L-K-type eld

158

16. Presentation of Work by T. Taniuchi

DE WITT: Does Professor Lichnerowicz know whether gravitational waves

form shocks? Do the characteristics cross?
LICHNEROWICZ: Not for a regular model.

References
[1] M. Born. Proc. Roy. Soc. Lond. Ser. A 143 (1933), 410.
W. Heisenberg. ZS. f. Phys. 113 (1939), 61; 126 (1949), 519; 133
(1952), 79.
[2] L. Landau. Izv. Akad. Nauk. S.S.S.R. 17 (1951), 51.
[3] J. I. Khalatnikow. Jour. Exp. Theor. Phys. 27 (1954), 529.
[4] R. Courant and K. O. Friedrichs: Supersonic Flow and Shock Waves.
Interscience Publishers, Inc., 1948.

Chapter 17
Negative Mass in General Relativity
Hermann Bondi

In general relativity there is no obvious reason why masses shouldnt be

negative, so we will investigate this concept. We can get three denitions
of mass: inertial mass, passive gravitational mass on which the eld acts
and active gravitational mass which is the source of the eld. In Newtonian theory, due to action and reaction, passive and active gravitational
masses must be the same. The fact that passive and inertial masses are the
same is the consequence of experiments. In general relativity, passive and
inertial masses are the same due to the principle of equivalence. The rst
place where active gravitational mass occurs is the m in the Schwarzschild
solution. There is no action and reaction principle, and we know that active and passive masses may not be equal. In Newtonian theory, the most
obvious notion of a negative mass is similar to charge; like masses attract
and unlike repel. In general relativity, a negative mass repels all masses,
a positive attracts all. If we have a system with one mass of each type,
the system will uniformly accelerate. I have constructed an exact solution
of the eld equations which shows these properties. The solution lls only
one-fourth of space-time. I have found a method of continuing the solution
into the other regions.
BERGMANN: I think one can show that within the general theory, it is
not possible to have the active and passive masses dierent.
BONDI: This is a question of conservation of momentum, which means
integrals over extended regions of space. But what do you say about the
Schwarzschild interior solution where the m is certainly not just the integral
of the g00 ?
GOLD: What happens if one attaches a negative and positive mass pair
to the rim of a wheel? This is incompatible with general relativity, for the
device gets more massive.
BONDI: The purpose of this is to show that negative mass is incompatible,
but I havent got there yet.

160

17. Negative Mass in General Relativity

PIRANI: I want to reply about active and passive mass. One can show
simply that they are dierent and that the density of one is got by taking
the density of the other and subtracting the principal stresses.
A Dynamic Instability of Expanding Universes
R. Mjolsness
Small deviations of the metric from exact sphericity are considered in order to investigate the stability of the standard Friedman solution. An
oscillating cosmological model is used with positive curvature and zero
cosmological constant. In order to solve the problem exactly, a relation
between the pressure and the density is needed. Two cases are considered
for which this relation is known; the dust-lled and the radiation-lled
universe. This problem is similar to one done by Lifschitz in 1946. He
wished to discover whether one can account for the formation of nebulae
in an expanding universe of negative curvature. He found that the perturbations do not grow suciently rapidly. Here, the question is reopened,
for the perturbations do grow suciently rapidly. Deviations of the metric from spherical symmetry are expanded in terms of tensors formed from
hyperspherical harmonics. Two of the four types of tensors formed give
no contribution to a change in the density, but two of them do. These
two result in two coupled ordinary dierential equations. The case of
the dust-lled universe is completely solvable and unstable. Work on the
radiation-lled case has been started. Here it appears that there is a solution in which the perturbations grow. There is a close analogy with the
problem of an oscillating underwater bubble which is well known to show
Rayleigh-Taylor instability against small departures from a spherical form.
BONDI: Have you tried this for the Newtonian cosmological models of
Milne and McCrae?
MJOLSNESS: No.
BONDI: They have an extremely close analogy; on the cosmological and
galactic scale we get the same equations. I have played about with that,
and I think that you always get that while the relative density uctuations
increases, the density goes down.
WEBER: In the topological models of Wheeler, one forms charges out of
elds. The question arises how one can measure elds without charges. I
will outline one method. If a neutral body is constructed out of elds and
placed in an external uniform eld, the body will become polarized and

17. Negative Mass in General Relativity

161

appear to have a dipole moment. If two such neutral bodies are placed
in a eld, one can determine the existence of an external eld from their
motion. In addition, the fact that a eld is capable of polarizing a neutral
body implies that a neutral body should repel a charged body. This may
have some cosmological signicance, but the forces are very small.
SCIAMA: Instead of studying all of the complexities that exist in the
symmetric theory of gravitation, I want to propose the possibility that the
theory of the pure gravitational eld should be based on a non-symmetric
potential if the sources have spin. When one tries to dene spinors in such
a scheme, certain denite statements about elementary particles may be
made, which can be checked within a year or two. The original motivation for introducing a non-symmetric potential is heuristic. From special
relativistic eld theory, we know that the energy momentum tensor is
non-symmetric if the system has spin. Belinfante has shown how this tensor can be symmetrized. This procedure constructs a complicated energy
momentum tensor whose moment contains the spin; but the spin is not
fundamentally a moment, as orbital angular momentum is.
As a purely heuristic argument, it might be less articial to keep the
energy momentum tensor non-symmetric, so that it is apparent that the
system possesses spin. In that case, one is forced to construct a theory
of gravitation with a non-symmetric potential. However, we make an exception of elds with zero rest mass, for in this case we cannot make a
distinction between spin and orbital angular momentum. We construct a
theory whose mathematics is similar to Einstein-Schrdinger type theories,
but has nothing to do with the electromagnetic eld. We nd that the orbital angular moment in the special relativistic limit is the moment of the
canonical stress energy tensor. The quantity gi j plays the same role in the
equations of motion of a scalar test particle as gi j in the symmetric theory. To introduce spinors in the non-symmetric theory, we use a Hermitian
metric with symmetric real part and non-symmetric imaginary part. This
allows us to introduce complex vierbeine, so we can perform unitary transformations rather than rotations and this leads to a Hermitian requirement
for the energy momentum tensor. To get something analogous to spinors,
we must discuss quantities that transform irreducibly with respect to the
unitary group. There are two main dierences between this group and the
Lorentz group. First, the Lorentz group is composed of four disconnected
parts, while the unitary group envelops all continuously. This suggests
that parity must be conserved in such a scheme. These representations
are complex, and thus describe charged particles; strictly neutral particles
must be described by real wave functions and thus are subject only to

162

17. Negative Mass in General Relativity

the Lorentz subgroup and will have symmetric energy momentum tensors.
Neutral particles may not conserve parity. Also, neutral particles must
have zero spin, zero rest mass, or both. The second dierence relates to
topological properties; the topological properties of the unitary group are
more complicated, and this allows one more freedom in constructing representations. A phase transformation is included under the unitary group.
If one has only one eld, the phase transformation cannot be determined;
but if several elds interact, this leads to selection rules. The question
now is can we arrange these rules to forbid certain interactions which are
known not to occur in nature, which are consistent with the present selection rules. This work is not yet complete; but one can presumably nd
rules which will prevent heavy particles from decaying into light particles,
and thus understand the conservation of heavy particle number.
KURSUNOGLU: In this theory with a complex representation of eld
variables, isnt the velocity of light greater than its normal value as it is
with Einsteins invariance?
SCIAMA: I presume that the motion of light will be given by the same
quantity that governs the equations of motion.
BERGMANN: This is a dierent theory, you dont have invariance.
A discussion of the recent discoveries regarding parity followed. Professor ROSENFELD spoke of the contents of a recent note from Landau
in which he suggested that in weak interactions, the coupling is not invariant for parity, or charge conjugation, but for the product of the two. By
introducing this combined parity, one is left with invariant couplings; one
cant have one of the transformations without the other.

Session VI Quantized General Relativity

Chairman: J. A. Wheeler

Chapter 18
The Problems of Quantizing the Gravitational Field
P. G. Bergmann

This session opened the second half of the conference, devoted to discussion of the problems of quantizing the gravitational eld, previous sessions
having been restricted to the classical domain. The rst contribution was
an introduction by P. G. BERGMANN outlining the present position of
the infant subject of quantum gravidynamics, indicating why one is interested in it in the rst place, and stating some of its outstanding problems.
The following is a summary of BERGMANNs introduction:
BERGMANN rst asked the question, Why quantize? His reply
was that physical nature is an organic whole, and that various parts of
physical theory must not be expected to endure in peaceful coexistence.
An attempt should be made to force separate branches of theory together
to see if they can be made to merge, and if they cannot be united, to try
to understand why they clash. Furthermore, a study should be made of
the extent to which arguments based on the uncertainty principle force
one to the conclusion that the gravitational eld must be subject to quantum laws: (a) Can quantized elementary particles serve as sources for a
classical eld? (b) If the metric is unquantized, would this not in principle
allow a precise determination of both the positions and velocities of the
Schwarzschild singularities of these particles?
These aims have not yet been achieved, but BERGMANN expressed
certain hopes as to the results of such a program: Quantization of the gravitational eld is likely to have a profound eect on our notions of space
and time, making all distance and volume concepts subject to uncertainty
relations. Thus, in spite of the quantitatively negligible character of the
gravitational forces between elementary particles, it is conceivable that (a)
the gravitational eld may help to eliminate the divergences of quantum
eld theory (which result from the compounded eect of singularities in
particle propagators) by smearing out the light cone, and (b) it may somehow contribute to the structure of elementary particles. In regard to the
latter point, however, BERGMANN expressed his opinion that one cannot

166

18. The Problems of Quantizing the Gravitational Field

hope to get the complete structure of the elementary particles from any
quantized unied eld theory that is principally motivated by the desire
to unify just the gravitational and electromagnetic elds.
BERGMANN emphasized the formidable nature of the problem of
quantizing generally covariant theories and expressed the conviction that
one will rst have to have a thorough clarication of the underlying conceptual problems in the classical theory. He then went on to outline the
principal methods of approach to the purely technical problems of quantization which we now possess: (1) The canonical Hamiltonian method
(Dirac); (2) The Lagrangian method (Schwinger); (3) The path-summation
method (Feynman). It appears that the need to identify the so-called true
observables, or coordinate-invariant quantities, arises in all three schemes.
Furthermore, this identication will be intimately related with the structure of the transformation groups under which the classical theory remains
invariant.
In the quantum theory the state vector of a generally covariant system
will be subject to various constraints which must, of necessity, be imposed
owing to the existence of the invariant transformation groups. A true
observable will be described by an operator which, when applied to the
state vector, produces another vector which satises the same constraints
as the original vector. A reduced Hilbert space is envisaged in which the
only canonical transformations which are physically meaningful are those
generated by true observables. The constraint operators themselves qualify
as true observables under this denition, but they are trivial, being simply
null generators. BERGMANN believes that it is immaterial whether the
Lagrangian or Hamiltonian approach is used to discover the nontrivial true
observables; the results will be the same in either case.
As remaining problems which must eventually be looked at, BERGMANN gave the following partial list:
1. The hyperquantized particle eld vs. the treatment of particles as
singularities in a quantized gravitational eld.
2. The interaction of the gravitational eld with fermions.
3. The interaction of the gravitational eld with other quantized elds.
4. The relation of elementary particle theory to unitary eld theories.
5. The relation of the law of conservation of energy and momentum arising from the coordinate transformation invariance of general relativity to other strong conservative laws of physics with their associated
invariance groups.

18. The Problems of Quantizing the Gravitational Field

167

Discussion then turned to the problems of measurement of the gravitational eld. This item was placed rst on the agenda in an attempt to
keep physical concepts as much as possible in the foreground in a subject
which can otherwise be quickly ooded by masses of detail and which suffers from lack of experimental guideposts. The question was asked: What
are the limitations imposed by the quantum theory on the measurements
of space-time distances and curvature? Since the curvature is supposed to
be aected by the presence of gravitating matter, an equivalent question is
to ask: What are the quantum limitations imposed on the measurement of
the gravitational mass of a material body, and, in particular, can the principle of equivalence be extended to elementary particles? (In the interest
of clarifying the importance of some of the following discussion, the editors
would like to point out at least one reason, which was not given sucient
attention at the conference, why the answer to the latter question is not
a simple matter of dimensional arguments. In quantum gravidynamics
there exists a natural unit of mass, namely ch/G 105 g. One might
be tempted to suppose that this represents a lower limit on the mass of a
particle whose gravitational eect can in principle be measured. That this
conclusion is wrong can be shown by the following arguments:
Consider just a classical test particle of mass m which is initially at rest
at the point x0 (we restrict ourselves to one dimension) in a gravitational
potential . At the time t the position of the particle will be
1 2
t .
2 x
The gravitational eld strength is given by / x and can immediately be
determined by a measurement of t and x. If however the particle is subject
to quantum laws its initial position and velocity are subject to (root mean
square) uncertainties related by
x = x0

v0 =

h
2mx0

leading to an uncertainty in position at time t > 2mx02 /h = h /2mv20 of

amount
x = v0t =

h t
.
2mx0

The initial position measurement may be made by a photon of momentum

uncertainty p = mv0 (or, in principle, by a graviton having the same
momentum uncertainty if one prefers to deal with absolute electrically

18. The Problems of Quantizing the Gravitational Field

168

neutral test particles!). The resulting uncertainty in the initial time may
be ignored since it is given by t0 = x0 /c = h /2mcv0 << t (assuming
v0 << c). The nal position and time measurements may be made with
arbitrarily high precision, using energetic photons, since the experiment is
then over.
The gravitational eld can now be determined from the classical equation if





 2
t >>

h
h t
>>
.
mx0
mc

However, one must be able to choose t small enough so that | / x| does

not change appreciably during the course of the motion. This imposes a
condition on the gravitational eld, namely




2
x2



 2

t << 

or







x
2
x2




h
 >>>>
.

mc


If the gravitational eld is produced by a point mass M then = GM/x,

/ x = GM/x2 , 2 / x2 = 2GM/x3 , and the required condition becomes x >>>> h /mc which can always be satised. Evidently the gravitational eld of any mass is measurable in principle because of the long tail
in the Newtonian force law. In fact, the long tail is what permits us to
measure the gravitational mass of, for example, a nucleon, by measuring
the gravitational mass of a large ball of lead and dividing by the known
number of nucleons in it.
It is, however, still a matter of basic interest to determine whether or
not the measurement could be carried out on a single elementary particle.
Here the main practical problem concerns the measurement of time. One
has the restriction t >> (h/mc)1/2 /| / x|1/2 or

t >>

h x2
>>>>
GMmc

h 3
.
GMm3 c3

If both the test particle and source has protonic mass then

18. The Problems of Quantizing the Gravitational Field

169

h 3
105 sec.,
GM 4 c3
and all conditions may seemingly be satised by choosing t to have the not
unreasonable value t = 1 sec. Moreover, no complications should arise due
to the nonlinear character of the gravitational eld (the linear Newtonian
approximation should be masses as small as a proton).
Nevertheless, a number of subtleties enter the problem at this point,
chiey concerning the nature of the recording apparatus, or clock, which
measures the requisite time intervals (independently of the photons which
interact with the test particle) and how the presence of the clocks mass
in the neighborhood may inuence results of the experiment.)

Chapter 19
Conceptual Clock Models
H. Salecker

H. SALECKER opened the discussion by reporting on some conceptual

clock models which he has analyzed in collaboration with E. P. Wigner.
He rst made some general comments on the fundamental nature of spacetime distance measurements: In ordinary quantum mechanics the spacetime point is specied by its four coordinates, but no prescription is given
as to how these coordinates are to be measured. This, however, is in
conict with the principles of the general theory of relativity, according
to which coordinates have no meaning independent of observation. A
coordinate system can be dened only if space-time distance measurements
can be carried out in principle without restrictions. SALECKER then
went on to point out that the use of clocks alone is sucient to measure
both space-time and time-like distances. This is important since measuring
rods, in contrast to clocks, are essentially macro-physical objects which will
strongly inuence other objects during the measuring procedure through
their gravitational elds. Moreover, the measurement of distances between
space-time points is considerably more complicated with measuring rods
than clocks because of the Lorentz contraction of rods.

Figure 19.1

19. Conceptual Clock Models

172

The method by which a clock can be used to determine a space-like

distance between two events A and B is as follows: Let the geodesic world
line of the clock pass through the event A at time tA , and suppose that
the clock emits a continuously modulated light signal, e.g., monotonically
changing color.1 Let this signal be reected at event B by a briey exposed
mirror. The time of emission t1 of the reected portion of the signal will
be determined by inspection of the color of this portion when it eventually
returns to the clock at time t2 . The invariant distance S between A and B
is then given by
S2 = c2 (t2 tA )(tA t1 )
provided S is small compared to the radius of curvature of the space-time
neighborhood of the two events.
A time-like distance can of course be measured by the obvious method
of letting the clock pass along a geodesic between the two events.
SALECKER then described the results of considering the clock itself,
and the process of reading it, in greater detail. Although his actual exposition suered from lack of time in which to present the material, he very
kindly gave the editors access to the manuscript of a paper which he will
shortly publish on the subject. The following is a resume of portions of his
paper. Certain simplications have been made, leading to slight changes
in numerical coecients, for which the editors are entirely responsible.
One must rst ask the question: What is the accuracy with which
a clock can be read, independently of inherent inaccuracies of the clock
itself? At the beginning of a time interval an observer may emit a light
signal to read the clock. If the length of the signals train is l and if the
pulse is properly shaped, the root mean square uncertainty in the initial
time measurement will be
tl = l/2c.
The clock may simultaneously be made to emit a photon to compensate
for recoil. However, an inevitable uncertainty in the clocks momentum
will remain, of amount
pl = h /l = h /2ctl
corresponding to an energy uncertainty in the light signal of amount
1 SALECKER let the position of event B be simply determined by the time of emission
t1 of a pulsed signal. The editors have inserted the idea of color modulation in order to
avoid objections based on the principle of causality.

19. Conceptual Clock Models

173

El = cpl = h /2tl .
The clock will generally have also an initial position uncertainty x
giving rise to a momentum uncertainty
px = h /2 x.
The total momentum uncertainty after the rst reading is therefore
(
)
1
1
p = px + pl = h
+
2x l
corresponding to a velocity uncertainty of amount
(
)
p
h
1
1
v =
=
+
M
M 2x l
where M the mass of the clock. (We neglect here certain relativistic corrections which SALECKER considered.) If x is the mean distance from
the observer to the clock then the time at which the observer receives the
initial reading from the clock will be
t1 = (x x)/c tl .
The time at which the observer receives a second reading at the end of a
(clocks) time interval t will be
t2 = (x x vt)/c tl + t,
giving for the total inaccuracy in the reading of the time interval
t = |t2 t1 t|max = 2 x/c + 2 tl + vt/c
1
h t 1
1
= (2x + l) +
(
+ ).
c
Mc 2x l
The minimum uncertainty is achieved by choosing 2x = l = (ht/M)1/2 ,
which yields
h
.
Mc2
Evidently the accuracy of reading is greater the larger the mass of the clock.
On the other hand, the gravitational eld of the clock will be disturbing if
tmin = 4(tct)1/2

tc =

19. Conceptual Clock Models

174

its mass is made too large. One has therefore to consider the problem of
how to construct a clock which shall be as light and as accurate as possible.
One must at this point take into account the inherent inaccuracies in the
clock itself, by considering its atomic structure.
A single atom by itself represents to a very high degree of accuracy
an oscillator, but in spite of this it is not possible to take a single atom
emitting radiation as a clock. Before an oscillator can be considered to
be a clock it must be possible to register its information; i.e., the atom
must be coupled with a device to count the number of times the emitted
electromagnetic eld strength reaches a certain value. Such a counting
device would be in contradiction with the principles of quantum mechanics.
As his rst model, therefore, SALECKER considered a statistical
clock, composed of a certain number N0 of elementary systems initially
in an excited state. The systems were assumed to go over directly to a
ground state and to be suciently well separated so as not to reexcite one
another. If the decay rate is , then the probability of nding N systems
remaining in the excited state after a time t is
P(N,t) =

N0 !
N (1 )N0 N
N!(N N0 )!

where

= e t .
The average value of N and the root mean square deviation at time t are
given respectively by

N = N0

N = N0 (1 ).
The registering device in the clock has only to count the number of atoms
remaining in the excited state (or, alternatively, the number of systems
which have decayed) in order to record a statistical time given by
tstat =
which has an uncertainty of amount

1 N0
ln
N

19. Conceptual Clock Models

175

1 N0
1 N
(e t 1)1/2
tstat = | ln | =
.
=
1/2
N
N
N0
The minimum uncertainty is achieved by choosing = 1.6t, which yields
tstat min = 1.2

t
1/2
N0

A nal uncertainty arises from the fact that the registering device
must distinguish between excited and unexcited systems, and for this purpose a time tm at least as great as h/(E1 E0 ) is required, where E1 and
E0 are respectively the elementary excited and ground state energy levels.
In the most favorable case the decaying systems would undergo complete
disintegration with, for example, the emission of two photons in opposite
directions so as to eliminate recoil. The registering device might then be a
counter to detect the photons, but however constructed it would have an
absolute minimum time inaccuracy of amount tm = h /mc2 , corresponding to photon energies of order mc2 where m is the mass of a decaying
elementary component of the clock.
The total inaccuracy in the measurement of a time interval t by means
of a statistical clock is therefore at least

ttot = tmin + tstat min + tm

t

= 4(tct)1/2 + 1.2 1/2 + (1 )1 N0tc

M
N0
where is the mass of the registering device plus the framework of the
clock, and M = N0 m + is the total clock mass. If one is clever enough
1
in the construction of the clock the number (1 M
) may be kept from
being excessively large. The minimum total inaccuracy for a given total
mass M is achieved by choosing
3/2

