364 RICHARD C.

TOL MAN
It will be noted that in addition to the ordinary half-width for transitions from 8 to A, the
absorbed radiation of half-width C/4m. , there is an consideration of the transitions A to 8
alone
additional absorption of radiation over the entire becomes a poor approximation to the actual
half-width p. The intensity of this additional state of affairs. The double transition A to to A 8
absorption is much less than that of the normal may be treated by a direct application of the
absorption, to which it bears the ratio of the foregoing analysis, provided the ordinary ap-
square of G/4~y'. When H equals zero, this is proximations are made for the spontaneous
the ratio of the normal half-width C/4m to the jumps 8 to A. In addition to the usual substi-
half-width 7 of the incident radiation. In general tution of C/47r+I' for C/4~ in (27), the most
this ratio is small and therefore the usual important result is that y+I' replaces y in the
analysis, which assumes F(s) equal to a constant square root in (22). Since I' will include a term
C, may be legitimately applied. This verifies G/y' because of the induced transitions to A,8
Weisskopf's9 conclusion. the square root will apparently be real whenever
The above formulas give the order of magni- G(s) is given by (19). Thus there will be no shift
tude of the effects involved. When y becomes of the absorbed line; this is again in agreement
much less than F, defined as the corresponding with Weisskopf's result' when the frequency of
his strictly monochromatic radiation coincides
9 V. Weisskopf, Ann. d, Physik 9, 23 (1931). with the line center, (Zs — Zg)/h.

FEBRUARY 15, 1939 PII YS ICAL REVIEW VOLUME SS

Static Solutions of Einstein's Field Equations for Spheres of Fluid

RICHARD C. TOLMAN
Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, California
(Received January 3, 1939)

A method is developed for treating Einstein's field equations, applied to static spheres of
a manner as to provide explicit solutions in terms of known analytic functions.
fluid, in such
A number of new solutions are thus obtained, and the properties of three of the new solutions
are examined in detail. It is hoped that the investigation may be of some help in connection
with studiesof stellar structure. (See the accompanying article by Professor Oppenheimer
and Mr. Volkoff. )

$1. INTRODUCTION drops from its central value to zero at the

'T IS boundary. ' To these, by regarding empty space
difficult to obtain explicit solutions of
as the limiting case of a fluid having zero density
~ ~ Einstein's gravitational field equations, in
and pressure, we can also add de Sitter's cos-
terms of known analytic functions, on account of
mological solution for a completely empty uni-
their complicated and nonlinear character. Even
verse, and Schwarzschild's so-called exterior
in the physically simple case of static gravi-
solution for the field in the empty space sur-
tational equilibrium for a spherical distribution
rounding a spherically symmetrical body, thus
of perfect fluid, there are only two explicit
giving four solutions in all.
solutions which are at present well known. These
The present paper has a twofold purpose. In
are Einstein's original cosmological solution for a
the first place, a method will be given for treating
uniform distribution of fluid with constant
' In addition to these explicit solutions for a spherical
density p and constant pressure p throughout the distribution of Quid, we also have Lemaitre's interesting
whole of space, and Schwarzschild's so-called explicit solution for a spherical distribution of solid, each
interior solution for a sphere of incompressible concentric layer of which supports its own weight by
purely transverse stresses. See Eq. (5.11), Ann. de la Soc,
fluid of constant density p and a pressure p which Scient. de Bruxelles A53, 51 (1933).
 SOLUTIONS OF FIELD EQUATIONS
the nonlinear diA'erential equations, applying to tensor R„, and its trace R are known functions of
the gravitational equilibrium of perfect Huids, in the potentials g„. and their erst and second
such a manner as to make it somewhat easy to derivatives with respect to the space-like and
obtain a variety of explicit solutions. In the time-like coordinates x'
- x'.
Owing to the
second place, it will then be shown that this complicated and nonlinear dependence of R„„and
method of treatment leads directly not only to R on the g„„and their derivatives, no general
the four well-known solutions mentioned above, precise, explicit solution for these equations is
but also to a number of others which may have a known, and special solutions are difficult to
measure of physical interest. In particular, it is obtain in explicit analytic form as has already
hoped that some of these new solutions