N0

= 0.6(1

t
)
M tc

which yields
ttot min = 4(tct)1/2 + 2.1(1

1/3 2 1/3
)
(tct ) .
M

19. Conceptual Clock Models

176

SALECKER considered also another type of statistical model, with

essentially the same results. It should be pointed out immediately that for
a given M and t (e.g., suitable for a certain experiment) it may be utterly
impossible to choose the optimum value of N0 , because of the very limited
range of elementary-system masses m occurring in nature. SALECKER
therefore considered, as a third model, a very hypothetical type of clock,
consisting of a single elementary harmonic oscillator, but operating at
suciently high quantum numbers for it to be read like a classical device. Permitting himself to imagine an elementary oscillator which could
function even at relativistic energies, he found for the minimum total uncertainty

h
h t
tot min 2 +
mc
m c2
where m = m + E/c2 , m being the rest mass of the oscillator and E is its
excitation energy.
The quantity tot min is seen generally to depend on the time interval
itself. When t becomes of the order of tot min then the time interval can
no longer be measured. In every case the smallest time interval which can
be measured is roughly
h
.
mc2
That is, the time associated with the mass of one of the elementary systems
out of which the clock is constructed represents an absolute minimum
for measurable time intervals. This minimum arises, of course, from the
coupling between the registering device and the elementary systems, and
would exist even if the registering device were removed from the immediate
vicinity of the clock proper. (For example, the registering device might
be the observer himself, but then, in the case of the statistical clocks,
the observer would have to send out a large number of photons so as to
read each elementary system separately, thus raising the reading error
from 4(ht/Mc2 )1/2 to 4(ht/mc2 )1/2 .) Since, for existing elementary systems,
m 1024 g, the corresponding time tmin 1023 sec. may, presently at least,
be regarded as an absolute lower limit to measurable time intervals.
SALECKER nished his discussion by reporting on the results of applying such clocks to the measurement of gravitational elds. This part
of his work has not yet been written up, so we can only record the nal
results which he quoted. Three typical measurements were considered by
him:
tmin

19. Conceptual Clock Models

177

1. The direct measurement of g .

} through the observation of geodesics.
2. The measurement of {

3. The measurement of the scalar curvature of a three-dimensional

space-like cross section of a quasi-static region of space-time. This is
dened by
1
K = lim
a0 a

i=1

where a is the area of a small triangle (or prism, in space-time)

around which a test body is passed, and the i are its angles. (He also
proposed that a direct measurement of the Riemann tensor should
be carried
by the )
method suggested in a previous session by
( 2out,
a
Pirani. t2 = Ra0b0 b . He had not been aware of this possibility
prior to the conference.)
Carrying out these conceptual measurements on the gravitational eld
of a particle of protonic mass, using a particle of electronic mass as a test
particle, and choosing the most favorable clock model to serve as measuring apparatus, SALECKER found for the uncertainty in the resultant
measurement of the mass producing the eld
M 1024 g .
That is, the uncertainty is of the same order as the mass itself. From this,
he concluded that the gravitational mass of a single proton is not strictly
an observable quantity.
FEYNMAN asked SALECKER if he could write down a formula for
M in terms of fundamental constants, omitting dimensionless factors.
SALECKER replied that this would be quite dicult. The diculty seems
to be that one has to account for the perturbing eect of the clock and,
in eect, solve a two-body problem. The uncertainty in the measurement
of time increases proportionally to some fractional power (t 1/2 or t 2/3 ) of
the interval being measured, and decreases with increasing clock mass. On
the other hand the bigger the clock the greater its perturbing (or masking)
eect. One has to carry out a minimizing procedure in which the interval
t, the protonic mass under observation, the clock mass M, and, in the
case of statistical clocks, the number N0 are all involved. In the resulting
numerical computations the fundamental constants get mixed up in a very

178

19. Conceptual Clock Models

complicated way. SALECKER did say, however, that M is proportional

to the mass being measured, the factor of proportionality depending on
the ratio of the gravitational radius of this mass to that of the clock.
(The editors would like to suggest that, in view of the long range
character of the gravitational force, discussed previously, the gravitational
constant may, in the last analysis, not enter into the factor of proportionality. Furthermore, the value actually found for M may in some way
reect the fundamental limitation in the clock, viz. that it cannot be
constructed out of elementary systems having masses smaller than those
found in nature, i.e., protonic.)
FEYNMAN remained unconvinced of Saleckers result. He suggested the
use of hypothetical atoms in which only two forces are operative: (1) the
gravitational force and (2) some force which does not have an inverse
square character. The relative phases of dierent time dependent states
will then dier for atoms of dierent masses, and by waiting a suciently
long time (e.g., 100 times the age of the universe!) one could measure the
phase dierences and hence infer the gravitational masses involved to as
high a degree of precision as may be desired.
SALECKER replied that Feynman would still have to take into account
the apparatus which measured this very long interval of time. Moreover,
he pointed out that his own result did not take into account the possibility
of pair production arising from necessity of measuring space-like intervals
having dimensions less than the electronic Compton wavelength.
ROSENFELD asked how the expression for M depends on h .
SALECKER replied that he was unable to say, since his computations
were started from the beginning with numerical values (e.g. 1024 g for
the proton mass, etc.).
ROSENFELD then took the oor to make some remarks about the diculties of trying to extend the theory of measurement to the gravitational
eld in the manner in which he and Bohr had done for the electromagnetic eld. He rst said that he had tried to see how far one can get,
following Salecker, in determining the gravitational potentials by metrical
measurements rather than by dynamical measurements. (By dynamical
measurements one can get only the derivatives of the potentials, metrical
measurements give the potentials themselves, apart from a gauge factor.)
Roughly speaking, the inaccuracy in the measurement of a gravitational potential - say g00 - will be proportional to the inaccuracy in the
determination of lengths. Therefore, if you can determine lengths with

19. Conceptual Clock Models

179

any accuracy, then you can also determine the potential with any accuracy. However, quantum considerations tell you that if the position of the
measuring rod or clock is known to an accuracy x then its momentum is
uncertain by an amount p > h /x. This gives rise to an uncertainty in
the value of the gravitational eld produced by the measuring instrument.
The factors which saved Bohr and Rosenfeld in the electromagnetic
case were:
1. Because of the existence of both negative and positive charge the
perturbing eld of the measuring instrument could be reduced to a
dipole eld.
2. The charge to mass ratio of the measuring instrument could be controlled.
Therefore, Bohr and Rosenfeld came to the conclusion that the measurement of any component of the electromagnetic eld could be carried out
with arbitrarily high precision in spite of the quantum restrictions.
These saving features are not present in the gravitational case. Whether one takes the measuring instrument heavy or light its perturbing eect
will be roughly the same (proportional to p). Therefore ROSENFELD
agreed with Salecker that there will be a fundamental limitation on the
accuracy with which one can measure the gravitational eld, although he
could give no numerical estimate of this limitation.
WHEELER suggested that perhaps one should simply forget about the
measurement problem and proceed with other aspects of theory. The
history of electrodynamics shows that it is always a ticklish business to
conclude too early that there are certain limitations on a measurement.
He would propose rather to emphasize the organic unity of nature, to develop the theory (i.e., quantum gravidynamics) rst and then to return
later to the measurement problem. He suggested that this was particularly appropriate when we dont even understand too much about the
classical measurement process! We dont know yet exactly what it is that
one should measure on two space-like surfaces, i.e. the specication of the
initial value problem.
He then went on to imagine what sort of ideas scientists might come
up with if they were put under torture to develop a theory that would
explain all the elementary particles and their interactions solely in terms
of gravitation and electromagnetism alone! He rst took a look a magnitudes and dimensions. In the Feynman quantization method one must

19. Conceptual Clock Models

180

sum over histories an amplitude which, in the combined electromagneticgravitational case has roughly the form
[
]
)
(
i
c4
g 2 4
2 4
exp
E d x+
d x .
h c
G
x
Therefore if one is making measurements in a space-time region of volume
L4 , contributions to this sum will be more or less in phase until variations in
the electromagnetic and gravitational eld amplitudes from their classical
values become as large as

h G

h c
1033 cm
3
E 2 , g c
.
L
L
L
These represent the quantum uctuations of the electromagnetic and gravitational elds. In the gravitational case, owing to the nonlinearity of the
eld equations truly new eects come into play at distances as small as
1033 cm where g becomes of the order of unity. WHEELER envisages
a foam-like structure for the vacuum, arising from these uctuations of
the metric. He compared our observation of the vacuum with the view of
an aviator ying over the ocean. At high altitudes the ocean looks smooth,
but begins to show roughness as the aviator descends. In the ease of the
vacuum, WHEELER believes that if we look at it on a suciently small
scale it may even change its topological connectedness, thus:

Figure 19.2

19. Conceptual Clock Models

181

In this way he has been led to the concept of wormhole in space. A

two-dimensional analog of a wormhole would look like this:

Figure 19.3
WHEELER pointed out another way in which a dimension of the order of 1033 cm could be arrived at, by considering only the uctuations in
the electromagnetic eld. A uctuation of amount E would correspond to
an energy uctuation of amount E 2 L3 = h c/L in a region of dimension L.
The gravitational energy produced in this region by this uctuation would
be of the order of G(
hc/L)2 /c4 L = h 2 G/c2 L3 . The two energies become
comparable when L = h G/c3 1033 cm. In WHEELERs dream of the
tortured scientists the wormholes may serve as sources or sinks of electric
lines of force, no charge being actually involved since the lines of force may
pass continuously through the wormholes. A surface integral of the lines
of force
over a region of dimension L would yield a value of the order of
EL2 = h c which would represent a rough average of the apparent charge
associated with each wormhole. No quantization of charge is implied here.
In fact the wormholes themselves have nothing directly to do with elementary particles. WHEELER envisaged an elementary particle as a vast
structure (1013 cm) compared to a wormhole. However, he left open the
possibility that elementary particles might somehow be constructed out
of wormholes. He compared the wormholes to undressed particles, their
continual formation corresponding to pair production which is going on in
the vacuum at all times. The electromagnetic
mass associated with each
wormhole is of the order of E 2 L3 /c2 = ch/G = 105 g, but this huge
mass is almost entirely compensated by an equivalent amount of negative
gravitational energy.
(Editors Note: No one at the conference thought to ask Wheeler why
wormholes corresponding to magnetic poles would not be just as likely to
occur as those corresponding to charges.)

19. Conceptual Clock Models

182

WHEELER outlined ve new concepts which have either already

arisen or may yet arise as a result of pushing the ideas of general relativity to the limit:
(1) Electromagnetism without electromagnetism. By this, WHEELER was
referring to MISNERs work which shows that the electromagnetic
eld can be completely specied by its stress tensor alone. But the
stress tensor is in turn specied by certain derivatives of the metric.
Hence, merely by looking at the metric one can tell all about the
electromagnetic eld. One might call this the unied imprint theory, since the electromagnetic eld leaves its characteristic imprint
on the structure of space-time.
(2) Mass without mass. This concept is illustrated by geons, of which several varieties are currently under consideration:
a) Electromagnetic geons.
b) Geons built out of neutrinos.
c) Geons built out of pure gravitational radiation alone.
(3) Charge without charge. Wormholes.
(4) Spin without spin.
(5) Elementary particles without elementary particles.
As for the last two ideas, WHEELER stated that so far the tortured
scientists have been unable to come up with anything plausible, except
for some vague suggestions that may temporarily stave o the punishment
which awaits them if they fail to produce an answer. He thought that
spin might arise in a Feynman quantization procedure owing to a possible

double-valuedness in the sign of g which occurs in the action.

FEYNMAN objected at this point, saying that as long as g is positive,

g is always of the same sign. A much more serious problem, he felt,

would be to determine what happens if g changes sign.
BELINFANTE pointed out that the square root does not occur in the
Lagrangian if the metric eld is described in terms of vierbeine.
WHEELER emphasized that he was not trying to give answers but merely
pointing out what one might try to do if motivated by the overriding idea
that physics is geometry and that everything can be derived from a
master metric eld.

19. Conceptual Clock Models

183

WEBER wondered how one could get some quantities possessing spin and
others which are spinless out of such a picture.
As for elementary particles without elementary particles, WHEELER suggested that one must rst study the vacuum. The vacuum is in
such turmoil, according to his picture, that it would be foolish to study
elementary particles without rst trying to understand the vacuum. He
drew the following schematic dispersion curve for a wave disturbance (e.g.,
a photon or a graviton) propagating through the vacuum:

Figure 19.4

Over a tremendous range the wave satises the simple relation = ck.
When the wavelength becomes of the order of the size of the universe (
1028 cm), however, the phase velocity is greater than c, as has been shown
long ago by Schrdinger (Papal Acad.). On the other hand, when the
wavelength becomes of the order of 1033 cm the disturbance will be slowed
down by the foam-like structure of the vacuum (and also by having always
to climb over the metric bump which it is itself continuously creating due to

184

19. Conceptual Clock Models

its large energy concentration - Ed.). WHEELER suggested wedding the

ultramacroscopic to the ultramicroscopic by nding the point of inection
on the dispersion curve. This would be the point at which disturbances
would tend to hold themselves together. He also drew an analogy with
superconductivity by suggesting that elementary particles might be the
product of long range collective motions of the vacuum foam, the scale
of elementary particles being much larger than the scale of the wormholes,
just as the scale of superconducting regions is much larger than that of
the elementary structures which give rise to it. In view of possibilities like
these, WHEELER concluded nally that one could not absolutely deny
the possibility of explaining all things in terms of metric.
GOLD suggested that this was an answer without an answer.
WEBER spoke for many by saying that he didnt understand how such a
large number of wormholes ( 1060 ) could give rise to such a well-dened
quantity as the observed charge on an electron, except possibly by some
statistical means.
FEYNMAN suggested that perhaps there was a unique stable ground state
for a wormhole and that this would represent a sharply quantized bare
charge.

WHEELER pointed out that the discrepancy between the charge

h c
associated with an undressed particle and the observed charge h c/137
on an electron could be accounted for by vacuum polarization.
ROSENFELD then took the oor to express some second thoughts which
he had had on the question of measurability. It seemed to him that, in
principle, one could determine the mass of an arbitrarily small body to
an arbitrarily high degree of precision by putting it on an ordinary spring
balance and waiting long enough. Therefore he thought there was perhaps
some doubt after all about Saleckers limitations on the measurability of
the gravitational mass of elementary particles.
FEYNMAN pointed out however that Salecker was trying to measure the
gravitational eld produced by a small mass whereas Rosenfeld was here
considering the response of that mass to a given gravitational eld.
ROSENFELD said, nevertheless, that perhaps his original pessimism in
regard to the measurability of gravitational elds, as compared to the electromagnetic case, might be unjustied. For example, it might happen that
the uncertainties x and p in the measuring instrument aect dierent

19. Conceptual Clock Models

185

components of the gravitational potential, leading to reciprocal relations

of the form
g00 x
g0i p
g00 g0i h
which would not, however, prevent the precise determination of the value
of a given component.
ANDERSON raised the question of the propriety of measuring quantities
} which have neither invariance nor tensor properties.
such as {
BARGMANN replied that the insistence on invariance has often been overdone. An invariant can always be dened provided one simply introduces
the measuring apparatus into the mathematical description.
ANDERSON said that gadgets then evidently introduce preferred situations into the scheme of things and permit one to measure non-invariant
quantities directly.
BERGMANN objected to this point of view.
PIRANI remarked that, in any case, if you are clever enough you should be
able to construct true invariants or tensors (which dont depend on your
gadgets) from the results of reading your gadgets.
This concluded the discussion of the problems of measurement. Attention next turned to the technical aspects of the purely formal problems
which arise when the attempt is made to apply mathematical quantization
procedures to the gravitational eld.

Chapter 20
The Three-Field Problem
F. J. Belinfante

F. J. BELINFANTE opened with an examination of the three-eld problem: gravitational and electromagnetic elds plus the Dirac electron eld.
He considered rst the classical theory, dened by him as the limit of
the quantum theory as h 0 so that gravitational and electromagnetic
elds commute and spinor elds purely anticommute. The notational developments which he then proceeded to outline go something like this:
He denes a modernized Poisson bracket (A, B ) = (1)uA uB +1 (B , A)
with u = 1 for Fermi-Dirac elds and u = 0 for Bose-Einstein elds. He
also distinguishes p f / q = (d f )(dq)1 and A f / q = (dq)1 (d f ), which
is a renement necessitated by factor ordering diculties.

He then uses vierbeine h( ) to describe the gravitational eld, and

introduces local ( ) components independent of x for the spinor elds so
that

(x) = h( ) (x) ( ) .

The canonical formalism can be developed following the method of Dirac1

There are 11 rst-class primary constraints , , ( )( ) arising from
gauge, coordinate, and vierbein-rotational invariance of the theory, which
lead to ve rst-class secondary constraints , . In addition there are
eight second-class constraints

p L/ ,0

p L/ ,0

arising from redundancy in the variables used to describe the spinor eld.

1 Can.

J. Math. 2, 129 (1950).

20. The Three-Field Problem

188

One nds

= P,ss + ie ( ) h0( )

( )

0 = H + [p0 hr( ) Pr A0 ],r +C grs

,rs
( )

s = Gs + [ps hr( ) Pr As ],r Cg0r

,rs
where C = c4 /16 G, and P is canonically conjugate to A , while H with Gs
is integrated over space to give the energy-momentum (free-) fourvector,
if one assumes vanishing elds and at space-time at innity.
In passing to the quantum theory one makes use of the modied Poisson bracket due to Dirac, which is dened by

1
d 3 x d 3 x (A, i )ci (x , x )( , B )
2

1
+ (1)uA uB d 3 x (B , )c i (x , x )(i , A)
2

(A, B )modif (A, B )

with
and

One sets

c i (x , x) (i , )

d 3 x ci (x , x )c k (x , x ) = ki 3 (x x ).
AB (1)uA uB B A = ihc(A, B )modif .

The dynamical equations then become

AF
F,O (F, H0 )modif + Oi
pi
| {z }
where the symbol |{z} means place the Oi factor where pi is taken out.
The Oi factor is given by
i
Gi
Oi ,O
where the i pi p L0 / qi,O .
( p )
pL
L0
G =

.
qi
qi, ,
i

20. The Three-Field Problem

189

Also
H0

(pi qi,O L0 )d 3 x
(i i + + O2 + H)d 3 x

the i and being certain coecients and H a function of the ps, qs,
and their space derivatives only. O2 is quadratic in the . The constraint
expressions and hereare meant as functions of the q and p and therefore
are expressions in the partially linear in them. (Note that the constraints
are identities when expressed in terms of q and L0 / q,O , but through the
vanish only weakly when expressed in the q and p as we do.) If one
works only with modied Poissonbrackets the i may be set to zero, and
then
F,O =

]
[F, H]modif + (F, i )modif i d 3 x.

Kennedy has veried the consistency of this scheme by direct computation

of the modied Poisson brackets (i , ), (i , ), ( i , ). They all

reduce to linear combinations of the s and s alone.
The modied Poisson brackets for the spinor variables lead to some
unwanted peculiarities which can be removed by redening the spinor variables according to

(g)1/4 [ g00 (0) 0 ]

2( g00 + h0(0) )

pi = i +

p L0
qi,O

One then nds

(qi , p )modif = i 3 (x x ) , (qi , q )modif = 0 , (pi , p )modif = 0.

Thence, modied Poisson brackets dened by partial derivatives with respect to q and p equal ordinary (though modernized) Poisson brackets
dened by partial derivatives with respect to q and p. Thus working with

190

20. The Three-Field Problem

L0 in terms of the q and q, from the beginning makes it possible to quantize without ever mentioning modied Poisson brackets.
BELINFANTE pointed out that in its present form his theory seems
to be incovariant. This is related to the fact that the Oi do not vanish
in the strict sense. (They only vanish weakly, in Diracs terminology.)
However he proposed simply to bypass this problem for the time being,
and, for the sake of being able to make practical computations, pass over
to what he calls a muddied theory, i.e., a theory obtained by throwing
in mud. In electrodynamics this is just Fermis procedure of adding a
non-gauge invariant quantity to the Lagrangian. The rst-class constraints
then disappear and one must replace them by auxiliary conditions (e.g. the
Lorentz condition). There is a certain arbitrariness here, since the forms
of the auxiliary conditions depend on the precise form of the mud which
has been thrown in. However, if the qs
are replaced by their expressions in
terms of the ps then the auxiliary conditions must reduce to the original
constraints. BELINFANTE has made certain special choices for these
conditions (e.g., De Donder condition), based on convenience, and he hopes
he can then do meaningful practical calculations, just as the Fermi theory
was long used for practical calculations in electrodynamics before all the
mathematical subtleties of various constraints were precisely understood.
BELINFANTE has shown by explicit computation that the constraints of
his muddied theory are conserved, and has calculated explicitly the
q(p).

With the muddied theory the commutation relations are covariant.

However, the equations of motion are no longer covariant. This means
that the qs and ps at time t2 will depend not only on the qs and ps
at a time t1 , but also on the coordinate system chosen to link t1 and t2 .
On the other hand, dierences in results of going from t1 to t2 are merely
accumulated mud; so, if a true theory is later developed in which the
mud can be made identically zero by altering the commutation relations as
proposed by Bergmann and Goldberg, the covariance will be restored. (In
the muddied theory, covariance will exist weakly unless the positions of
the weakly vanishing factors in products due to noncommutativities would
cause troubles.)
One can introduce annihilation and creation operators (for photons,
gravitons, etc.), although the Fourier transformation procedure on which
they are based is a non-covariant procedure.
BELINFANTE has gone on to see if he can nd the unitary transformations which will separate the gravitational and electromagnetic elds
into their so-called true and untrue parts (e.g., transverse and longi-

20. The Three-Field Problem

191

tudinal parts). This is rather dicult, particularly when spinor elds are
present. But there is also a more serious problem, connected with the constraints (above). True observables have been dened by Bergmann
and Goldberg as those which commute with the rst-class constraints as
well as with the canonical conjugates to the rst-class constraints. BELINFANTE prefers to call them true variables. If the constraints are
only linear in the momenta it is not dicult to nd the true variables.
However, the constraint 0 involves the quantity H which is quadratic in
the momenta, being essentially the energy density of the combined elds.
This means that the only true variables which will be easy to nd are
the constants of the motion. (Since 0 diers from H by a divergence one
might at rst sight conclude, by integrating 0 over all space, that the
total energy must always be zero. However, the surface integrals cannot
be ignored here Ed.)
BELINFANTE concluded by suggesting that a theory in which only
true variables appear may be mathematically nice but somewhat impractical. From the point of view of a scattering calculation, for example,
there may be some truth even in an almost true untrue variable. In any
event the true theory still eludes us at present.
Following BELINFANTEs remarks, there was considerable discussion
as to whether or not all true observables are necessarily constants of the
motion in a generally covariant theory. No progress was made on this
question, however, and the answer is still up in the air as of this moment.
NEWMAN next reported on some work he has been doing to try
to obtain the true observables by an approximation procedure. Instead
of dealing directly with the gravitational eld he considered a particle
Lagrangian of the form
1
L = gi j qi q j + ai qi v (i, j = 1, ..., n)
2
for which the equations of motion are nonlinear but invariant under a
transformation group analogous to the gauge group of electrodynamics or
the coordinate transformation group of general relativity. The gravitational eld is embraced by this example when n becomes transnite. In a
linear theory the true observables are easy to nd. In NEWMANs approximation procedure the search for the next higher order terms (with respect
to some expansion parameter) in the true observables is no worse than
nding the exact expression in the linear case, and can actually be carried
out, even when some of the constraints are quadratic in the momenta. In
the case of the gravitational eld the true observables evidently become

192

20. The Three-Field Problem

more and more nonlocal (i.e., involving higher order multiple integrals) at
each higher level of approximation. NEWMAN could say nothing about
the convergence of his expansion procedure.
BERGMANN remarked that NEWMANs procedure was quite dierent
from the more common dierential-geometric approach which is specically tailored to the gravitational case. He mentioned in this connection
the work of Komar and Ghniau on metric invariants constructed out of
the curvature tensor. It is thought that these invariants have some close
connection with the true observables.
MISNER advocated at this point that one simply forget about the true observables, at least as far as quantization is concerned. He suggested starting with a metricless, eld-less space, dened simply in terms of quadruples
of real numbers, then performing certain formal mathematical operations
and nding the true observables later, if one desires them.
SCHILLER outlined still another way of looking at the problem of nding
the true observables: The eld equations of Einstein are not of the CauchyKowalewski type. That is, the metric eld variables and their time derivatives cannot all be specied on an initial space-like surface. However, Einsteins equations can be replaced by a Cauchy-Kowalewski set provided one
imposes coordinate conditions. A standard canonical formalism can then
be set up in which the coordinates and momenta are expressed as functions
of 2n constants of the motion c and the time t : qi = qi (c,t) pi = pi (c,t) (n is
transnite). In the canonical formalism the coordinate conditions take the
form Z (p, ,t) = 0 for certain functions Z . These three sets of equations
can then (in principle) be solved to eliminate some of the constants c from
the theory. The remaining constants, when, reexpressed as functions of
the ps, qs and t, will be the true observables. This, of course, assumes
that the true observables are constants of the motion.
WHEELER remarked that all of these discussions lead to the conclusion
that the problems we face are problems of the classical theory.
BERGMANN agreed, and expressed his conviction that once the classical
problems are solved, quantization would be a walk.
WHEELER, however, still felt that in the Feynman quantization procedure
the whole problem is already solved in advance.
GOLDBERG and ANDERSON concluded the afternoon session with a
discussion of Schwinger quantization, that is, the procedure which uses

20. The Three-Field Problem

193

a q-number Lagrangian as a starting point. In making variations of such

a Lagrangian one must pay careful attention to the ordering of factors.
GOLDBERG outlined a problem of trying to nd the unitary operators
which generate various invariant transformations, within the subspace dened by the constraints.
ANDERSON pointed out some diculties, connected with the factor ordering problem, of dening a unique q-number Lagrangian.
BELINFANTE suggested that, if an interaction representation could be
found, at least some of the factor ordering problems might be avoided with
the use of Wick brackets which reorder the annihilation and creation
operators, even though these brackets, being dened with respect to a at
space, would have no generally covariant signicance.

Session VII Quantized General Relativity, Continued

Chairman: A. Schild

Chapter 21
Quantum Gravidynamics
Bryce DeWitt

DE WITT opened the session by expressing the hope that one would soon
be able to compute something in quantum gravidynamics. He felt that
formal matters should be settled as soon as possible so that one could
get down to physics. One of the most pressing problems to his mind was
the question of what measure to use in the quantization of a non-linear
theory, particularly in quantization by the Feynman method. In previous
conversations with other conferees he had become aware of some dierences
of opinion on this point. In his view the appropriate measure or metric
was in every case (including that of the gravitational eld) already given
by the Lagrangian of the system. As an illustration he considered a system
described by a Lagrangian function of the form
1
L = gi j qi q j ,
2
where the gi j may be functions of the qs and the indices i, j run from 1
to n. The fact that n may be nondenumerably innite is ignored. For an
actual system the Lagrangian may possess other terms, but the essential
diculties are contained in the term considered. Inclusion of the other
terms modies the following discussion in no essential way.
If (gi j ) is a nonsingular matrix with inverse given by (gi j ) then the
system possesses a Hamiltonian function given by
1
H = gi j pi p j ,
2
and the action
S(q ,t |q ,t ) =
satises the Hamilton-Jacobi equations

q ,t
q ,t

L dt

21. Quantum Gravidynamics

198

S 1 i j S S
+ g (q ) i j = 0,
t 2
q q
S 1
S S
+ gi j (q ) i j = 0.
t
2
q q
According to DE WITT, the structure gi j , which is already contained in
the Lagrangian to be taken as the metric for the space of the qs and
provides the appropriate measure for a Feynman summation. Following
Pauli1 one may introduce a classical kernel of the form

q ,t |q ,t c = (2 ih)n/2 g1/4 (q )D1/2 (q ,t |q ,t )g1/4 (q )e h S(q

i

,t |q ,t )

where g |gi j |, and where the quantity D is a determinant originally introduced by Van Vleck in an attempt to extend the WKB method to systems
in more than one dimension.2
D |D ji | ,

D ji

2S
q j q i

and satises an important conservation law:

S
[Dgi j (q ) j ] = 0
+

q
t

which can be obtained by dierentiating the rst Hamilton-Jacobi equation

with respect to qi and q j and multiplying by the inverse matrix D1 i j .
From the quantum viewpoint this law expresses conservation of probability.
It is easy to see that q ,t |q ,t c is an invariant under point transformations of the qs. With the aid of the Hamilton-Jacobi equations one
may show that it satises the dierential equations

1 Feldquantisierung,

lecture notes, Zrich, 1950-1951.

H. Van Vleck. Proc. Nat. Acad. Sci. 14, 178 (1928); C. Morette. Phys. Rev.
81, 848 (1951).
2 J.

21. Quantum Gravidynamics

199

H )q ,t |q ,t c
t
h 2
= g 1/4 D1/2 g i j (g1/4 D1/2 ).i j q ,t |q ,t c
2

(ih H )q ,t |q ,t c
t
h 2
= g 1/4 D1/2 gi j (g 1/4 D1/2 ).i j q ,t |q ,t c
2

(ih

where for brevity we write g i j gi j (q ), g i j gi j (q ), etc.; where the dot

followed by indices denotes covariant dierentiation with respect to either
the q i or q i (as indicated by the context) and where the operator H is
dened by
H

h 2 i j
h 2 1/2
1/2 i j
g
(g
g
)

g .i j .
2
qi
qj
2

DE WITT pointed out that if it were not for a certain peculiar phenomenon
which occurs when the space of the qs is curved, the operator H could
immediately be regarded as the Hamiltonian operator for the quantized
system. In order to discuss this phenomenon some further development is
necessary:
If one recalls that the classical action denes a canonical transformation by the equations
pi =

S
,
q i

pi =

S
q i

then one easily sees that the Van Vleck determinant is just the Jacobian
involved in transforming from a specication of the classical path by means
of the variables q i , q i , to a specication in terms of initial variables q i , pi .
From the Hamilton-Jacobi equations one sees furthermore that the action
may be expressed in the form
S=

t t i j t t i j
g pi p j =
g pi p j .
2
2

Therefore, noting that as t t the pi become innite except when q i = q i

(for all i), one may write, for an arbitrary function f ,

21. Quantum Gravidynamics

200

lim

t t

f (q )q ,t |q ,t c g1/2 (q )dq 1 ...dq n

n/2

= (2 ih)

1/2

lim D

t t
n/2

= lim (t t )
t t

(q ,t |q ,t ) f (q )

i t t g i j p p
i j
2

e h

d p1 ...d pn

D1/2 (q ,t |q ,t )g1/2 (q ) f (q ).

In order to evaluate this last expression one must evaluate the Van
Vleck determinant. This is easily done by expanding the action about the
point substituting it in the Hamilton-Jacobi equation. One nds, after a
straightforward computation,

S(q ,t |q ,t )
1
1
= { gi j (q i q i )(q j q j )
t t 2
1
+ (gi j,k + g jk,i + gki, j )(q i q i )(q j q j )(q k q k )
12
1
+ [gi j,kl + gik,l j + gil, jk + gkl,i j + gl j,ik + g jk,il
72
g mn ([i j, m] [kl, n] + [ik, m] [l j, n] + [il, m] [ jk, n] )]
(q i q i )(q j q j )(q k q k )(q l q l ) + O(q q )5 }

D(q ,t |q ,t )



= 
=

where


2 S 
q j q i 

g 1/2 g 1/2
1
[1 + Ri j (q i q i )(q j q j ) + O(q q )3 ]

n
(t t )
6

21. Quantum Gravidynamics

201

1
[i j, k] = (gik, j + g jk,i gi j,k )
2
Ri j = gkl Rik jl
1
Rik jl = (gi j,kl gil,k j ) gk j,il + gkl,i j )
2
+ gmn ([i j, m][kl, n] [k j, m][il, n]),
commas followed by indices denoting dierentiation with respect to the
qs. Here a convention has been chosen so that the scalar R = gi j Ri j is
positive for a space of positive curvature.
Using the nal expression for the Van Vleck determinant one infers
lim q ,t |q ,t c = (q , q )

t t

and

h 2 1/4 1/2 i j 1/4 1/2

D ).i j q ,t |q ,t c
g
D
g (g
t 2
h 2
= R (q , q )
12

lim

where (q , q ) is the invariant delta-function in q-space. Referring to the

dierential equations satised by the classical kernel q ,t |q ,t c , one sees
that it equals the true quantum transformation function q ,t |q ,t + of a
system possessing the Hamiltonian operator
H+ H +

h 2
R,
12

up to the rst order in (t t ). That is,

q ,t |q ,t c = q ,t |q ,t + + O(t t )2
where
(ih

H+ ) = q ,t |q ,t + = 0
t

satisfying the boundary condition

21. Quantum Gravidynamics

202

q ,t|q ,t+ = (q , q )

for all t.

The rst order contact between q ,t |q ,t c and q ,t |q ,t + is

already sucient to determine the behavior of wave packets for the quantized system. If the system has a Hamiltonian operator H+ then its wave
packets will more approximately along the classical paths of a classical system which has simply H = 21 gi j pi p j as Hamiltonian function. Conversely
if the quantized system has Hamiltonian operator H then the motions of
its wave packets will be along the classical paths for a classical system
h 2
which has the Hamiltonian function H 12 gi j pi p j 12
R. Evidently there
is an ambiguity here in the choice of Hamiltonian operator when the space
is curved, and there is nothing in the classical theory to resolve it for us.
The Feynman formulation of quantum mechanics follows immediately
from the order contact between q ,t |q ,t c and q ,t |q ,t + . Breaking
the transformation function up into innitely many pieces by means of the
composition law

q ,t |q ,t + =

q ,t |q ,t + d n q q ,t |q ,t +

where d n q g1/2 dq1 . . . dqn , one may write

q ,t |q ,t + = lim

N
t0

d n q(1)

d n q(N) q ,t |q(N) ,t (N) c

q(2) ,t (2) |q(1) ,t (1) c q(1) ,t (1) |q ,t c

where t = max(t t (N) , . . . t (2) t (1) ,t (1) t ),t > t (N) > > t (2) > t (1) >
t . It is evident, from the expression for q ,t |q ,t c and the form of
the expansion for the action S, that the (in the limit) innitely multiple
integral receives signicant contributions from the integrand only when the
dierences q i q(N)i , . . . q(2)i q(1)i , q(1)i q i are of the order of (ht)1/2

21. Quantum Gravidynamics

203

or smaller. Therefore, if the symbol =

is used to denote equivalence as far
as use in the innitely multiple integral is concerned, one may write
q ,t |q ,t c =(2
ih) 2
n

=(2
ih) 2
n

=(2
ih) 2
n

1
1 + 12
Ri j (q i q i )(q j q j )

(t t )n/2
1 i j
1 + 12
R pi p j (t t )2

(t t )n/2

e h S(q

e h S(q

,t |q ,t )

,t |q ,t )

ih
1 + 12
R (t t )2 i S(q ,t |q ,t )
e h
(t t )n/2

since
lim [2 ih(t t )]n/2 (t t )

t t

= lim
t t

t t
2 ih

)n/2

f (q )pi pj e h S d n q
i

(t t )g1/2 (q ) f (q )

i t t g i j p p
i j
2

pi pj e h

d p1 . . . d pn

= ihgi j (q ) f (q ) ,
and therefore
q ,t |q ,t c =[2
ih(t t )]n/2 e h S (q
i

,t |q ,t )

where S is the action function for the system with classical Hamiltonian
function
1
h 2
H gi j pi p j R, satisfying
2
12
S (q ,t |q ,t ) = S(q ,t |q ,t ) +

h 2
R (t t ) + O[(q q )(t t )].
12

The Feynman formulation now becomes

q ,t |q ,t +

= lim

N
t0

[2 ih(t t (N) )]n/2 d n q(N) [2 ih(t (2) t (1) )]n/2 d n q(1)

i
[2 ih(t (1) t )]n/2 exp [S (q ,t |q(N) ,t (N) ) +
h
+ S (q(2) ,t (2) |q(1) ,t (1) ) + S (q(1) ,t (1) |q ,t )]

21. Quantum Gravidynamics

204

and the dependence of the sum-over-paths on the metric gi j is seen to

occur through the invariant volume elements d n q(1) ...d n q(N) . This last expression is sometimes written in the symbolic form
q ,t |q ,t + =

q ,t
q ,t

[q] exp

)
( t
1
L_ dt
h t

where the symbol [q] indicates that a functional integration is to be

performed. In linear examples (i.e., linear equations of motion) the functional integration can often be easily performed without even referring to
its rigorous denition as an innitely multiple integral. This is not likely
to be so simple in the present nonlinear case because of the occurrence of
a variable metric in the invariant volume elements d n q(1) ...d n q(N) .
It will be noted that the last result is even more curious than the previous result obtained with the Pauli method using q ,t |q ,t c . In order to
obtain the transformation function for a quantized system having Hamiltonian operator H+ one must use, in the Feynman summation, the action
corresponding to a classical system having Hamiltonian H or Lagrangian
h 2
L = 21 gi j qi q j + 12
R. When DE WITT rst discovered this phenomenon
he thought that he had made an error in sign and that the occurrences
h 2
of 12
R actually cancelled each other when one passed from the Pauli form
on to the Feynman representation. J. L. Anderson later convinced him,
however, of the reality of the phenomenon, by carrying out a computation
patterned directly after Feynmans original paper.
Instead of stating the result in a symmetric manner one may also
state it in the following forms: If the true classical action L dt (h 0)
is used in the Feynman summation then one generates the transformation
function q ,t |q ,t ++ satisfying
(ih

H++
)q ,t |q ,t ++ = 0
t

where
h

H++
= H + R.
6
On the other hand, in order to generate the transformation function
q ,t |q ,t satisfying
(ih

H )q ,t |q ,t = 0
t

21. Quantum Gravidynamics

205

one must use in the Feynman summation the action for a classical system
possessing the Lagrangian function
1
h
L = gi j qi q j + R
2
6
and Hamiltonian function
1
h
H = gi j pi p j R.
2
6
FEYNMAN remarked that quantization of a system like L = 21 gi j qi q j is
necessarily ambiguous when the space is curved. One must rst imbed
the space in higher dimensional space which is at, since only in spaces
which are at (at least to a very high degree of approximation) do we have
experiments to guide us to the appropriate form of the quantum theory.
The original space must then be conceived of as the limiting form of a thin
shell, the particle which moves in the original space being constrained
to the shell by a very steep potential. But then the Schrdinger equation
for the particle will depend on the precise manner in which the thickness
of the shell tends to zero, and this is arbitrary.
DE WITT agreed, but pointed out that the manner in which the thickness of the shell tends to zero can be described by the addition of a simple
potential-like term to the Lagrangian function appearing in the Feynman formulation. To illustrate this he considered the case of a particle
constrained to move on the surface of an ellipsoid. He rst took for the
constraining shell the region between two confocal ellipsoids:

Figure 21.1
In order to eliminate, right at the start, the degree of freedom transverse to the shell - which must have no reality in the end, anyway - one
may suppose that the wave function has a simple node on each ellipsoid
and none in between (i. e., transverse ground state). Since we know that
confocal ellipsoids form a separable system for Schrdinger equation, we

21. Quantum Gravidynamics

206

may immediately factor out the transverse part of the wave function and
at the same time subtract a constant term proportional to (x)2 from the
energy, where x denotes the thickness of the shell at some point. The remaining part of the wave function then satises the Schrdinger equation
with the simple operator H appropriate to an ellipsoid. This corresponds
to a Feynman formulation using the classical Lagrangian L , for which
the classical paths are attracted to the region of greatest positive curvature,
2
the attraction being described by a potential h6 R. In the present case
the region of greatest positive curvature is the nose of the ellipsoid and
here the shell is thinnest. The three-dimensional wave function undergoes
a crowding at this point. Since the transverse part of the wave function
is constant, this crowding must be borne by the active two-dimensional
part. The tendency of the amplitude to increase in the nose region is describable in classical terms as an attraction. A wave packet would actually
display the eect of this attraction.
The Feynman formulation which starts with the classical Lagrangian
L, on the other hand, corresponds to the use of a shell of uniform thickness:

Figure 21.2
All points of the limiting ellipsoid are here weighted equally. DE
WITT conjectured that the use of any other shells of varying thickness
would quite generally be describable in terms of additions to the classical
Lagrangian of potentials proportional to h 2 .
(Editors Note: - DE WITTs argument is not entirely rigorous. He, of
course, avoided discussion of a spherical shell since the curvature is then
constant and has only the eect of uniformly shifting all energy levels.
However, there is a diculty for ellipsoidal systems in that the separation
constants are not themselves separable, and hence the transverse part of
the wave function is not rigorously factorable for arbitrary behavior of the
non-transverse part. It may, however, be approximately factorable when
the shell is thin.

21. Quantum Gravidynamics

207

Attention should also be called to the fact that the behavior of wave
packets is not described by the Lagrangian appearing in the Feynman
formulation but by the Lagrangian of the Pauli formulation, which is halfway between that of Feynman and the corresponding quantum form. Thus,
when Feynman uses L, Pauli uses L+ to obtain the same quantum theory.
Or when Feynman uses L , Pauli uses L . It is the Van Vleck determinant
of the Pauli formulation which gives the key to the motion of wave packets,
through its conservation law. Remembering that qi = gi j p j , one may write
that conservation law in the form

+ i (Dq i ) = 0.

t
d
If a wave packet is replaced by an ensemble of classical particles then D
gives a measure of the density of these particles at any time and place.)
WHEELER remarked that it had been suggested that an experimental
determination of the Lagrangian to be used in the Feynman formulation
in curved spaces might be achieved through observations on molecules
which have ags on them - for example ethyl alcohol:

Figure 21.3
If some of the bonds are regarded as rigid (e.g., at room temperature)
then the Lagrangian for such a molecule contains a metric corresponding
to a space which is not at. However, WHEELER pointed out that this
system, in the last analysis, has its existence in a at space - that the
rigidity of the bonds is an idealization, and that, in fact, the binding forces
themselves provide the specication of the shell on which this system is
constrained to move. Hence such a system could tell us nothing about

21. Quantum Gravidynamics

208

which Lagrangian to use, for example, in the Feynman quantization of the

gravitational eld where the ambiguity does arise.
ANDERSON pointed out that at any rate DE WITTs approach leaves no
ambiguity in choice of metric. In fact, no other choice is possible if one
requires lim
q ,t |q ,t c = (q , q ). This relation will not generally be

t t

satised if one chooses a metric for q-space which is unrelated to the gi j

appearing in the Lagrangian.
DE WITT then went on to discuss the second important and pressing
problem which arises in the quantization of nonlinear systems, namely, the
factor ordering problem: How should one order non-commuting operators to obtain appropriate quantum analogs of various classical equations?
When the matrix (gi j ) is non-singular the factor ordering Hamiltonian
operator is easily solved. The Hermitian requirement on the momenta pi
leads to the quantum representation requirement
pi = ihg1/4

1/4

1
g = ih[ i + (ln g),i ]
i
q
q
4
2

in arbitrary curvilinear coordinates. Hence, in order that H = h2 gi j .i j ,

one must write
1
H = g1/4 pi g1/2 gi j p j g1/4
2
The Hermitian character of H is obvious from the symmetry of this expression.
For covariant theories, on the other hand, (gi j ) is singular, constraints
are present, and the analysis becomes much more complicated. Moreover,
since (gi j ) is singular it is not obvious immediately what measure to use
in a Feynman formulation.
(Editors Note: - The following is taken from a set of mimeographed
notes which DE WITT distributed to the conferees. The actual exposition of it was broken up by many questions, mostly on the necessity of
going through a careful layout of identities and constraints. These questions are best answered by the more complete uninterrupted presentation
given below. One may add, to the purposes of an unambiguous derivation
already mentioned in the previous paragraphs (nding the correct factor
ordering and the measure), the search for the true observables of the
theory.)

21. Quantum Gravidynamics

209

As a prototype of a covariant theory DE WITT considered a system

possessing a Lagrangian function of the form
1
L gi j qi q j + ai qi v
2
for which the equations of motion are
w

gi j q j + [ jk, i]q j qk fi j q j + v,i = 0

where fi j = a j,i ai, j . These equations of motion are, of course, invariant
under point transformations of the qs among themselves as well as under
phase transformations ai ai +P,i where P is an arbitrary function of the
qs, The covariance of the theory is described by an additional transformation group under which the equations of motion remain invariant, whose
innitesimal elements have the general form qi qi + qi , where
.
qi (i + i j q j ) + i
The i , i j , i are certain denite functions of the qs, while the
are arbitrary innitesimal functions of the time and of the qs and
any of their time derivatives. Such a transformation may be called a
gauge transformation; in general relativity it is an innitesimal coordinate
transformation, the qi being the metric eld variables.
The Lagrangian function must be altered under a gauge transforma It is not hard to see that F must have
tion by a total time derivative F.
the general form
1

F (A + B i qi + C i j qi q j ) + D
2
where the A , B i , C i j , D are functions of the qs only. DE WITT
assumed that C i j 0, as this simplies the analysis in certain respects.
This assumption is actually incorrect for the gravitational eld, but is
not expected to alter the main qualitative features of the analysis. (It
will be discussed more fully at a later point.) By performing a gauge
transformation on the Lagrangian and making a comparison with F one
nds that the following identities must be satised:

21. Quantum Gravidynamics

210

A v,i i + ai i
B i ai j i
D ai i
gi j j 0
gi j j k 0
[i j, k] k 0
[i j, l] l k + [ jk, l] l i + [ki, l] l j 0
fi j j gi j j
fik k j + f jk k i gi j,k k + gik k , j + g jk k ,i
fi j j (v, j j ),i v, j j i
v,i i 0
fi j i j 0
gik (k , j j k , j j ) 0.
DE WITT called these identities of invariance.
It is the identity gi j j 0 which shows that (gi j ) must be a singular
matrix in a gauge invariant theory. This has the consequence that the
momenta of the Hamiltonian formalism are not all independent. It also
has the consequence that the initial conditions on the motion must be
subject to constraints
in order that the motion actually be able to make

the action integral L dt an extremum. Some, at least, of these constraints

are obtained by multiplying the equations of motion by i . Using identities
of invariance one nds
w

i gi j q j + v,i i = 0.
These will constitute the complete set of constraints if their time derivatives to all orders vanish solely as a consequence of the equations of motion.
If the time derivatives do not automatically vanish they must be made to
vanish by adding extra constraints, whose time derivatives, in turn, must
be made to vanish, etc.

21. Quantum Gravidynamics

211

DE WITT introduced a second set of identities by assuming that the

i form a complete and independent set of null eigenvectors of gi j . One
may then infer that

i j i S j

i , j j i , j j i t

i , j j i , j j i S j j i u

where the S j , t , u are certain coecients. DE WITT called these

identities of completeness.
A nal set of identities are obtained from the group property of gauge
transformations. Requiring that the dierence between the results of applying two innitesimal gauge transformations in dierent orders be also
a gauge transformation (of the second innitesimal order) one nds

2 1 qi 1 2 qi (pi + i j q j )1 2 + i 1 2

2 1 F 1 2 F (A + B i )1 2 + D 1 2
where

1
1 2 r (1 2 2 1 )
2
1

+ (S j j S k S j j S k )qk (1 2 2 1 )
2

2 1
)
+ S j (1 2
j

+ 2 1 1 2
only if

i , j j i , j j i r

for certain coecients r , and if

21. Quantum Gravidynamics

212

ti , j j ti , j j ti S j j + i S j j 0

S j,k S k, j S j S k + S k S j 0

i , j j i , j j i S j j + i S j j i v
i , j j i , j j i S j j + i S j j i v

fi j i j + v,i i v 0

where the v are certain coecients satisfying

S i v S i v S i 0
v ,i + v

r S i i S i i + v .
DE WITT called these identities of integrability. It will be observed that
they lead to the identications

t S i i S i i

u r S j j .
DE WITT then went on to remark that the above three groups of identities comprise all the identities that are available, or needed, for the study
of the dynamical behavior of the system under consideration, and he called
attention to the fact that they are obtained entirely within the framework
of the classical theory. In developing the Hamiltonian formulation for the
system DE WITT followed the method of Dirac3 which distinguishes bew
s
tween weak equations (=) and strong equations (=). Equations of motion
and constraints (reduced to their lowest order) are weak equations. Equations of identity () are strong equations. Also, the product of two weakly
vanishing functions is strongly equal to zero.
The canonical momenta for the system are
w

pi =
3 Can.

J. Math. 2, 129 (1950).

L
gi j q j + ai .
qi

21. Quantum Gravidynamics

213

Multiplication by i gives immediately the set of constraints to which the

momenta are subject:
w

= 0
(pi ai )i .
Completing the set of vectors i through the introduction of a set of in
dependent vectors i , and introducing also inverses i , i satisfying

i
i
i
j + j = j , one may write the energy function in the form
E pi qi L
1
1
s
= pi qi gi j qi q j ai qi + v + gi j (pi ai gik qk )(p j a j g jl ql )
2
2
i
H + i q
where
1
H gi j (pi ai )(p j a j ) + v
2

ij
g i j g , g g , g gi j i j .
The nal expression for the energy function provides an illustration of
Diracs theorem to the eect that the energy is always strongly equal to a
function H of the qs and ps, which may be called the Hamiltonian function, plus a linear combination of the s with coecients which depend
on the velocities qi . In a theory without constraints the energy is identical
with the Hamiltonian function. When constraints are present the two are
only weakly equal.
If the coecients multiplying the s are completely undetermined by
the equations of motion, and hence completely arbitrary, the s are said
to be of the rst class. This point always requires special investigation.
Using the equations of motion together with the constraints one may easily
verify that the time rate of change of any function F of the qs and ps is
given by
w
F = (F, H) + (F, )i qi

21. Quantum Gravidynamics

214

where ( ) denotes the Poisson bracket. In particular, the time rate of

change of a is given by
w

( , H) + ( , )i qi .

Since the s vanish their time derivatives must also vanish. This will
happen automatically if the Poisson brackets ( , H) and ( , ) vanish,

at least weakly. If these Poisson brackets do not all vanish then the i qi
cannot all be completely arbitrary and the qi will be subject to additional
constraints.
Using the relation gik ji i j together with identities of invariance,
one nds, after a straightforward computation,

( , H) [( j,k k, j )k gi j (pi ai ) + i i ] = 0
where
w

(pi ai )i + v,i i = 0.
The latter equations are simply the velocity constraints reexpressed in
terms of the momenta. The -equations, existing only in the canonical formalism, are sometimes known as primary constraints. The -equations,
which follow from them, are then called secondary constraints. The s
as well as the s must have vanishing time derivatives. This leads one to
investigate also the Poisson brackets ( , H), ( , ).
Using only the identities of invariance and completeness one nds

( , ) t = 0

( , H) (i,k k + k k ,i k k S i )gi j (p j a j )

+ S i gi j (p j a j ) = 0

( , ) u + S j j = 0.
The vanishing (in the weak sense) of all these Poisson brackets means
that there exist no further constraints and the s are all of the rst
class. There still remains, however, one further Poisson bracket which
it is necessary to examine, namely ( , ). Dirac has shown (op. cit.)
that if this Poisson bracket does not vanish (at least weakly) then some
of the canonical variables may be eliminated from the theory through
the introduction of a new type of bracket, a modication of the ordinary

21. Quantum Gravidynamics

215

Poisson bracket, which DE WITT has called the Dirac bracket. It is the
Dirac bracket which corresponds to the commutator (or anticommutator)
in the quantum form of the theory.
In the evaluation of ( , ) identities of integrability are needed for
the rst time. One nds

( , ) r = 0.
Evidently Dirac brackets are here the same as ordinary Poisson brackets.
Under these circumstances the s are said to be of the rst class. It is
characteristic of any theory in which the constraints arise as a result of a
gauge invariance principle that all of the s and s are of the rst class.
DE WITT called attention to the fact that up to this point not all
of the identities of integrability have been used. The classical theory,
which is completed at this point, can in fact get along without the unused
identities. In the quantum theory, however, the unused identities turn out
to be crucial.
DE WITT rst considered the and equations, which, in the quantum theory, are to be regarded as supplementary conditions on the state
vector of the system:
= 0 , = 0.
Since the classical expressions for the and involve quantities which,
in the quantum theory, do not commute, the factor ordering problem here
makes its appearance. According to DE WITT the quantum analogs of
and should be taken as

1
1
= {pi ai , i } + ihS i i
2
2
1
1
i
= {pi ai , } + v,i i + ihS i i
2
2
where { } denotes the anticommutator bracket. With the aid of the previously unused identities of integrability one may then show that

21. Quantum Gravidynamics

216

[ , ] = ih t

[ , ] = ih u + ih S i i

[ , ] = ih r
where [ ] = ih ( ) denotes the commutator bracket and where the ordering
of factors on the right is now important. Since the ps and s all stand
to the extreme right the corollaries
[ , ] = 0 , [ , ] = 0 , [ , ] = 0
of the supplementary conditions are automatically satised. DE WITT
pointed out that the appropriate quantum analog of a classical quantity

may have one or more terms in it (such as the 12 ihS i i above) which
are proportional to h or powers of h , and which do not appear in the
corresponding classical expression (since h 0).
Similar considerations are involved in nding the quantum analog of
the function H. It is convenient to return for a moment to the classical
theory: Since the s and s are all of the rst class, the quantities i qi
are completely arbitrary.
appearing in the dynamical equation (i.e., for F)
One is at liberty to set them equal to arbitrary functions f of the qs and
ps through the addition of extra supplementary conditions:

i qi = f .
w

The energy function E is then strongly equal to

H H + f .
Furthermore, the dynamical equation may be replaced by
w
w
F = (F, H) = (F, H) + f (F, ).
w

= 0 , = 0 are, of course, unaected.

The conditions
In the quantum theory the dynamical equation may be taken as
ihF = [F, H]
in which the quantum form of H is written as above with the s standing
to the right. The time derivatives of the supplementary conditions, namely

21. Quantum Gravidynamics

217

= 0 ,

= 0

will then be satised provided

[ , H] = 0 , [ , H] = 0.
The quantum form of H will be chosen in such a way that these equations
are automatically satised by virtue of the supplementary conditions. The
supplementary conditions when combined with the dynamical equations
for qi will generally imply
(i qi f )Q = 0
where ( )Q indicates some suitable quantum analog for i qi f , and
in this way the consistency of the quantum scheme can be checked for
arbitrary choices of the f .
It is to be noted that one is here working in the Heisenberg picture
in which the state vector is time independent. The arbitrariness in the
choice of the functions f shows that there are many dierent Heisenberg
pictures, all equally valid. They all lead, however, to a single unique
Schrdinger picture:

(t) = e h Ht = e h Ht .
i

One may arrive directly at the Schrdinger picture by starting, in

the classical theory, from the homogeneous velocity formalism (Dirac,
op. cit.) in which the time is introduced as a canonical variable with a
w
conjugate momentum . This leads to a new equation, namely = 0
where + H. In the quantum theory, representation of ih t puts
the corresponding condition on the state vector in the form
ih = H
which is just the Schrdinger equation.
Construction of the quantum form of H is more dicult than the
construction of the quantum forms of and for two reasons: (1) H is
quadratic rather than linear in the momenta. (2) No metric has yet been
dened in q-space. If there were no gauge transformation group for the
system, gi j would be nonsingular and would, as DE WITT had already
pointed out, provide the natural metric. In the present case DE WITT

21. Quantum Gravidynamics

218

suggested building up a nonsingular metric gi j as a sort of superstructure

with gi j as a core, in the following manner:
[
][ ]
[ ] h
k
j
(gi j ) = i i

k g + k k j
matrix multiplication as indicated, where h , k (= k ) are arbitrary
functions of the qs. However, h is assumed to have an inverse h
which is used to raise the indices , , , etc., while g is used to raise
the indices , , , etc. (gi j ) will then have an inverse given by
[
][ ]
[ i i ] h + k k k j
ij
(g ) =
k
g
j
and a determinant given by
g = 2 hg
where = |i i | , h = |h | , g = |g |. The metric gi j may be used to
make a special choice for the function H, namely
1
H gi j (pi ai )(p j a j ) + v
2
corresponding to the choice

f 2i k (pi ai ) + (h + k k ) .
The equations of motion generated by this Hamiltonian may be derived
from a Lagrangian function of the form
1
L gi j qi q j + ai qi v.
2
The quantum theory generated by this Lagrangian function is not identical with that which has been developed here, but becomes so upon the
addition of the equations (i qi f )Q = 0 as supplementary conditions.
This is a standard procedure in quantum electrodynamics in which these
supplementary conditions become the Lorentz conditions.
A choice of metric having been adopted, one may now express the
quantized momenta explicitly in the form

21. Quantum Gravidynamics

219

1/4
g
qi

pi = ihg1/4

and one nds that the supplementary conditions become

i
1
i i + [ ai i + i i ] = 0
h
2
i
1
i
i
i + [ (ai vi i ) + i i ] = 0
h
2
a dot followed by an index denoting covariant dierentiation with respect
to the metric gi j , with, however, the modication
{ }
i

k i S j
kj
{ }
i

i j i , j +
k i S j .
kj

i j

{i}
kj

i , j +

being the Christoel symbol dened by gi j . A special signicance

possessed by the coecients S j is here emphasized. The S j provide a kind

of ane connection in terms of which one may dene parallel displacements
which are invariant not only under point transformations but also under
transformations associated with the indices , , etc., of the form

i = i X

i = i X

ij = i j X + i X , j
= X 1

1
S i X

1
v X X

S i = X

v = X

+X

1
X ,i

where the X are arbitrary functions of the qs, the matrix (X ) possess1
ing, however, an inverse (X ). These transformations, which DE WITT
called X-transformations, leave all the identities of invariance, complete-

21. Quantum Gravidynamics

220

ness, and integrability unchanged. The X-transformation procedure may

be extended to the indices , , , etc., by the denitions

i = i X + i X

i = X 1 i X 1 X X

=X

1
i

1
i

where the X , X are also arbitrary functions of the qs, with (X ) pos1

sessing as an inverse (X ). The whole theory (in particular, the metric

gi j ) will then be invariant under X-transformations provided

h
= h X X

k = k X X + h X X .
In order to show the invariance of the quantum theory under point,
phase, gauge, and X-transformations one does not actually have to use the
dierential representation of the momenta, convenient though it generally
is. For example, the point transformation law for the momenta is
{
}
1 qj
i
p =
, pj ,
2 q i
and, remembering that the indices i, j, etc., on the quantities gi j , gi j , i ,
i , pi , Si j , etc., are all tensor indices under point transformations, one

nds by straightforward computation that H = H, = , = ,

provided one writes
1
H = g1/4 (pi ai )g1/2 gi j (p j a j )g1/4 + v + h 2 Z
2
where Z is a scalar function of the qs, as yet undetermined. Similarly,

under X-transformations one nds H = H and

= X ,

= X .

Since the s and s stand to the right the supplementary conditions

remainunchanged: = 0 , = 0. Invariance of the quantum theory
under gauge transformations is not immediately apparent at this point,

21. Quantum Gravidynamics

221

but becomes so subsequently. Phase transformations, on the other hand,

are trivial:
ai = ai + P,i ,

pi = pi + P,i ,

or, if one is using the xed dierential representation for the momenta,
one places the burden of keeping the theory phase-invariant on the state
vector (or wave function) by the law

= e h P .
i

Having introduced X-transformations, DE WITT then pointed out

that the anity S i j satises the very important condition of being integrable. This means that one can carry out an X-transformation which will

make the S i j vanish everywhere. The required transformation is given by

the solutions of the simultaneous equations

X ,i = S i X .
The solubility of these equations is guaranteed by the identity of integra

bility S i, j S j,i S i S j + S j S i 0. After this transformation has been
carried out one has

i , j j i , j j 0,
which implies that there exists a point transformation qi q , q such
that

i =

qi
.
q

Making this point transformation, together with an additional X-transformation

X = , X = k , X = ,
and letting the i be given by

i =

qi
,
q

one arrives at a great simplication in the formalism. Using a simple

replacement of indices i , ; j , , etc., to denote quantities in the
new coordinate system, one has

21. Quantum Gravidynamics

222

= , = 0 , = 0 , =

g = h , g = k = 0 , g = g
g = hg .
Moreover, the identities of invariance and integrability take the forms

g = 0 , g = g = 0
g , = 0
f = 0 , f = g

g , + g , + g , = 0

f = v, , f + f = v,

v, + v, = 0

, = v , , = 0

, , = v v + v

, , = v

f ( ) + f + v, v = 0

v , = 0 , v , = 0.
The identity f = 0 implies that a phase transformation (P, = a ) can
be carried out which makes a vanish. Assuming that it has already been
performed and noting that
1
1

i i = + = (ln g), = (ln h),

2
2
1
1

i
i = , + , + (ln g), + (ln g),
2
2
1
1
1

= v + (ln h), + (ln h), + g (g , + g , + g , )

2
2
2
1
1

= v + (ln h), + (ln h),

2
2
one may now write the supplementary condition in the forms

21. Quantum Gravidynamics

223

1
, + (ln h), = 0
4 [
]
i
1
1

, + (a v, ) + v + (ln h), = 0.
h
2
4
The rst supplementary condition has the important result that the
arbitrary quantities h can now be actually eliminated entirely from the
quantum theory, just as they are not needed in the classical theory. That
is, having carefully built up a superstructure around the metric gi j (or
g ), one then proceeds to throw it away. To do this one simply introduces
a new wave function

1 = h1/4
satisfying

1, = 0

[
]
i
1

1, + (a v, ) + v 1 = 0.
h
2
This denition removes from the wave function that part of the metric
density which refers exclusively to the q , and corresponds to the use of
g1/2 dq as invariant volume element. The condition 1, = 0 means that

the new wave function does not depend on the variables q . The q are
non-observable or non-physical variables of the system. This was, in fact,
already true in the classical theory, since the q can be made to undergo
arbitrary changes by means of gauge transformations.
In the new representation g by itself provides a natural metric in
the reduced space of the q . To see how this works for the Hamiltonian
operator, one can multiply the Schrdinger equation
ih = H
by h1/4 . Using the explicit form for H, one nds, after a straightforward
calculation,
1
1
1
ih 1 = h 2 g 1 + ih(a 1 + a 1 ) + ( a a + + h 2 Z)1
2
2
2

21. Quantum Gravidynamics

224

where the dots now denote covariant dierentiation with respect to the
metric g , and where
1

Z = Z + h 1/4 g1/2 (g1/2 g h1/4 ) + h1/4 (h h1/4 ).

2
q
q
2
q
q
The quantity Z is invariant under point transformations which are restricted to the variables q alone.
The new Schrdinger equation may be written in the form
ih 1 = H 1
where
1
H = g1/4 (p a )g1/2 g (p a )g1/4 + v + h 2 Z,
2
the explicit dierential form for the momenta p in the new representation being
p = ihg1/4

1/4
g .
q

The existence of the second supplementary condition suggests that

there are further non-physical variables in addition to the q . To show this

one must rst eliminate the term in (a v, ) from the supplementary

condition. This can be done by carrying out a phase transformation P
satisfying

P, = v, a
and at the same time satisfying
P, = 0

so as not to disturb the condition a = 0. The integrability of these equations is insured by the identities of invariance and integrability, as one
may verify by straightforward computation, showing that the expressions

for (P, P, ) and (P, P, ) vanish. One may then write

1

1, + v 1 = 0.
2

21. Quantum Gravidynamics

225

Next, by dierentiating the identity , , = v one can show

that

v v + v v + v v
= 0.

The coecients v are then immediately recognizable as the structure

constants of a Lie group. The secondary constraints are, in fact, the
innitesimal generators of the group. In the general case the structure
constants will be non-vanishing and the Lie group will be non-Abelian.
This immediately leads one to various possibilities depending on the classication of the Lie group in question. As an illustration, suppose that
the Lie group is the irreducible rotation group in m dimensions. The
indices must then range over 12 m(m 1) dierent values. This is not
however the number of further non-physical variables. A point transformation q xa , qA [a = 1...m, A = 1...n 21 m(m + 1)] can be made such that
the new variables xa describe the subspace in which the rotations actu
ally take place. The quantity v vanishes for the rotation group and the
second supplementary condition is found to reduce to a 1,a = 0 which,
when analyzed, is seen to state simply that the wave function is invariant
under rotations in the subspace of the xa . Therefore, in this case only m 1
further variables can be eliminated from the theory as non-physical. The
wave function can still depend on the rotation-invariant combination xa xa
as well as on the qA .
The primary constraints also, of course, generate a Lie group.
But this group is necessarily Abelian. DE WITT assumed, for simplicity,

that the group generated by the is also Abelian (so that v = 0) and
furthermore that the vectors i are linearly independent of the i and

of each other. In this case, when S i = 0, one may carry out a point

i
A
transformation q q , q , q such that

i =

qi
qi
, i = .

q
q

(Here, in order to avoid confusion one must use a prime to distinguish

indices related to the i from those related to the i .) In the new coordinate system one has

21. Quantum Gravidynamics

226

= , = 0 ,

= 0 ,

A = 0 ,

= , A = 0 ,

and the identities of invariance and integrability become

g , = 0 ,

g A, = 0 ,

gAB, = 0

g , = 0 , g A , = 0 , gAB, = 0
f = 0 ,

f = 0

g f = v,
g A = fA
fA = v, A
v, = 0.
If the phase is adjusted so that
a = 0 and a = v,
then one also has

g A = aA,
aA, = a ,A fA = v, A v, A = 0.
Moreover, the supplementary conditions become

1, = 0 , 1, = 0

so that the q , like the q , are now non-physical variables.

The quantity v is seen to be independent of the q and to have a dependence on the q which is no more than quadratic (since v, = g , = 0).
One may choose the origin of the unobservable coordinates q in such a
way that
1
v = w q q + u
2

21. Quantum Gravidynamics

227

where w and u are independent of the q and q . Furthermore, since

aA, = 0 and aA, = g A, = 0, one may write
aA = bA q + bA

where bA and bA are independent of the q and q . In order to complete

the elimination of the q from the theory, a natural metric gAB for the
reduced space of the qA must be introduced. DE WITT showed that it
must be dened in the following manner:
[
g
gA
[
g

gA

]
] [
w
bB
g
=
gAB
bA gAB bA w bB
]
[

w + w bC gCD bD w
g B
=
gAC bC w
gAB

w bC gCB
gAB

in which use has been made of the relations g = v, = w , gA =

aA, = bA , and in which the matrix (w ) is assumed to have an inverse
(w ).
With this denition one has gAC gCB = BA and
g = wg
where w = |w | and g = |g |. The determinant w forms that part of
the metric density g which refers exclusively to the non-physical variables

q . It may be removed from the normalization of the wave function 1

by introducing a new wave function

2 = w1/4 1 .

Since w, = 0, w, = 0, 2 is independent of the q and q . It may

becalled the true physical wave function of the system referring only to
the physical variables qA . Its Schrdinger equation may be found by
multiplying the Schrdinger equation ih 1 = H 1 by w1/4 making use of
the relations
a = v, = w q

gA a = gAC (aC bC )

g a a = v + u + (aA bA )gAB (aB bB ).

21. Quantum Gravidynamics

228

One nds, after a straightforward computation,

ih 2 = H 2
where
1
H = g1/4 (pA bA )g1/2 gAB (pB bB )g1/4 + u + h 2 Z ,
2
1

Z = Z + w1/4 g1/2 A (g1/2 gAB B w1/4 ) ,

2
q
q

pA = ihg1/4 A g1/4 .

The operator H remains invariant under point transformations which are

restricted to the variables qA alone.
The operators qA , pA and any quantities constructed out of them
are the true observables of the theory. The theory is now reduced to its
simplest terms. In this form it is quite easy to see that the Schrdinger
equation is consistent with the constraints

2, = 0 ,

2, = 0.

One has only to require that Z be independent of the q and q . H

will then be a true observable (the true energy operator, depending only
on the qA and pA ) and the wave function 2 will remain independent of

the q and q at all times.

The gauge invariance of the quantum theory is now also immediately

evident. In the coordinate system q , q , qA , gauge transformations are

given simply by

, q = , qA = 0
q =
and hence

H = 0 , 2 = 0.
There remains only the term h 2 Z which is unknown. However this
term, as Feynman pointed out previously, depends on how the system is
constrained to a shell which passes over to the space of the qA in the
limit, and this is arbitrary.

21. Quantum Gravidynamics

229

It is of interest to examine the Feynman quantization method in the

present context. The path summation may be dened by breaking the
time interval into small pieces and integrating over the physical variables
qA only, using the measure dened by the metric gAB . The variables q
may be chosen as arbitrary functions of the time (corresponding to the
original arbitrariness in the i qi ) and the original Lagrangian function L

may be used to compute the action, provided the variables q are made
to vary in time in such a way as to satisfy the velocity constraints which
in present notation, become

g q + g A qA + v, = 0
or

q = q w bA qA .
For the Lagrangian function then becomes

1
1
L = gAB qA qB + g A q qA + g q q + aA qA + a q v
2
2
1
= gAB qA qB + bA qA u
2

which is precisely the form which gives rise to the Hamiltonian H. One
may therefore express the transformation function in the symbolic form
q A ,t |q A ,t =

q A ,t
q A ,t

[qA ] exp(

i
h

t
t

L dt)

with the understanding that the functional integration is to be carried out

over all paths between the points q A ,t and q A ,t which satisfy the velocity constraints with q given, each path being assigned a weight according
to the measuredened by gAB . (Here, potential-like terms proportional to
h 2 , which may be added to L, are ignored.)
It is not actually necessary to restrict the summation to the variables
qA only. For example, one may work with the transformation function

,q

,t |q

, q ,t =

q ,q A ,t

q ,q A ,t

i
[q , q ] exp(
h

t
t

L dt)

21. Quantum Gravidynamics

230

which connects the values of the wave function 1 at two dierent times.

Since 1 is independent of the q for physical states, one may write

2 (q A ,t ) =
1/4

=w

= w 1/4

1 (q

q A ,t |q A ,t g 1/2 dq A 2 (q A ,t )
A

,t )

q A , q ,t |q A , a ,t g 1/2 dq A w 1/2 dq 1 (q A ,t )

which implies
q A ,t |q A ,t =

dq w 1/4 q A , q ,t |q A , q ,t w 1/4 .

The expression on the right must be independent of the q . Hence

one may integrate over these variables provided one divides the result
by an innite normalization factor. Moreover, in the path summation

expression for q A , q ,t |q A , q ,t the choice of time-like behavior for

the variables q is immaterial. Hence one may write
q A ,t |q A ,t = N 1

q A ,t
q A ,t

[qi ] exp(

i
h

t
t

L dt)

where N is a suitable innite normalization factor, the functional integration being now carried out over all paths for which qA = q A , q A at
t = t , t respectively (all possible end-point values for the variables q ,

q being included in the summation) each path being given a weight according to the measure dened by metric density g1/2 multiplied by the

Jacobian of the transformation qi q , q , qA . In calculation of actual

physical quantities one never needs to mention the normalization factor N
explicitly, since what is involved is always a ratio of the form
q A ,t |F(t)|q A ,t
q A ,t |q A ,t
where F is some true observable (function of the qA , pA alone), and the
normalization factors in front of the functional integrals cancel.
DE WITT concluded his remarks with some comments about his simplifying assumption, made at the beginning, that the coecients C i j , occurring in the expression for F, vanish. In the case of the Lagrangian
for the gravitational eld the C i j do not vanish. This has the result that

21. Quantum Gravidynamics

231

the secondary constraints are no longer linear in the momenta, but

quadratic. In fact, it had been pointed out earlier in the conference that
the quadratic term, in the gravitational case, is just the energy density for
the gravitational eld. This created a certain amount of perplexity, leading some of the participants to think that perhaps all true observables are
simply constants of the motion. However, this problem is not yet settled.
DE WITT pointed out that the discovery of the non-physical variables associated with the will be considerably more dicult in the case C i j = 0.
The resulting quadratic dependence of the on the momenta means that
the non-physical variables can no longer be found simply by a point transformation. A more complicated type of canonical transformation will be
required.
DE WITT suggested that the use of parametrized space-like surfaces
might help this situation. But he also pointed out that the equations like
i = qi / q become variational dierential equations in eld equations
in eld theories, and that there is some uncertainty about the ease with
which they can be solved in nonlinear contexts. The search for answers to
these questions will provide a large program for the future.
FEYNMAN raised the question of the interest of the true observables.
Commenting on Belinfantes remark that true observables are very useful
to get meaningful answers, FEYNMAN proposed to formulate the problems not in terms of incident gravitons, photons, etc., but in terms of
fermions or other sources of these incident and outgoing elds only, in an
action-at-a-distance fashion.
LICHNEROWICZ remarked that the classical treatment may have the
following geometrical interpretation:
(1) It may be simpler to consider directly the conguration spacetime of the system, i.e., to treat t as a local coordinate. Let Vn+1 be the
conguration space-time.
(2) The geometrical frame is then the ber bundle space of 2n + 1
dimensions - say U2n+1 - consisting of the tangents of Vn+1 at all points.
(3) The gauge transformations have, for innitesimal generators, vector elds Z of U2n+1 . The precise hypothesis for the Lie algebra of the Z
elds is the following: Z can be decomposed into a direct sum X +Y where
Y is a tangent to the bers. This decomposition is such that Y depends
on the points of Vn+1 only, and X depends linearly on the directions. Y
denes a eld of plane surfaces on which the pseudo-metric gi j vanishes.

21. Quantum Gravidynamics

232

(4) By introducing an extra dimension, one can have a Lagrangian

corresponding to a Riemannian pseudo-metric (degenerate quadratic form),
and it seems that the explicit calculations in the classical part could then
be simplied.
ANDERSON and LAURENT reported on the use and extension to the
gravitational case of Feynmans method in its eld theoretic form.
In order to investigate the question of what variables to employ in
the Feynman integral in the gravitational theory, ANDERSON has looked
at the corresponding problem in electrodynamics. There the situation is
somewhat simpler because of the linearity of the theory and consequently
it is possible to see in a fairly straightforward fashion what one is to do in
this case. Adopting the Feynman prescription in toto means that one must
look for solutions to the eld equations which are then in turn substituted
into the Lagrangian for the electromagnetic eld and integrate over the
time interval.
The chief diculty in this procedure rests in the fact, of course, that
one of the equations of motion for the eld is free of second time derivatives
and therefore represents a restriction on initial and nal values which one
may impose on the eld conditions. However, if one transforms to a set
of eld variables which include Atr , Alo , and the scalar eld , then the
equations of constraint do not depend on Atr . In fact, the Lagrangian
originally looks like
1
1

L = ( A + )2 ( A )2
2
2
and the equations of constraint appear as

( A + ) = 0
so in terms of Atr , Alo , equations of constraint look like

A lo + = 0.

21. Quantum Gravidynamics

233

Thus solutions satisfying all eld equations yield for L

1

2 1
A tr ( A )2 .
2
2
Thus the construction of the Feynman integral is then seen to be in
terms of Atr alone, and the measure is given uniquely by the Lagrangian in this case simply 1.
If one were to extend this line of reasoning to the gravitational case,
it is seen that one must look for a set of variables which have the property
that they do not appear in the constraint equations and thus are capable
of being given arbitrary initial and nal values. The Feynman integral
would then be over these variables and the measure would derive from the
Lagrangian expressed in terms of these variables. As yet this is only a
program of approach. However, one can see that it is possible to eliminate immediately as unphysical variables the g4 which corresponds in the
electromagnetic case to the scalar potentials. The elimination of the longitudinal part of the gravitational eld is of course a much more dicult
problem and has in no way been solved as yet, although it appears to be
the central problem to the quantization.
L=

LAURENT showed how the Feynman action method works gauge-invariantly

in quantum electrodynamics and then suggested an extension to gravitation theory whereby the use of extended gauge-invariance is forced upon
the formalism. (The following is the reproduction of his notes. For a more
complete exposition of these and related questions, the reader may consult
a recent article of B. E. Laurent.4
In the following I wish to explain why I think that it is worthwhile to
try Feynmans action method in covariant theories. This is merely a sketch
and does not pretend to be either complete or mathematically rigorous.
It has been of great help to me to be able to discuss these problems with
Professor O. Klein and Dr. S. Deser.
I shall start with the simplest generally covariant theory known, that is
electrodynamics, which is covariant with respect to gauge transformations.
Feynmans well-known action principle5 has been used by several authors6 to calculate ratios of time-ordered vacuum matrix elements in noncovariant eld theories. These authors have shown that such ratios may
be written as follows:
4 Nuovo

Cimento IV, 1445 (1956).

Feynman: Rev. Mod. Phys. 20, 367 (1948).
6 P. T. Matthews, A. Salam: Nuov. Cim 2, 120 (1955); W. K. Burton, A. H.
DeBorde: Nuov. Cim 4, 254 (1956); P. W. Higgs: Nuov. Cim 6, 1263 (1956).
5 R.P.

21. Quantum Gravidynamics

234

(0, +|TA0 (1) . . . Z0 (n)|0, )

(0, +|0, )

(A(1) . . . Z(n)eiI A . . . Z

=
eiI A . . . Z

A(1) . . . Z(n)0 =

(21.1)

Here I is the action-functional for the elds (or eld) and all A(1)...Z stand
for the eld-functions in the n space-time points x1 ...xn . The integrals are
functional integrals over the functions in all space-time, performed after a
certain small imaginary term has been added to the action (as Burton and
DeBorde show, this picks out a vacuum state). An extended class of operators may be constructed as sums of time-ordered products TA0 (1)...Z0 (n).
To be general I shall in the following write F(A, ..., Z) instead of A(1)...Z(n)
in (21.1). F( ) is here a functional of the elds A, ..., Z.
In the usual Lorentz-gauge electrodynamics a vacuum propagator is
to be written as follows:

F(A, , )0 =

F(A , , )eiIL A ,

eiI A

(21.2)

A special device must here be used to obtain the wanted anticommuting properties of , 7

d 4 x(g

IL =

A A + i { eA } )

(21.3)

where g
is the Lorentz-metric and the Dirac-matrices.
(21.2) is exactly the ordinary vacuum propagator of electrodynamics
if we restrict F( ) so that we have nothing but electrons and transversal
photons in the initial and nal states.8 F( ) may for instance have the
following form:

F(A ) = N(|k|)

t=t2

A 2 eik2 x d 3 x

t=t1

Here N(|k|) is a normalizing factor

k4 = |k|, 4 = 0 and mk = 0 (m = 1...3).
7 Feynman,
8 Switching

A 1 e+ik1 x d 3 x

op. cit., and Burton and DeBorde, op. cit..

o of the electron charge e is here assumed.

(21.4)

21. Quantum Gravidynamics

235

The operator corresponding to (21.4) gives merely transversal photons

in the initial and end states of the propagator.
Observe that in a gauge-transformation

.
(21.5)
x
F( ) in (21.4) is invariant. We shall assume throughout that our Fs
have this property.
Let us now re-form (21.2)
A = A +

F0 =

F(A , , )eiIL A

eiIL A

F(A , , )eiIL A
=

(21.6)

eiIL A

We have split up the integrals over all possible A -functions in sums

of integrals over part-regions, of which the rst is around the A -functions
satisfying the Lorentz-condition. The other regions are obtained from the
Lorentz-region by gauge-transformations. The regions cover completely
the region of all A -functions and they do not overlap. The part-regions
can be made as narrow as we like.
Remembering that F( ) is invariant and that the gauge-transformation is linear we see that the following is true:

F(A , , )eiI A
( )

= det
F(A , , )eiI A
L L

(21.7)

I is the complete gauge-invariant action

I = IL +

0 0

d 4 xg

A A

(21.8)

( )
det L is the Jacobian of the transformation
from the -th region to the
( )
Lorentz-region. In this case det L is 1 but we shall keep it in in spite of
that to be able to use the result later in gravitation theory.

21. Quantum Gravidynamics

236

We may now write

( )
iI A ei(IL I) det
F(A
,

)e

L
L

( )
F0 =

i(IL I) det
iI
L e A e
L

(21.9)

0 0

IL I = d 4 xg g A A depends on the region only (when the

region is narrow). In the Lorentz-region, IL I = 0. Hence

F0 =

iI
L F(A , , )e A
iI
L e A

(21.10)

Further we see that

( )
iI A det
F(A
,

)e

L
L
( )
F0 =

( L eiI A ) det
L

F(A

F0 =

, , )eiI A

(21.11)

iI
e A

F(A , , )eiI A

eiI A

(21.12)

From this form it is evident that the propagator is invariant.

We see from this that under the supposition made about F( ), that
it must not change its form when A is replaced by A + x , the two
demands are fullled that quantization shall, as in (21.10), be performed
over the real degrees of freedom only and that the procedure shall be
covariant as in (21.12).
According to the foregoing it is close at hand to guess that a propagator in a theory that is covariant with respect to linear transformation shall
be written in the form (21.12), where F( ) and I must be invariant in the

21. Quantum Gravidynamics

237

sense mentioned above9,10 (in the general case we call it extended gaugeinvariance). If in gravitation theory, for instance, F( ) is a functional
of the g only it must not change its form in the linear transformation

g (x) = xx xx g (x (x)),where x = x (x) is an arbitrary (non-singular)

transformation. We observe that a scalar does not necessarily fulll the
requirement, because, in general, it changes its functional form of x in the
transformation. If F( ) is a function as here described and the integration
variables transform linearly, we see from the procedure followed in the electromagnetic case that we may always come from the covariant propagator
of type (21.12) to the propagator of type (21.10) , where only so much of
the eld is quantized as there are degrees of freedom.11
A diculty in the gravitation case is that we do not know over which
variables we ought to integrate, and which measure should be chosen in
the functional integral.12
The choice of integration variables that seem most natural at rst

sight is perhaps that of the g or gg (g = det g ), but as Professor

J. A. Wheeler told me in a letter, a better choice is probably one which Dr.
C. Misner has suggested, that is the coecients in the transformation from
the consideredmetric in a certain point to the Lorentz metric. However,
the question cannot be regarded as solved yet.
(MISNER pointed out that this choice of variables gives an invariant
measure which preserves the signature.)
Finally I should like to say a few words about how functionals of the
type mentioned earlier may be constructed in gravidynamics.13 Integrals
of scalar densities over all space-time is of this type but they do not seem
to be of much use in this connection. Professor O. Klein has pointed
out to me that the total energy-momentum pseudo-vector may be useful
9 This does not mean that we have no use for other types of F in intermediate calculations. In electrodynamics with truncated Lorentz-gauge action we may, for instance,
calculate (21.2) with F( ) = A(1)A(2) and that is very useful.
10 I is not necessarily the classical action-functional for the system (also when such a
functional exists), but it should go over into the classical function when h 0.
11 In fact, a little more than the degrees of freedom are quantized in (21.10) because
the gauge is not uniquely determined by the Lorentz-condition. This does not matter
for the practical use of (21.2) and in principle we could of course have restricted the
gauge more closely.
12 If the chosen volume element in the functional integral is a certain factor times the
product of dierentials (the A in (21.12)), the factor ought to be invariant and we
may include it in the action (see footnote 10).
13 In electrodynamics this question is easily answered: Every gauge-invariant functional
is of the wanted type. If the gravitational eld constant is switched on and o, the
matter can to some extent be made equally simple in gravidynamics.

21. Quantum Gravidynamics

238

here, and in fact we see that it is of the correct type if we choose it in the
following form:

g(G 4 + t 4 )d 3 x.

(21.13)

(I do not think that this formula needs any explanation.)

{ g(G + t )}, = 0 is identically satised which gives (21.13) the

desired property for g -transformations which are identity-transformations on the boundaries of the integration-volume. Now all the propagators
in the foregoing are limits when the space-time region involved becomes
innitely large14 and nothing hinders us from prescribing, for instance, the
Lorentz-metric outside the region and on its boundaries. This means that
every interesting transformation is of the type mentioned. In this case the
propagator is dependent on the boundary conditions chosen in innity.
During the discussion which sandwiched and followed LAURENTs
exposition, FEYNMAN asked which
action was used in the gravitational

case. To LAURENTs answer gRd 4 x - FEYNMAN raised the diculties brought in by the presence in R, of second derivatives.
BELINFANTE suggested that Laurent uses, in Feynmans sum over elds,
the action from the muddied theory, together with the determinant derived from the muddied theory. However, it was pointed out that Laurents proof of the equivalence of a true and a muddied treatment in
the electromagnetic case may not hold in the gravitational case, because
the determinant depends on the variables of integration.
FEYNMAN asked also what is the determinant in the path integral ex
pressed in terms of the variables proposed by Misner (the variables c
0

dened by the equation g = c g c )? More precisely, what is the expression of the determinant which gives a value to the path integral independent of the mesh introduced for its denition? This question remained
unanswered but led to the discussion of the denition of Feynmans path
integral in a generally covariant theory. Originally, Feynmans path integral was dened in terms of a time slicing as each time interval goes to
zero. In the gravitational case, it is preferable not to single out the time,
and to redene Feynmans method accordingly.
WHEELER pointed out the various advantages of the latent variables
(such as A in electrodynamics) and of the true (or physical) variables
(such as Atr ). At present we have three schemes:
14 Burton

and DeBorde, op. cit.; Higgs, op. cit..

21. Quantum Gravidynamics

239

Sum over elds expressed in terms of latent variables. The

innities cancel out.
Add a term to the Lagrangian so as to make the problem
non-singular.
Use only physical variables.

The whole art is working back and forth between latent and true variables.
It is important to show the equivalence of the various schemes and to
show the equivalence of the various slicings, but it is a task yet to be
accomplished.
BERGMANN summarized the session as follows: It appears that there
are two dierent methods of Feynman quantization - one principally represented by Bryce DeWitt and Jim Anderson, the other by Bertel Laurent
and Stan Deser, who is unfortunately not with us this morning. The dierence appears to be that one group wants to settle the question of covariant
measure in the function space rst and then integrate, whereas the other
group prefers to integrate rst and ask questions afterwards. To me as
an innocent bystander it appears that one group knows what they are doing but dont know how to do it, and the other group is able to proceed
immediately, but with some question as to the meaning of their results.
One word about why the two approaches appear to lead to mutually
consistent results in electrodynamics: in electrodynamics the components
of the metric tensor, that is the second derivative of the Lagrangian with
respect to the velocities, are constants. It appears natural that in such a
situation there should be no disagreement; but I feel very doubtful, without availability of further strong arguments, about anticipating a similar
agreement in general relativity, where the same coecients are complicated
functions of the eld variables.
I think it is obvious to all of us that the dierence between the two
approaches again amounts to an estimate of the relative importance of
true observables vs. all dynamical variables. Therefore, permit me to give
one more argument that indicates the probable importance of the true
observables, or dynamical variables as John Wheeler has called them. In
general relativity, it is sometimes useful, for reasons of convenience, to
introduce parameters and thus to upgrade the ordinary coordinates into
dynamical variables. Alternatively, it is sometimes desirable to introduce
the so-called vierbeine (quadruples). If you do that without eventually
going back to true observables, then you have a proliferation of variables

240

21. Quantum Gravidynamics

that must eventually be represented by quantum operators, and I think if

you try to do this you will go crazy.
I would like to close with a comment on the motivation for Feynman
quantization as such. The proponents of Feynman quantization claim as
its virtue that once a completely satisfactory classical theory is available,
all they have to do is to follow an algorithm, and the task of quantization
will be completed. I must admit to having some lingering doubts about the
uniqueness of the procedure, in view of the fact that the integrals converge
only conditionally, not absolutely; but presumably these doubts can be
resolved. A more physical question is this: If you think for a moment
about the classical Lagrangian for a free particle, we all know that there
is a multiplicity of corresponding quantum theories, not all of which could
have been produced by the unique device of Feynman quantization. There
remains thus the possibility that we may have eventually to guess at a
complete quantum theory rather than carry out the program in two steps,
rst classical theory then quantization. I do not think that we shall be able
to resolve this question in the immediate future; hence I feel that we are
completely justied in following both approaches - to work out methods of
quantizing a given c-number theory, and examining the intrinsic structure
of a full-blown covariant q-number theory.

Session VIII Quantized General Relativity, Concluded

Chairman: V. Bargmann

Chapter 22
The Possibility of Gravitational Quantization

KURSUNOGLU began by asking that those who had presented techniques

for quantizing the gravitational eld in the morning session make it clear
whether or not it has been decided that quantization is possible according
to these techniques. He suggested that before one builds a house, he should
buy the ground rst. WHEELER asked, Dont we believe that the that
the theory should be quantized? BARGMANN asked if Kursunoglus
question had to do with matters of convergence. He pointed out that it is
not easy to decide whether or not one can quantize, because many things
may be meant by this question. WHEELER said in a quantum theory of
gravitation we would expect new, fascinating physics to come out, because
the physical constants that appear in the quantum theory are the gravitation constant G, the velocity of light c, and the Plancks constant h , but
whether the results of such a theory would agree with experiment is another question. In the classical theory we have no natural mass or length.
BONDI suggested that, when one does construct natural units of mass and
length from these constants they may turn out to be much smaller than
anything present-day physics is interested in. WHEELER replied that in
a session of the day before, he had suggested a way in which one could get
quantities which have a range of size inside the interesting realm. GOLD
said the theory of electrodynamics required quantization because there was
a broad range of phenomena that had to be accounted for; for example, the
infrared catastrophe had to be somehow explained. With gravitation, one
could say that either it must be quantized because otherwise one might get
into a contradiction with the logical structure of quantum theory, or else
that there exists a broad range of phenomena which the classical theory
(without quantization) is unable to explain. But this latter range of phenomena seems to be missing, unless there is gravitational radiation and
some kind of infrared divergence occurs there. WHEELER said that
it might turn out that the elementary particles depend on gravitational
uctuations for their stability. GOLD said that this was only a hope, and
WHEELER agreed that a proof was missing. ANDERSON asked if there

244

22. The Possibility of Gravitational Quantization

was anything wrong with the argument that if it were possible to measure
all gravitational elds accurately, that one would have enough information
to violate the uncertainty principle. WHEELER answered that this was
Golds rst point. GOLD said that he wanted still to be convinced that
one gets into a contradiction by not quantizing the gravitational eld. DE
WITT said that that there is a conceptual problem here in that if the
matter elds are quantized, one must decide what to use as the source of
the gravitational eld. If our experiments lead us to believe that the gravitational eld is produced by what one may call the expectation value of
the stress-energy tensor, there is still a diculty because the expectation
value depends on the measurements we make on the system. After the
system has been prepared in a denite state, a measurement can change
the expectation value and so the gravitational eld suddenly changes because of a measurement performed on the system. Classical gravitation
theory works only because the experimental uctuations are so small on
the scale at which gravitational eects become noticeable.
FEYNMAN then proposed the following experiment.

-%..*/)0"&%'($

!"##$%&'"()*+%'',$)

Figure 22.1
If one works in space-time volume of the order of L3 in space and
in time, then the potentials are uncertain by an amount

g =

hG
c3 L2

L
c

22. The Possibility of Gravitational Quantization

245

and g = MG
is the potential produced by a mass M in the region L3 . An
Lc2
uncertainty in the potential is then equivalent to an uncertainty in the
measurement of M, and one gets M 105 grams. This means that if
the time of observation is restricted to be less than Lc , then the mass cant
be determined to better accuracy than this. Of course, if one is allowed
an innite time, M can be determined as accurately as one wants. If,
however, this is not possible because the particle must be allowed to pass
through the equipment, then unless the mass is at least of the order of 105 ,
the apparatus will be unable to discover the diculty (no contradiction).
One can conclude that either gravity must be quantized because a logical
diculty would arise if one did the experiment with a mass of order 104
grams, or else that quantum mechanics fails with masses as big as 105
grams.
UTIYAMA then presented a technique for quantizing the gravitational
eld using Guptas method. One of the most attractive points of Einsteins theory of general relativity seems to be that the fundamental tensor
g can be interpreted as a potential of the gravity on the one hand, and
as a metric tensor of the space-time world on the other hand. In trying
to quantize the gravitational eld, however, the metric interpretation of
g will cause some troubles, especially from the observational viewpoint.
Therefore, contrary to Einsteins original intention, the separation of this
geometrical concept from the interpretation of g seems convenient for
the quantization of the gravitational eld. Guptas [1] approach seems
adequate to this line of thought.
We have investigated the cancellation between contributions from
the electromagnetic and the gravitational interactions by using Guptas
method of quantization of the weak gravitational eld [2].
Under the assumption of the weak eld, we can put
g (x) = + s , hk (x) = k + bk (x)
s = b + b , a = b b ,

1 0 0 0
0 1 0 0

=
0 0 1 0 =
0 0 0 1
where hk (k = 1, 2, 3, 4) is the so-called vierbeine. s(x) or b(x) is, in our
approximation, considered as a eld quantity in Minkowski-space, and the

22. The Possibility of Gravitational Quantization

246

dual interpretation of g is abandoned. The quantization of s or b is made

by following Guptas method [2].
Let us consider a system of electrons, photons and gravitons interacting with each other. The interaction Lagrangian is
(e)
(p)
L = {T + T }s ,
2
where the energy-momentum tensor of the electron-eld T (e) includes the
interaction between the electron and the photon elds, and T (p) is the
energy-momentum tensor of the photon eld. In order to avoid the formal diculties due to the derivative couplings appearing in T (e) and T (p) ,
Schwingers dynamical principle [3, 4] is suitable to derive the S-matrix in
the interaction representation. The S-matrix is written in the form

S=

in
n!
n=0

T (L(x1 ), L(x2 ), ...L(xn ))d 4 x1 d 4 x2 ...d 4 xn .

Following Dysons analysis of the divergences appearing in any element of

S-matrix, we see that there occur so many types of primitive divergences
owing to the derivative couplings that this system is unrenormalizable.
In our present treatment the most essential feature of the gravitational
eld, namely, the non-linearity is not taken into account. Accordingly it
is not clear how far our conclusion is reliable.
References
[1] S. N. Gupta: Phys. Rev. 96, 1683 (1954).
[2] S. N. Gupta: Proc. Phys. Soc. (London) 65, 161 (1952).
[3] J. Schwinger: Phys. Rev. 82, 914 (1951).
[4] T. Imamura, S. Sunakawa, R. Utiyama: Prog. Theor. Phys. 11, 291
(1954), especially cf. p. 300, Eq. (3.5)L .

Chapter 23
The Necessity of Gravitational Quantization

DE WITT called attention to the fact that there is an ambiguity in the

choice of eld variables in terms of which one may make an expansion about
the Minkowski metric . For example, one might use either or
where g = + , g = + . To be consistent in a self-energy
calculation one should expand out to the second order, and the dierence in
choice leads to a dierence in the trace of the non-gravitational (or matter)
stress tensor. There are some elds, notably the electromagnetic eld, for
which this trace vanishes. In this case, you get the same result in the
second order, no matter what you expand in terms of. It is a curious thing
in the electromagnetic case (although, as Utiyama has pointed out, you do
have the derivative coupling and therefore the divergence is of the second
or third order), that if you include second order terms in the interaction,
the old-fashioned self-energy will be exactly compensated. MISNER asked
if the computation has also been done for the neutrino eld. DE WITT
replied that although it should be done, it has not been done for two
reasons: (1) The mathematics of the spinor problem up to the second
order is considerably more complicated and has not yet been fully worked
out. (2) Interest has in the meantime shifted to the problem of tackling
the complete nonlinear gravitational eld.
WHEELER pointed out that the linear starting point is incompatible with
any topology other than a Euclidean one; that if one has curved space
or wormholes, you just cant start expanding this way. BELINFANTE
asked if anyone had found it possible to have hole theory in a curved space;
i.e. can one make a covariant distinction between positive and negative
energy states?
ROSENFELD said that Dirac, just after he had formulated hole theory
and was faced with the objection that there is an innite energy associated
with these states, had attempted unsuccessfully to overcome this diculty
by introducing a closed universe.

248

23. The Necessity of Gravitational Quantization

MISNER said that one could get a quite good qualitative idea of what happens in curved space (the metric being externally impressed) by using the
Feynman prescription. Since the action is still quadratic in the interesting
eld variables, there is no diculty. In a spherical space there exist states
of excitation which do not scatter each other; but as soon as the space has
bumps in it photons in one state get scattered into other states. Hole
theory is possible if the metric is static; however, a time dependent metric
causes electrons to go to positive energy states.
BERGMANN pointed out that, on this account, one does not have to
exclude hole theory, because if the electrons get excited, there is occasional
pair production.
SALECKER introduced a thought experiment, involving a stream of particles falling on a diraction grating. On account of the de Broglie relation
h
for the waves associated with the stream, = mv
, one expects that particles of dierent mass will scatter dierently if they fall from a given height.
According to general relativity, one expects the same behavior for dierent masses with the same initial state of motion. Therefore, we arrive at
a contradiction with the principle of equivalence.

Figure 23.1
FEYNMAN asked if the grating is here allowed to exert forces on the particles which are non-gravitational. DE WITT said that one needs rather
a grating (made, for example, out of planets) which acts only through its
gravitational eld on the stream. FEYNMAN then said that he did not
believe that the principle of equivalence denies the possibility of distinguishing between two dierent masses. Of course, the principle of equivalence would prevent one from distinguishing between masses by means of
this particular experiment if only classical laws were operative. However,
the introduction of Plancks constant into the scheme of things introduces
new possibilities, which are not necessarily in contradiction with the principle of equivalence. As far as this particular experiment is concerned,

23. The Necessity of Gravitational Quantization

249

all that the principle of equivalence would say would be that if one performs the experiment in an elevator, he will obtain the same result as in
a corresponding gravitational eld. FEYNMAN also emphasized that the
quantities G and c by themselves do not lead to a unit of mass, whereas
such a unit exists if h is included.
WHEELER pointed out that the principle of equivalence only denies the
possibility of distinguishing between the gravitational and inertial masses
of a single body, but denitely does not prevent one from distinguishing
the masses of two dierent bodies, even when only gravitational forces are
involved. For example, we know the relative sizes of the masses of the
sun and the various planets solely from observation of their gravitational
interactions. BERGMANN added that the principle of equivalence makes
a statement about local conditions only. Therefore you can do one of two
things: either (1) use a small diraction grating that is not gravitational,
or (2) use a diraction grating made of planets. In this case, the conditions
are certainly not local.
FEYNMAN characterized the point which Salecker had raised as an interesting point and a true point, but not necessarily a paradoxical one. If the
falling particles are not allowed to react back on the grating, then according to the classical theory they will all follow the same paths. Whereas,
in the quantum theory they will give rise to dierent diraction patterns
depending on their masses.
SALECKER then raised again the question why the gravitational eld
needs to be quantized at all. In his opinion, charged quantized particles
already serve as sources for a Coulomb eld which is not quantized. (Editors Note: Salecker did not make completely clear what he meant by this.
If he meant that some forces could be represented by actions-at-a-distance,
then, although he was misunderstood, he was right. For the corresponding
eld can then be eliminated from the theory and hence remain unquantized. He may have meant to imply that one should try to build up a
completely action-at-a-distance theory of gravitation, modied by the relativistic necessities of using both advanced and retarded interactions and
imbedded in an absorber theory of radiation to preserve causality. In
this case, gravitation per se could remain unquantized. However, these
questions were not discussed until later in the session.)
BELINFANTE insisted that the Coulomb eld is quantized through the
-eld. He then repeated DeWitts argument that it is not logical to
allow an expectation value to serve as the source of the gravitational

250

23. The Necessity of Gravitational Quantization

eld. There are two quantities which are involved in the description of
any quantized physical system. One of them gives information about the
general dynamical behavior of the system, and is represented by a certain
operator (or operators). The other gives information about our knowledge
of the system; it is the state vector. Only by combining the two can one
make predictions. One should remember, however, that the state vector
can undergo a sudden change if one makes an experiment on the system.
The laws of nature therefore unfold continuously only as long as the observer does not bring extra knowledge of his own into the picture. This
dual aspect applies to the stress tensor as well as to everything else. The
stress tensor is an operator which satises certain dierential equations,
and therefore changes continuously. It has, however, an expectation value
which can execute wild jumps depending on our knowledge of the number
and behavior of mass particles in a certain vicinity - if this expectation
value were used as the source of the gravitational eld then the gravitational eld itself - at least the static part of it - would execute similar
wild jumps. One can avoid this subjective behavior on the part of the
gravitational eld only by letting it too become a continuously changing
operator, that is, by quantizing it. These conclusions apply at least to the
static part of the gravitational eld, and it is hard to see how the situation
can be much dierent for the transverse part of the eld, which describes
gravitational radiation.
FEYNMAN then made a series of comments of which the following is a
somewhat condensed but approximately verbatim transcript:
Id like to repeat just exactly what Belinfante said with an example because it seems clear to me that were in trouble if we believe in quantum
mechanics but dont quantize gravitational theory. Suppose we have an
object with spin which goes through a Stern-Gerlach experiment. Say it
has spin 1/2, so it comes to one of two counters.
Connect the counters by means of rods, etc., to an indicator which is either
up when the object arrives at counter 1, or down when the object arrives
at counter 2. Suppose the indicator is a little ball, 1 cm in diameter.

23. The Necessity of Gravitational Quantization

251

Figure 23.2
Now, how do we analyze this experiment according to quantum mechanics? We have an amplitude that the ball is up, and an amplitude that
the ball is down. That is, we have an amplitude (from a wave function)
that the spin of an electron in the rst part of the equipment is either up
or down. And if we imagine that the ball can be analyzed through the
interconnections up to this dimension ( 1 cm) by the quantum mechanics, then before we make an observation we still have to give an amplitude
that the ball is up and an amplitude that the ball is down. Now, since the
ball is big enough to produce a real gravitational eld (we know theres
a eld there, since Coulomb measured it with a 1 cm ball) we could use
that gravitational eld to move another ball, and amplify that, and use
the connections to the second ball as the measuring equipment. We would
then have to analyze through the channel provided by the gravitational
eld itself via the quantum mechanical amplitudes.
Therefore, there must be an amplitude for the gravitational eld,
provided that the amplication necessary to reach a mass which can produce a gravitational eld big enough to serve as a link in the chain does not
destroy the possibility of keeping quantum mechanics all the way. There
is a bare possibility (which I shouldnt mention!) that quantum mechanics
fails and becomes classical again when the amplication gets far enough,
because of some minimum amplication which you can get across such a
chain. But aside from that possibility, if you believe in quantum mechanics up to any level then you have to believe in gravitational quantization
in order to describe this experiment.
You will note that I use gravity as part of the link in a system on
which I have not yet made an observation. The only way to avoid quantization of gravity is to suppose that if the amplication gets big enough
then interference eects can in principle no longer play a role beyond a
certain point in the chain, and you are not allowed to use quantum me-

252

23. The Necessity of Gravitational Quantization

chanics on such a large scale. But I would say that this is the only out if
you dont want to quantize gravity.
BONDI: What is the dierence between this and people playing dice, so
that the ball goes one way or the other according to whether they throw
a six or not?
FEYNMAN: A very great dierence. Because I dont really have to
measure whether the particle is here or there. I can do something else:
I can put an inverse Stern-Gerlach experiment on and bring the beams
back together again. And if I do it with great precision, then I arrive at a
situation which is not derivable simply from the information that there is a
50 percent probability of being here and a 50 percent probability of being
there. In other word , the situation at this stage is not 50-50 that the die
is up or down, but there is an amplitude that it is up and an amplitude
that it is down - a complex amplitude - and as long as it is still possible to
put those amplitudes together for interference you have to keep quantum
mechanics in the picture.
It may turn out, since weve never done an experiment at this level,
that its not possible - that by the time you amplify the thing to a level
where the gravitational eld can have an inuence, its already so big that
you cant reverse it - that there is something the matter with our quantum
mechanics when we have too much action in the system, or too much mass
- or something. But that is the only way I can see which would keep you
from the necessity of quantizing the gravitational eld. Its a way that I
dont want to propose. But if youre arguing legally as to how the situation
stands...
WITTEN: What prevents this from becoming a practical experiment?
FEYNMAN: Well, its a question of what goes on at the level where
the ball ips one way or the other. In the amplifying apparatus theres
already an uncertainty - loss of electrons in the amplier, noise, etc. - so
that by this stage the information is completely determined. Then its a
die argument.
You might argue this way: Somewhere in your apparatus this idea
of amplitude has been lost. You dont need it any more, so you drop it.
The wave packet would be reduced (or something). Even though you dont
know where its reduced, its reduced. And then you cant do an experiment
which distinguishes interfering alternatives from just plain odds (like with
dice).

23. The Necessity of Gravitational Quantization

253

Theres certainly nothing to prevent this experiment from being carried out at the level at which I make the thing go clink-clank, because
we do it every day: We sit there and we wait for a count in the chamber
- and then we publish, in the Physical Review, the information that weve
obtained one pi meson - And then its printed (bang!) on the printing
presses - stacked and sent down to some back room - and it moves the
gravitational eld!
Theres no question that if you have allowed that much amplication
you have reduced the wave packet. On the other hand it may be that we
can think of an experiment - it may be worthwhile, as a matter of fact,
to try to design an experiment where you can invert such an enormous
amplication.
BERGMANN: In other words, if it is established that nobody reads the
Physical Review, then there is a denite 50 percent uncertainty...
FEYNMAN: Well, some of the copies get lost. And if some of the copies
get lost, we have to deal with probabilities again.
ROSENFELD: I do not see that you can conclude from your argument
that you must quantize the gravitational eld. Because in this example at
any rate, the quantum distinction here has been produced by other forces
than gravitational forces.
FEYNMAN: Well, suppose I could get the whole thing to work so that
there would be some kind of interference pattern. In order to describe it I
would want to talk about the interaction between one ball and the other.
I could talk about this as a direct interaction like 2 /ri j . (This is related
to the discussion of whether electrostatics is quantized or not.) However,
if you permit me to describe gravity as a eld then I must in the analysis
introduce the idea that the eld has this value with a certain amplitude, or
that value with a certain amplitude. This is a typical quantum representation of a eld. It cant be represented by a classical quantity. You cant
say what the eld is. You can only say that it has a certain amplitude to
be this and a certain amplitude to be that, and the amplitudes may even
interfere again... possibly. That is, if interference is still possible at such a
level.
ROSENFELD: But what interferes has nothing to do with gravitation.
FEYNMAN: Thats true ... when you nish the whole experiment and
analyze the results. But, if we analyze the experiment in time by the
propagation of an amplitude - saying there is a certain amplitude to be
here, and then a certain amplitude that the waves propagate through there,

254

23. The Necessity of Gravitational Quantization

and so on - when we come across this link - if youll permit me to represent

it by a gravitational eld - I must, at this stage in time, be able to say
that the situation is represented now not by a particle here, not by a result
over there, but by a certain amplitude for the eld to be this way and a
certain amplitude to be that way. And if I have an amplitude for a eld,
thats what I would dene as a quantized eld.
BONDI: There is a little diculty here (getting onto one of my old
hobby horses again!) if I rightly understand this, which Im not sure that
I do: The linkage must not contain any irreversible elements. Now, if my
gravitational link radiates, lve had it!
FEYNMAN: Yes, youve had it! Right. So, as you do the experiment you
look for such a possibility by noting a decrease of energy of the system. You
only take those cases in which the link doesnt radiate. The same problem
is involved in an electrostatic link, and is not a relevant diculty.
BONDI: Oh yes, because in the electrostatic case I can put a conducting
sphere around it ...
FEYNMAN: It doesnt make any dierence if it radiates. If every once
in a while the particle which is involved is deected irreversibly in some
way, you just remove those cases from your experiment. The occurrence
could be observed by some method outside.
BERGMANN: Presumably the cross section for gravitational radiation
is extremely...
FEYNMAN: And furthermore, we can estimate what the odds are that
it will not happen.
BONDI: Im just trying to be dicult.
GOLD: But that need not mean that there is some profound thing wrong
with your quantum theory. It can mean merely that when you go into the
details of how to make an op...
FEYNMAN: There would be a new principle! It would be fundamental!
The principle would be: - roughly: Any piece of equipment able to amplify
by such and such a factor (105 grams or whatever it is) necessarily must
be of such a nature that it is irreversible. It might be true! But at least
it would be fundamental because it would be a new principle. There are
two possibilities. Either this principle - this missing principle - is right,
or you can amplify to any level and still maintain interference, in which

23. The Necessity of Gravitational Quantization

255

case its absolutely imperative that the gravitational eld be quantized ...
I believe! or theres another possibility which I havent thought of.
BUCKINGHAM: The second possibility lands you back in the same difculty again. If you could amplify to any factor, you could reduce to a
negligible proportion an additional signal to take an observation on, say,
those balls.
FEYNMAN: No!
BUCKINGHAM: Because you only need one light quantum.
FEYNMAN: No!
BUCKINGHAM: If you could amplify up to any factor this becomes
negligible.
FEYNMAN: It depends! ... You see (pointing to a blank space on the
blackboard) this statement that I have written here is not written very
precisely as a matter of fact if you look at it you probably cant even see
the words. I havent thought out how to say it properly. It isnt simply a
matter of amplifying to any factor. Its too crude - Im trying to feel my
way. We know that in any piece of apparatus that has ever been built it
would be a phenomenally dicult thing to arrange the experiment so as to
be reversible. But is it impossible? Theres nothing in quantum mechanics
which says that you cant get interference with a mass of 105 gram - or
one gram.
BUCKINGHAM: Oh, yes. What Im saying, though, is that the laws
have to be such that the eect of one light quantum is sucient to determine which side the ball is on, and would be enough to disturb the whole
experiment.
FEYNMAN: Certainly! Thats always true. Thats just as true no matter
what the mass is.
ANDERSON: Suppose a neutral elementary particle really has a gravitational eld associated with it which you could actually use in the causal
link. The thing that bothers you is that you may be getting something
that is too small to produce a gravitational eld.
FEYNMAN: Its a question of design. I made an assumption in this
analysis that if I make the mass too small the elds are so weak I cant
get the experiment to operate. That might be wrong too. It may be that
if you analyze it close enough, youll see that I can make it go through a
gravitational link without all that amplication - in which case theres no

256

23. The Necessity of Gravitational Quantization

question. At the moment all I can say is that wed better quantize the
gravitational eld, or else nd a new principle.
SALECKER: If you assume that gravitation arises as a sort of statistical
phenomenon over a large number of elementary particles, then you also
cannot perform this experiment.
FEYNMAN: Yes, it depends what the origin is. One should think about
designing an experiment which uses a gravitational link and at the same
time shows quantum interference - what dimensions are involved, etc. Or
if you suppose that every experiment of this kind is impossible to do, you
must try to state what the general principle is, by trying a few examples.
But you have to state it right, and that will take some thinking.
DE WITT then remarked that there is still another type of experiment
which might some day help to decide the question of whether or not the
gravitational eld is quantized - namely, producing (or nding in cosmic
rays) particles having energies of 1019 Bev and observing their interactions.
At the level of such structures as wormholes these are the energies in
which one is interested.
ROSENFELD then gave an amusing historical survey in which he presented some of the ideas which Faraday and Maxwell had on gravitation.
Faraday attempted to measure the ability of a moving gravitational eld
to induce electric current by moving a coil of wire or a heavy mass up and
down. Although he detected no eect, he was convinced that such an effect must nevertheless exist. It struck Faraday that an important dierence
between the gravitational and electric elds was that, apparently, gravitational energy was not absorbed by matter. It also puzzled Maxwell that
the gravitational lines of force in the vicinity of two interacting masses
exhibited a behavior which did not seem to permit the introduction of
stresses in the ether to explain the attraction of the masses; the lines of
force have the masses for their sources, but do not end on them: they
exert a push, and not a pull, on the masses. It seems that they have to
end on some far-away mass distribution, which they can so to speak take
as support to push the masses nearer to each other and this gives rise to
their apparent attraction.
ROSENFELD then raised the question of the existence of gravitational waves. Formally, one can get a spherical wave which depends on
the third time derivative of the quadrupole distribution of masses. This
solution has been obtained, in the linear program, by expanding around
the Minkowski metric. However, Rosenfeld does not know if this wave has

23. The Necessity of Gravitational Quantization

257

a denite physical meaning because the energy transported in this way is

not described by a tensor but by an expression which has meaning only
with reference to the space chosen for background. This uncertainty about
the existence of waves has long prevented people like Gupta, for instance,
from quantizing them. It seems to me that the question of the existence
and absorption of waves is crucial for the question whether there is any
meaning in quantizing gravitation. In electrodynamics the whole idea of
quantization comes from the radiation eld, and the only thing we know
for sure how to quantize is the pure radiation eld.
Now the arguments that Feynman has just raised about the necessity
of quantizing gravitation, if one has to avoid contradiction with the uncertainty principle, did not convince me completely - though I must confess
I cannot refute them. In his famous device of the two holes, the idea is
that the motion of the mass will show through which hole the electron
has passed. I do not know if it is as easy as that, because if you put the
condition that the exchange of momentum between the electron and this
mass does not disturb the interference pattern, it means that the deviation
of the electron due to the momentum it has received must lead to a deviation which is less than the distance between the fringes. This distance
between the fringes is x = 2al . Now if you give an extra momentum p to
the electron,this would produce a displacement x = pp l and this must be

h
smaller than 2al . This means that p < 2ap = 2a
. But this means that the
uncertainty in positionof the mass must be larger than 2a and we dont
know whether the mass is at this hole or that hole.

Figure 23.3

258

23. The Necessity of Gravitational Quantization

FEYNMAN: Uh, yes ... I know. I might have to use gravitons to scatter
the particle, and then make some kind of assumption that I can measure
the gravity wave, no matter how weak it is. Now if I consider only gravitostatics, I still have a problem. I still have a quantum theory of gravity.
Although it is said that there is no quantum theory of electrostatics, there
2
is really, I think. The writing of rei j in the Schrdinger equation removes
electro-statics from a eld theory and makes it into the quantization of a
rather simple eld. I still think, in a certain sense, that when you represent
it as a eld, and not as a solution which is the result of a eld, that we
still have to quantize it in order to get it to work.
GOLD: It could be that inaccuracies are always introduced because no
experiment can be, nally, gravitational only. It is possible that the existing quantum theory will already always make sure that nothing can be
measured without sucient accuracy.

FEYNMAN: If I write down a term G mm

ri j in the equation for the universe
and if I have only particles in the system with no question of eld variable,
is this a quantum theory of gravitation or is it a classical theory? Thats a
question of denition. In other words, when this term is included has the
gravitation eld been quantized?
ROSENFELD: The classical theory of the gravitation eld would give
you this term.
FEYNMAN: No, it would not, because it cannot produce a Schrdinger
equation; it gives you a force between two points, and then you interpret
that energy as a Hamiltonian for the particles.
ROSENFELD: That is only if you impose upon me the obligation to start
from a Hamiltonian.
FEYNMAN: Yes - So if we consider Newtonian gravitation alone we
would not have to argue whether to quantize or not (whatever you call
this process of including a term in the Hamiltonian); but if we consider
the dynamic aspect and if we do not consider this process as quantization,
then my experiment does not prove the necessity for quantization.
BONDI: Does this mean, then, if there were no gravitational waves, you
would not feel that this experiment would prove the need for quantization?
FEYNMAN: It depends on your denition, as I tried to explain, of quantization. If I can use my other experiment - which is just exactly the same,

23. The Necessity of Gravitational Quantization

259

but is a little clearer in a certain respect, because the mass moves back
and forth - the question is: How do we analyze the situation? Now listen,
we can analyze as we go along and cut the thing in the middle if we want
to, and say that this produces a eld and the eld acts on the other one.
Thats one way of representing it. If we do it that way, then we have to
have an amplitude for the eld being here and an amplitude for the eld
being there. The gravitational eld has to be quantized. Incidentally,
this uses only gravito-statics. But theres another way of representing the
same thing, and that is there is an action at a distance between the two
particles: then we do not have to analyze it in the intermediate range as
a function of time. I am sorry I did stop at these subtleties when I made
up this example.
BONDI: That is very much clearer now. It does seem to me that this
vexed question of the existence of gravitation waves does become more
important for this reason.
FEYNMAN: Yes. Theres a delay in this equipment. If theres real
delay in this equipment, the information is stored in the eld and cant be
extracted by looking at the particles of the equipment; and if a quantum
theory of the gravitational eld is not necessary, there is another possible
theory: the action at a distance theory of gravity. That is another way
out. It may not be necessary to have a gravitational eld at all.
ROSENFELD: Well, the last point about which I worry very much is
this: It is dicult for me to imagine a quantized metric unless, of course,
this quantization of the metric is related to the deep-seated limitations
of the denitions of space and time in very small domains corresponding
to internal structures of particles. That is one prospect we may consider.
The whole trouble, of course, which raises all these doubts, is that we have
too few experiments to decide things one way or the other.
BELINFANTE then raised the question of the validity of Guptas successive approximations. These approximations (though possibly valid in a
dierent scheme) cannot be made using Papapetrous formula in the way
proposed by Gupta, for the following reason: This method mixes the various orders of approximation and thus leads to nonsense if xed up (by
adding adjustable stresses after each step) in such a way as to avoid the
other contradictions to which the method leads when used uncorrected the reason being that intermediate approximations lead to sources for the
next approximation which do not satisfy conservation laws and thus contradict the dierential equations of Papapetrou. (The trouble may be due

260

23. The Necessity of Gravitational Quantization

to Papapetrous use of a gravitational energy density depending on second

derivatives of the eld.)
The question of the absorption and production of gravitational waves
was raised again. FEYNMAN discussed a device which would absorb
gravitational energy, provided one assumes the existence of gravitational
radiation (but as he pointed out, My instincts are that if you can feel it,
you can make it.). For this purpose one can use a result already presented
at an earlier session that the displacement of a particle in the path of a
gravitational wave satises the dierential equation
d2
a
.
= R0ba
dt 2
A particle situated initially near a long light rod, oriented parallel to the
propagation direction, could be made to scrape against the rod by the
transverse-transverse wave amplitudes.
FEYNMAN then presented the result of a calculation of the energy radiated by a double star system in a circular orbit:

Energy radiated in one revolution 16 mM ( u )5

=
,
Kinetic energy content
15 m + M c
where the masses of the stars are m and M, and u is the magnitude of their
relative velocity. The eect is extremely small, the lifetime of the earth
going around the sun being of the order 1026 years.
WHEELER said that, from the point of view of expanding the eld about
the Minkowski metric, the behavior of the Schwarzschild solution might
be described by saying that the eld away from the singularity itself acts
partly as a source of gravitation. In this connection, one might conceive of
a gravitational geon - an object in which the gravitational waves traveling
around in circles provide the energy to hold the system together.

Figure 23.4

Closing Session
Chairman: B. S. DeWitt

Chapter 24
Divergences in Quantized General Relativity
S. Deser

The possible physical eects of general relativity on the elementary problem have usually been considered as negligible in view of the fact that
energies at which the former might have a bearing are much too high. It
is added that the eects of the new particles and the energies at which
current theory loses its validity occur very much below this range and
therefore a correct future theory will have solved the present diculties
in a much lower domain. In any case, eld theory will have been altered
so radically that arguments from the present one will lose their validity.
Further, it is sometimes felt that general relativity is a purely macroscopic
theory which loses its meaning in microscopic domains, where the concept
of metric is not very transparent.
To these classic objections there exist several levels of replies. On
a general level, it may be pointed out that the eect of a theory is not
always rst felt through its gross direct dynamical contributions (as, for
example, spin). Since general relativity is needed, at least in a formal
way, to provide a correct denition of the energy-momentum tensor, it
underlies any theories which deal with the energies of elds and particles.
The principle of general covariance on which the general theory is based
is not in any way restricted a priori to macroscopic considerations and
it is thus necessary to explore its consequences for any theory. The fact
that the metric may not be so simply measurable in microscopic domains
(say within an elementary particle) is no more an argument against the
relevance of relativity than the denition of position measurement in a
hydrogen atom is an argument against the use of coordinates in quantum
mechanics. Finally, that a future correct theory will exclude the relevance
of relativity is not an argument but a wish.
To these arguments of general principle, some very considerable quantitative ones can be added. The whole scheme of local eld theory is
plagued with divergences occurring because there is no upper limit to the
energies involved in it. No satisfactory cuto method is known which pre-

24. Divergences in Quantized General Relativity

264

serves the basic requirements of physical meaningfulness and there is no

strong doubt that any can be found within a Lorentz-covariant framework.
At this point general relativity may be invoked. Classical considerations
show that for any kind of matter coupled to the metric eld in the Einstein way, there are limitations to the energy densities and masses which
can be concentrated or built up in a given region. Similarly, lower bounds
exist on the sizes of wave-packets built up from linear wave equations.
M<

c2 d
, d = dimensions of region
4

For a high frequency wave packet one has roughly

M=

h
, d = dimensions of wave packet
cd

Therefore, the dimensions of the wave packet must be greater than the fundamental length (l0 = 1032 cm). In all these investigations it appears that
space-time loses its physically meaningful character beyond such limits. It
is true that these limits always involve l0 and are thus quite small, but
they do occur at nite energies and their eects may well be felt sooner.
A still more signicant point is the following. It is known that under very wide assumptions any theory of coupled elds leads, near the
light cone, to singularities in the propagators of the clothed particles
and to the existence of at least some innite renormalization constants,
independent of perturbation theory. All the known Lorentz-invariant eld
couplings are of this type, in particular electrodynamics and the renormalizable meson theories. It is just here that the coupling of the gravitational
eld diers so profoundly from the usual couplings between matter elds.
First of all a basic condition for the proof of these theorems in at spacetime is that there exists an energy-momentum vector P , and that the
usual commutation relations hold for any operator.
iO(x), [O(x), P ]
This is not the case in the general theory, and it is a very basic point there.
In fact, a P can, as is well known, only be dened so as to be independent
of any inner coordinates and therefore it cannot tell one the change of
a quantity that is located somewhere in that region in question. It is
connected with the non-linearity of the system and the lack of translational
properties of parts of the system by themselves. One cannot simply x the
position of an arbitrary component of the total system, for this would clash

24. Divergences in Quantized General Relativity

265

with the energy density bounds mentioned above. Furthermore, matter

and gravitational energies are not invariantly separable.
A second related departure from usual couplings is the form of the
interaction terms. Contrary to normal eld theory, there simply is no
free-particle part of the matter Lagrangian. Here, the coupling enters
in the kinetic energy, in a multiplicative fashion, and one cannot really
disentangle a non-interacting particle here. This is rightly so, since the
particles mass and energy are dened precisely by the coupling. The
equal time matter commutation relationships also would depend upon the
positions in space-time involved. This fundamentally dierent mode of
coupling will occur most critically at high energies and forbid any approximation from free-particles. In particular, each matter eld by itself will
no longer be represented by a linear wave equation in this domain. Free
motion varies with the geometry of the space involved, which is in turn
conditioned by the matter present which is doing the moving. Later, it will
be seen that the arguments obtained in this investigation for convergence
and non-singularity of propagators are based in good part on just this type
of coupling, it may well be that a more adequate future theory of particles will incorporate some of these aspects of the non-linear gravitational
coupling. Certainly, if it is a eld theory, it must escape the pessimistic
conclusions for the current theories by some such channel. If so, it may
be all the more instructive to see how an improvement seems to emerge in
this case.
Before one proceeds to further treat the extended theory, he must
decide whether it is necessary to quantize the gravitational eld. This will
not be discussed here; and the Feynman method will be used, in which the
question of which components of the gravitational eld are to be quantized
seems to be solved by itself. In any case, one need write down whatever
physical quantities are of interest as functional integrals, which in principle
go over only independent classical congurations of the system and yield
the quantum result.
This method will be used to determine the behavior of the matter elds as modied by gravitational interactions. It may be expected,
roughly, that the contributions of the scalar and transverse parts of
the gravitational eld, though not really separable will limit the density of
matter in a given region for the former, while for the latter will smear out
the light cone. This phrase refers to the expectation that the summation
over all possible Riemann spaces (corresponding to the various transverse
modes) will average away the uniqueness of the Minkowski light cone, and
with it the singularities of the matter propagators of such a cone.

24. Divergences in Quantized General Relativity

266

The eects of gravitation on propagators of various matter elds will

now be considered. Rigorously one should treat, for example, the Feynman
function for an electron interacting with the electromagnetic and metric
elds. However as is known, in ordinary electrodynamics in at space the
behavior of this propagator cannot be less singular on the light cone than
the free one, SF .
SF SF

1
p+m

SF is already singular enough to give the innite renormalization constants

of electrodynamics. Therefore, if one can show that SF , the propagator of
the electron modied by gravitation alone, behaves better than SF , this
is a step in the right direction and is an indication that the three-eld
problem will stay good. This is due to the fact that the three-eld interaction energy, which has been omitted, presumably would not make things
worse. If one gets promising results here then the whole problem should
be investigated. Alternately, the use of electrons and photons clothed
by gravitation as the non-interacting elements of quantum electrodynamics seems to be a very good approximation to the total problem. If
one calls the S functions the matter eld functions corrected only for the
gravitational eld, and if S reduces to S this is all right. It may also
be expected that many-particle propagators will behave, if anything, better than the one-particle ones since there will be a bias against too close
approach. Gravitational eld propagators themselves should also behave
non-singularly, i.e., the Schwarzschild type of singularities should vanish
(the matter now being regularly spread) while the transverse propagation should also be free of singularity, just as the photon Greens function
is expected to be. Also, it seems that the question of renormalization of
gravitational coupling should never appear since the mass, m, plays the
double role of charge and mass, and the two renormalizations, if any,
must be equal. Now, since all charges are positive, we expect no vacuum
polarization, Z3 = 1, and m should also be zero in view of the absolute
denition of mass and energy levels in the gravitational theory, and of the
fact that the mass is dened automatically from the gravitational coupling.
From the above arguments it would appear that the simplest case of
a scalar eld coupled to the metric is a sucient example to consider. In
the usual eld theory the vacuum expectation value
0| (x) (x )|T 0
=
S =
0|0

(x) (x )eiS( ,) ( , )

,
eiS ( , )

24. Divergences in Quantized General Relativity

267

where S is the total action and the integration is over the meson, electron,
and positron elds. The action,

S=

( p + m) +

[p2 2 ] +

Now, one can eliminate the electron eld and the vacuum expectation
value becomes equal to
i(S
e meson +S0 ) G(x, x ; )

N
where Smeson is due to the pure meson eld and S0 is due to closed loops.
The Greens function G(x, x ; ) satises (p2 + m + )G = (x, x ), if we
replace the spinors by p2 to simplify the calculation. Now one knows how
G behaves near the light cone.
1
F(x, x ; ),
(x x )2

where F is a well behaved function (aside from possible harmless logarithmic terms). Thus one nds, near the light cone, that S varies essen1

tially as (xx
)2 . This, propagation function, S , is singular enough to give
one divergences.
Now consider the parallel problem when a matter eld is interacting
with the gravitational eld. The Lagrangian is
L = Lgrav +

1
g{2 2 2 } ,
2

where the DAlembertian depends upon the metric. The mass term will, as
usual, not be relevant to the singularity questions. One wants to compute
(x, x ) =

0 | (x) (x ) |T 0
0|0

where x and x are to be dened with respect to a at background frame

which is used to make a classical measurement. The functional integration
used to dene is

(x, x ) =

(x)(x ) eiS (g . . .)

eiS (g . . .)

24. Divergences in Quantized General Relativity

268

where (g...) means a summation over all Riemann spaces which preserve
the signature of (+ + +). Now, as before, the matter variables are eliminated giving

(x, x ) =
where

g(x, x ) ei(SEinstein +S) (g . . .)

N

[2 g 2 ]g(x, x ) = (x, x )

and S comes from integration over matter variables. One is interested

only in the solution for two points which are on each others light cones.
Again, the general theory of dierential equations tells one that near the
light cone
g(x, x )

1
F(x, x ; g) ,
s2 (x, x , g)

where s2 (x, x , g) is the square of the interval between the two points and F
is a regular function. Here the dierence between these propagators and
the usual ones of eld theory is obvious since the s2 (x, x , g) depends upon
the g variables. Now,
(x, x )

ei(SEinstein +S)
F (g . . .) .
s2 (x, x , g)

At this point the more or less rigorous mathematical treatment stops and
things that are said cannot be proved mathematically.
There will be very few spaces where null geodesics will connect x
and x in comparison to the number of integrations possible. It is possible
to make a vague analogy to the usual theory of poles in a nite number of
dimensions and in such cases one expects the singularities to be smoothed
out - with any measure which makes sense. For large distances, on the
other hand, the exponential may be expected to vary wildly and thus cancel
the eects of the various spaces summed over. It is felt that one should
not make the linear approximation in the action since this will reinstate
the privileged character of the Minkowski metric and the whole point of
non-linear gravitation is that the propagator depends upon the space one
is in.
A corroborative argument from quite a dierent direction, which was
mentioned earlier, is the P commutation relation. In a at space-time,
and eectively (by suitable modication) in a non-at but externally given

24. Divergences in Quantized General Relativity

269

geometry this commutation rule holds. P determines the translation of

the system as a whole and, because of superposition, of any part of it. In
a space-time which interacts with matter, however, this relation no longer
holds. P does have as a canonical conjugate the center of mass, X , of
the system with respect to the at external frame. However, it cannot
be used to locate closely any component of the system. The existence
of a P in Minkowski space-time is essential to the pessimistic theorems
mentioned before. It is obvious that if P does not have all the properties
required there, the proof that any component eld, if suciently focused
on, acts as if it is uncoupled, loses support. It may be argued that the
propagators, , mentioned above, also lose some of their meaning by the
same argument. This is true only to the extent that physical quantities
always involve space-time integrals of such functions. The fact that the
are now non-singular is merely due to the fact that one can never force a
eld to behave as if it is too concentrated - the induced geometry would
force a repulsion.
If it is granted that this action introduces an eective cuto, at about
l0 , then such high energy approximations as Landaus could be examined
again. As Landau has noted, using l0 as a cuto, theory predicts roughly
the experimental charge regardless of the value of the bare charge and
similarly for the mass. Of course, our results, being generally covariant,
cannot be directly translated into at spacetime language without apparent paradoxes.
One thing that should be done is to look into the vacuum expectation
value of the total energy-momentum of the system since this should be zero
in general relativity due to the absolute denition of energies; similarly,
T should be nowhere singular.
In the discussion which followed DESERs presentation, LICHNEROWICZ and FOURES commented on the need for an exact denition of the
measure. ANDERSON suggested that the integration diculties might
be circumvented by rst solving the classical problem of motion of two
matter elds. Then, one can eliminate the non-physical variables from
the action. WHEELER rst brought up the point that the uctuation
of the metric at very small distances, which brings about change in the
topology of the space, may make a at space-time unusable, even as a
rst approximation. In reference to the work he previously discussed, the
use of a matter propagator as a starting point is foreign to that point of
view. Also, the whole nature of quantum mechanics may be dierent in
general relativity since in every other part of quantum theory, the space in
which physics is going on is thought of as being divorced from the physics.

24. Divergences in Quantized General Relativity

270

General relativity, however, includes the space as an integral part of the

physics and it is impossible to get outside of space to observe the physics.
Another important thought is that the concept of eigenstates of the total
energy is meaningless for a closed universe. However, there exists the
proposal that there is one universal wave function. This function has
already been discussed by Everett, and it might be easier to look for this
universal wave function than to look for all the propagators. FEYNMAN
said that the concept of a universal wave function has serious conceptual
diculties. This is so since this function must contain amplitudes for all
possible worlds depending on all quantum-mechanical possibilities in the
past and thus one is forced to believe in the equal reality of an innity
of possible worlds. The following argument was proposed to challenge the
conclusions of Deser. If one started to compute the mass correction to,
say, an electron, one has two propagators multiplying one another each of
which goes as 1/s2 and are singular for any value of the gs. Therefore, do
the spatial integrations rst for a xed value of the gs and the propagator
is singular, giving m = . Then the superposition of various values of the
gs is still innite. DESER replied that this is partly due to an unallowed
interchange of limits.
A resolution was passed at this conference to the eect that there is
to be another gravitational conference in Europe during the summer of
1958.
ERNST reported on a paper on Corinaldesi1 Here, higher order terms
are obtained through a quantum treatment of the linearized theory.

1 Proc.

Phys. Soc. 69, 189 (1956).

Chapter 25
Critical Comments
R. P. Feynman

The real problems in gravitational theory are:

1. To understand classical relativistic gravity without the complications
of other things.
2. The general theory of cosmology because we may be able to work
out the shape of the world.
3. To study the theory to see if anything new is in it which is not
contained at rst sight.
4. To learn from the philosophy of gravity something to use in other
elds such as we have made use of the invariance principles and the
ideas of Einstein.
5. Curiosity
There exists, however, one serious diculty, and that is the lack of experiments. Furthermore, we are not going to get any experiments, so we have
to take a viewpoint of how to deal with problems where no experiments
are available. There are two choices. The rst choice is that of mathematical rigor. People who work in gravitational theory believe that the
equations are more dicult than in any other eld, and from my viewpoint
this is false. If you then ask me to solve the equations I must say I cant
solve them in the other elds either. However, one can do an enormous
amount by various approximations which are non-rigorous and unproved
mathematically, perhaps for the rst few years. Historically, the rigorous
analysis of whether what one says is true or not comes many years later
after the discovery of what is true. And, the discovery of what is true is
helped by experiments. The attempt at mathematical rigorous solutions
without guiding experiments is exactly the reason the subject is dicult,
not the equations. The second choice of action is to play games by intuition and drive on. Take the case of gravitational radiation. Most people

272

25. Critical Comments

think that it is likely that this radiation is emitted. So, suppose it is and
calculate various things such as scattering by stars, etc., and continue until
you reach an inconsistency. Then, go back and nd out what is the diculty. Make up your mind which way it is and calculate without rigor in
an exploratory way. You have nothing to lose: there are no experiments.
I think the best viewpoint is to pretend that there are experiments and
calculate. In this eld since we are not pushed by experiments we must be
pulled by imagination.
The questions raised in the last three days have to do with the relation of gravity to the rest of physics. We have gravity - electrodynamics
- quantum theory - nuclear physics - strange particles. The problem of
physics is to put them all together. The original problem after the discovery of gravity was to put gravity and electrodynamics together since that
was essentially all that was known. Therefore, we had the unied eld
theories. After quantum theory one tries to quantize gravity. As far as
the two methods of quantization are concerned, I believe that if one works
the other will work. The crucial problem is to be able to tell when we have
succeeded in quantizing the gravitational eld. We should take the same
attitude as in other branches of physics and compute the results of some
experiments. We dont have the experiments and thus we do not know
which results to calculate. However, if someone brought an experimental
fact could we check it? Certainly, we can take a linearized approximation
and the answer will be right or wrong. The reason for stressing experiments is because of quantum electrodynamics which was worked out in
1928 but was full of diculties such as the innite energy levels of the hydrogen atom. When Lamb discovered the spacing between two levels was
nite, Bethe then got to work and calculated the dierence and introduced
the ideas of renormalization. We knew that the number was nite, but
someone measured it and thus forced the computation. The real challenge
is not to nd an elegant formalism, but to solve a series of problems whose
results could be checked. This is a dierent point of view. Dont be so
rigorous or you will not succeed.
Quantum mechanics and gravity do have something in common. The
energy in quantum mechanics is best given by describing how the wave
function changes if one solves the coordinate system a little bit, and gravity
is connected with just such transformations of coordinates. Thus, the
group-theoretic denition of energy and momentum in quantum theory is
not very far away from the geometric connection between energy and what
happens when you move the coordinate system.

25. Critical Comments

273

The connection of gravity with the other parts of physics (nuclear and
strange particles) was not mentioned here. This is interesting and strange
because from the point of view of a non-specialist there is just as much
physics in these other elds. From the experimental side we have much
more detail there but have no beautiful theory.
Instead of trying to explain the rest of physics in terms of gravity
I propose to reverse the problem by changing history. Suppose Einstein
never existed, and his theory was not available, but the experimenters
began to discover the existence of the force. Furthermore, suppose one
knows all the other laws known now including special relativity. Then
people will say we have something new, a force like a Coulomb force.
Where did it come from? There will be two schools of thought. First,
some people will say this force is due to a new eld and second some
people will say that it is due to some eect of an old eld which we do not
recognize. I have tried to do this forgetting about Einstein.
First I will do the case of the new eld. The force is proportional
to r12 and thus it must be mass zero eld. Also, I assume that later one
gets experiments about the precession of orbits so that the rate of motion
does not appear to be proportional to the mass. Someone would try scalar
elds, vector elds, ..., and so on. Sooner or later one would get to a spin
two eld and would say perhaps it is analogous to electrodynamics. Then
he would write down
(

A A

x
x

d4x + e

z A ds +
+

m
2

z2 ds +

1
2

z z h ds

)
second power of first derivatives of h s

or in place of the second term

A j d x , j = e

z 4 (x z)dS,

and in place of the fourth term

T h d 4 x , T =

z z 4 (x z)dS.

In all this h is the new eld under investigation.

From this one gets some second order equations roughly of the form

25. Critical Comments

274

2
h = T.
x x
Also the equation of motion of the particles is
g z = [ , ]z z
where g is just a shorthand for + h . The eld equations of electrodynamics have a property that the current conservation is an identity
from the eld equations.
Now one asks what is T equal to such that we get automatically T , = 0?
Now, when one puts in the second power of the rst derivatives of h in
the action he gets something denite. Incidentally, one gets the linearized
form of the gravity equations. If now these equations are solved to see if
any progress has been made, one deduces that light is deected by the sun.
One at this point might say that we know too much, that a eld theorist
would never have thought of this important conservation theorem. This is
not true. Pauli deduced this equation without looking for gravity, but by
asking himself what must the eld equations be for elds of arbitrary spin.
Soon someone would realize that something is wrong, for if particles move
according to the equation of motion, which they must do from the given
action, then the T doesnt satisfy the correct equations. A suggestion,
then, would be to add the eld energy into the stress energy tensor
and say

that this also is a source of gravity. Then, th would become h(T +t(h)).
However, this would not work because a variation of the hs gives not only
what one wants, but some new terms. Finally one asks if there exists an
expression of third order in the hs which can be used in the action of the
form (h)(h, )(h, ) and will give any relief from the diculty. It is possible
to prove from the denition of T and the equation of motion that the true
T must satisfy some equation of the form
g T , = [ , ]T .
Then one can go to the next higher order of approximation and this approximation will explain the perihelion of Mercury. Although one is on
the right track this process is just an expansion. It is possible, however,
to solve the problem mathematically by nding an expression which is
invariant under
g = g + g

g
A
A
+ g + A
.

x
x
x

25. Critical Comments

275

To get the solution of this problem one asks a mathematician. However,

it could have been solved by noticing that it is a geometric transformation
in a Riemannian space. Finally, someone might suggest that geometry
determines the metric. This would be a marvellous suggestion but it would
be made at the end of the work and not at the beginning.
What does one gain by looking at the problem in this manner? Obviously, one loses the beauty of geometry but this is not primary. What
is primary is that one had a new eld and tried his very best to get a
spin-two eld as consistent as possible.
I think quantization would proceed in the same direction as the original solution of the problem. One would consider this just another eld to
be quantized. From the other viewpoint the geometry is important, but
from this viewpoint gravity is just another eld. I am sure that an enormous amount of formulae would be collected without having the generally
covariant quantum theory. I advertise that this new point of view may, in
fact, succeed in the end. Certainly people would not think that the rest
of physics could be deduced from gravity. They may well be wrong, but
it also may be true that gravity is just one more of a long list of dicult
things that some day have to be put together.
Now let us go to the second possibility, old elds. The physicists
might try to explain the new force on the basis of the incomplete cancellation of electric charges of the order (number of particles)1/2 or some such
scheme. However, there are a number of interesting possibilities which are
not completely impossible. One important fact is that this eld has an
innitesimal coupling constant and already one knows about one weakly
coupled eld, the neutrino. Also, the neutrino has zero rest mass which is
needed for 1/r dependence. The neutrino equation is modied, for convenience, to read
2 = j ,
and the neutrino propagators are assumed to be p12 . A rst trial might be
the single exchange of a neutrino between the two interacting bodies. In
this case, however, a 1/r law does not result because the initial and nal
states are orthogonal. Another possibility is a two neutrino exchange, but
this gives a potential which falls o faster than 1/r. Next, one could try
one neutrino exchange between the two bodies with each body exchanging
one neutrino with the rest of the universe. The rest of the universe is
assumed to be at some xed distance, R. This does give rise to a potential
which varies as rR13 . However, the amount of matter varies as R2 so that

276

25. Critical Comments

upon integration over the universe one gets a logarithmic divergence. The
most serious diculty with this approach is, however, the large eect the
sun would have on the earth-moon system. This last trial was so much
better than the others one can go a step further and try a four neutrino
process with the additional neutrino being exchanged in the rest of the
universe. This will also give a 1/r potential and a higher order divergence
and one must worry about the density of matter, etc. Higher order terms
are possible but they are much more dicult to handle. Therefore, the
possibility exists that the material from the outside is making the source of
gravitation here through the exchange of well known particles, the neutrinos. This is obviously no serious theory and is not to be believed. For one
thing the use of the correct propagators would change the r dependence.
If one did put in things correctly and got a nite amount for a result, then
this is very curious and has not been noticed before. That is one of the
advantages of looking at something from a dierent direction. Anyway,
this is what the people who believe in old theories would do.
This dierent point of view has been advertised in the hope that a few
people will start looking at gravitation from a dierent direction. I think
really that it is more likely and more interesting to go from the geometric
side, but if a few investigations change directions we may get somewhere.
ROSENFELD comments that the possible connection between neutrinos
and gravitation was discussed a long time ago. Also, one says that gravitational energy is carried away by gravitons while energy is conserved in
beta decay by the neutrino. These might be alternate ways of saying the
same thing.

Chapter 26
Summary of Conference
P. G. Bergmann

Our agenda consisted essentially of two main problems with a connecting

link between them. The two main problems are classical theory and quantum theory and the connecting link is the theory of measurement which
stands with its feet in both camps.
A call for more experiments at the present time is not likely to produce
more experiments, but more experiments would be a wonderful thing. The
only two types of experiments that have been suggested at this conference
fall in two classes. First, do all old experiments over again with more accuracy and increased care. And as Dicke has pointed out, if an experiment
reported an accuracy great enough to check on the energy of interaction of
weak interactions it might give a result dierent, in principle, from those
done up to now. The other type of experiments which are going on, and
ought to be going on, are those intended to shed light on cosmological
problems at all levels. Actually, there exists a third type of experiment
which is apparently not feasible, and is not going to be feasible for a long
time. This type considers the detection of gravitational waves.
The classical theory must be concerned with a number of questions.
First of all we have the mathematical questions. I think that Lichnerowicz
and his group have cleaned up the question of the propagation of innitesimal disturbances so that we now know what the light cone is, not only in
the gravitational theory, but also in a non-symmetric theory. As far as the
true Cauchy problem is concerned, how can we develop an algorithm which
gives us self-consistent conditions on an initial hypersurface from which we
can continue? This problem is not solved and is essentially another wording of the search for true observables. The third type of problem which
one might classify as mathematical are the global theories which have been
investigated by the French group.
Now to the problems that are more concerned with the special question, where things having physical or model signicance are tried out. The
most important of these questions which must be settled is, are there grav-

278

26. Summary of Conference

itational waves? At the present there is no general agreement. The other

things to be mentioned are interesting but are of less crucial signicance.
People have studied model universes and stability questions and these are
of major importance for cosmology. The geons, which are not to represent
particles, may be of considerable use in studying model questions, i.e., how
will such a thing behave? One would prefer, however, not to use the dynamical or statistical averagings. Among these classical problems are the
unied eld theories, and my own inclination is to suspend for a few years
the search for bigger and better ones. Even if one believes in exhausting
the classical problems rst and also believes in unication, there is some
question as to whether the unied theories are the correct starting point.
At the end of the list of classical problems there is the real problem of the
feasibility of separating the strictly classical questions from the quantum
questions.
The work of Pirani which gives a simple classical observation of the
components of the Riemann-Christoel tensor should be accepted and
made part of our equipment. It enables us to set up a conceptual experiment to measure a specied component of this tensor. This result
should be a basic component in the design of new experiments. As far
as quantum measurements are concerned, we should separate them into
quantum theoretic measurements with and without gadgets. The man
who does an experiment in gadgets believes that he can bring clocks and
measuring rods into his physical space time without aecting it too much.
I remind you of Saleckers experiment. It is dangerous, however, to use
gadgets in an experiment without careful thought.
The rst point in the discussion of quantum theory is the necessity
for quantization. We have had two types of arguments for quantization
here. The rst is for the essential unity which quantization will bring.
This is a very vague and general argument. And the second is related
to measurement by gravity and has been discussed by both Feynman and
Rosenfeld. As far as technology is concerned, we have talked very little
about the non-Feynman methods of quantization. I agree with others that
one need not wait until all the mathematical diculties are cleared up in
connection with the Feynman method. If this can be carried through it
must also be possible to give a mathematically acceptable form to it. Very
little has transpired on the problem of elementary particles. We have been
given several word pictures and we must build a framework to hang the
pictures on.

Chapter 27
An Expanded Version of the Remarks by R.P. Feynman on the Reality of Gravitational Waves1

I think it is easy to see that if gravitational waves can be created they can
carry energy and can do work. Suppose we have a transverse-transverse
wave generated by impinging on two masses close together. Let one mass A
carry a stick which runs past touching the other B. I think I can show that
the second in accelerating up and down will rub the stick, and therefore
by friction make heat. I use coordinates physically natural to A, that is
so at A there is at space and no eld (what are they called, natural
coordinates?). Then Pirani at an earlier section gave an equation for the
notion of a nearby particle, vector distance from origin A, it went like,
to 1st order in

a + Ra0b0 b = 0 (a, b = 1, 2, 3)
R is the curvature tensor calculated at A. Now we can gure R directly, it
is not reasonable by coordinate transformation for it is the real curvature.
It does not vanish for the transverse-transverse gravity wave but oscillates
as the wave goes by. So, on the RHS is sensibly constant, so the equation
says the particle vibrates up and down a little (with amplitude proportional
to how far it is from A on the average, and to the wave amplitude.) Hence
it rubs the stick, and generates heat.
I heard the objection that maybe the gravity eld makes the stick
expand and contract too in such a way that there is no relative motion of
particle and stick. But this cannot be. Since the amplitude of Bs motion
is proportional to the distance from A, to compensate it the stick would
have to stretch and shorten by certain ratios of its own length. Yet at
the center it does no such thing, for it is in natural metric - and that
means that the lengths determined by size of atoms etc. are correct and
unchanging at the origin. In fact that is the denition of our coordinate
system. Gravity does produce strains in the rod, but these are zero at the
1 See

Feynmans remarks, p.252.

27. Expanded Remarks by Feynman

280

center for g and its gradients are zero there. I think: any changes in rod
lengths would go at least as 3 and not as so surely the masses would
rub the rod.
Incidentally masses put on opposite side of A go in opposite directions.
If I use 4 weights in a cross, the motions at a given phase are as in the
gure:

Figure 27.1
Thus a quadrupole moment is generated by the wave.
Now the question is whether such a wave can be generated in the
rst place. First since it is a solution of the equations (approx.) it can
probably be made. Second, when I tried to analyze from the eld equations
just what happens if we drive 4 masses in a quadrupole motion of masses
like the gure above would do - even including the stress-energy tensor of
the machinery which drives the weights, it was very hard to see how one
could avoid having a quadrupole source and generate waves. Third my
instinct is that a device which could draw energy out of a wave acting on
it, must if driven in the corresponding motion be able to create waves of
the same kind. The reason for this is the following: If a wave impinges
on our absorber and generates energy - another absorber place in the
wave behind the rst must absorb less because of the presence of the rst,
(otherwise by using enough absorbers we could draw unlimited energy
from the waves). That is, if energy is absorbed the wave must get weaker.
How is this accomplished? Ordinarily through interference. To absorb, the
absorber parts must move, and in moving generate a wave which interferes
with the original wave in the so-called forward scattering direction, thus
reducing the intensity for a subsequent absorber. In view therefore of
the detailed analysis showing that gravity waves can generate heat (and
therefore carry energy proportional to R2 with a coecient which can be
determined from the forward scattering argument). I conclude also that
these waves can be generated and are in every respect real.

27. Expanded Remarks by Feynman

281

I hesitated to say all this because I dont know if this was all known
as I wasnt here at the session on gravity waves.

Index

Absorber theory of radiation, 95,

142, 249, 280
Age of Earth, 58
Anderson, J. L., 59, 84, 150, 185,
192, 193, 204, 208, 232,
243, 255, 269
Anthropic reasoning, 125
Babson, R., 810
Bahnson, A., 711, 16, 18
Bargmann, V., 59, 142, 185, 243
Bass, R., 11, 89
Belinfante, F. J., 11, 100, 126,
143, 161, 182, 187, 190,
191, 193, 231, 238, 247,
249, 250, 259
Muddied theory, 190
Beltrami operator, 82
Bergmann, P. G., 4, 16, 17, 58,
63, 74, 80, 84, 102, 127,
128, 142, 148, 149, 159,
162, 165, 166, 185, 190
192, 239, 248, 249, 253,
254, 277
Betti numbers, 89
Bianchi identities, 107, 109, 147
Birkhos fundamental theorem,
89
Bondi, H., 17, 59, 95, 99102, 114,
116, 126, 127, 129, 142,
150, 159, 160, 243, 252,
254, 258, 259
Born-Infeld theory, 144, 145, 148
Brill, D.R., 142

Buckingham, M. J., 255

Cartan calculus, 80
Cauchy Problem, 66, 67, 69, 82
84, 277
Christoel symbols, 141, 219
Classication of 3-manifolds, 78
Classication of metrics, 79
Clock models, 171
Commutation relations, 265
Cosmology
Cosmological Principle, 122,
129
Cyclic cosmology, 114
experimental, 136
Coupling constants, 53
Cygnus A, 135
Cylindrical symmetry, 100, 101
Davis, W. R., 139, 145
Debever, R., 46, 79
Deser, S., 8, 233, 239, 263, 269,
270
DeWitt, B. S., 7, 74, 143, 197
199, 204206, 208212,
215217, 219, 221, 225,
230, 239, 244, 247249,
256
Dimensionless numbers, 53
Dirac, P. A. M., 53, 214, 217,
247
Dirichlet problem, 68
Distance measurements, 171
Eddington, A. S., 53, 114

284

Einstein-Rosen topology, 74, 78,

101, 104
Etvs experiment, 5153, 58, 59
Equivalence Principle, 51, 52, 59,
95, 140, 249
and anti-matter, 59
Ernst, F. J, 143
Ether drift, 54
Euler-Poincar characteristic, 89
Feynman quantization, 179, 182,
192, 208, 229, 232, 233,
238240
Feynman, R. P., 4, 13, 15, 19,
145, 149, 150, 177, 178,
182, 184, 228, 231, 238,
244, 248250, 252260,
270, 271, 278, 279
Formation of Moon, 56
Fours, Y., 67, 68, 72, 80, 83, 87,
269
Friedman solution, 160
Functional integration, 204, 229,
230, 265
Fundamental length, 145, 147
Gauge theoretic constraints, 4, 18,
83, 84, 87, 166, 189191,
193, 208, 210, 212215,
225, 228, 229, 231233
Diracs method, 166, 187, 212
Diracs Theorem, 213
Fermis method, 190
Ghniau, J., 16, 46, 79, 192
General covariance, 84
General relativity
analogy with electromagnetism,
44, 46, 82, 95, 100, 104
and atomic clocks, 44, 59
classic tests of, 43
divergences, 246, 264, 267

Index

elimination of, 147, 265

on a computer, 73, 83, 87
variational principle, 69, 84,
86, 142
Geons, 48, 49, 142144, 148, 149,
182, 260, 278
thermal geons, 139
George Rideout, 810
Glenn Martin Company, 7, 8, 10,
11
Global solutions, 70
Gold, T., 55, 57, 59, 121, 126
129, 159, 184, 243, 244,
254, 258
Goldberg, J., 16, 18, 109, 190
193
Gravitational waves, 3, 66, 67, 95,
99, 103, 107, 116, 117,
142, 158, 256, 258, 260,
277279
Gravity as a regulator, 165
Gravity Research Foundation, 8
10
Greens functions, 82, 266, 267
Gupta, S., 245
Hodges theorem, 90
Hubble radius, 53
Hydrodynamical analogy, 48, 83,
129
Initial value problem, 35, 45, 84,
114, 139, 179
Invariants of the Riemann tensor,
104
Involution property, 67, 69
Isothermal coordinates, 67
Jansky, K., 131
Kervaire, M., 19, 91

Index

Komar, A., 139

Kursunoglu, B., 16, 147, 148, 150,
162, 243
Lagrangian coordinates, 83
Lagrangian mesh, 83, 238
Lamb shift, 272
Laurent, B., 16, 18, 232, 233, 238,
239
Lichnerowicz, A., 45, 63, 67, 72,
73, 78, 81, 82, 87, 90,
102104, 142, 148, 158,
231, 269, 277
Lilley, A. E., 131
Lindquist, R. W., 114
Liquid Earth theory, 57
Local invariants, 79
Machs Principle, 45, 46, 139
Measurability of the electromagnetic eld, 179
Measurement of gravitational eld,
80
quantum limitations, 167169,
184
Misner, C. W., 18, 47, 48, 71,
74, 77, 79, 83, 86, 87,
91, 182, 192, 237, 238,
247, 248
Mjolsness, R., 160
Mller, C., 18
Natural coordinates, 279
Necessity of quantization of gravity, 165, 245, 265, 278
Neutrinos, 142, 143, 182, 276
connection to gravitons, 276
Newman, E. T., 17, 191, 192
Non-simply connected universe, 89
Non-symmetric theory of gravitation, 110, 144, 161, 277

285

Parity violation, 58
Particles as singularities, 109
Petrovs classication, 103
Pirani, F. A. E., 17, 74, 102,
106, 115, 128, 141, 142,
160, 177, 185, 278, 279
Planck length, 264
Planck mass, 167
Pseudo-tensor formalism, 101
Quadrupole, 108, 256, 280
Quantum eld theory in curved
spacetime, 205
Rainich, Y., 48, 71, 72
Regge, T., 139
Regularity problem, 71
Renormalization, 264, 266, 272
RIAS, INC, 7, 11, 16
Rosen, N., 16, 74, 98, 106, 107
Rosenfeld, L., 19, 149, 162, 178,
179, 184, 247, 253, 256,
258, 259, 276, 278
Salecker, H., 148, 171174, 176
179, 184, 248, 249, 256,
278
Schild, A., 17, 104
Schiller, R. S., 192
Schwarzschild solution, 71, 74, 101,
104, 113, 114, 139, 144,
159, 165, 260, 266
Sciama, D., 45, 127, 128, 161, 162
Singularity of propagator, 266
Spinors, 161, 189, 191, 247, 267
Steady-state theory, 127, 129
Thom, R., 79
Thought experiments, 80
dangers of, 278

286

Time-varying constants, 53, 54,

58
Tonnelat, M. A., 69, 110, 144,
145, 148
Toy models, 278
True Observables, 166, 191, 192,
208, 228, 230, 231
Two-body problem, 68
Ultraviolet divergences, 263
Unied eld theory, 15, 35, 48,
49, 70, 74, 78, 110, 144,
147150, 166, 182, 272,
278
already unied eld theory,
71
problem of other elds, 149
Utiyama, R., 16, 17, 151, 245,
247
Vierbein, 102, 239, 245
Weber, J., 101, 102, 116, 160, 183,
184
Wheeler, J. A., 10, 1316, 19,
43, 57, 67, 71, 74, 82,
84, 86, 89, 91, 100, 114,
128, 129, 139, 142, 143,
148, 149, 160, 179184,
192, 207, 237239, 243,
247, 249, 260, 269
Wigner-Seitz approximation, 113
Witten, L., 11, 59, 89, 150, 252
Wormholes, 47, 74, 77, 79, 181,
182, 184, 247, 256

Index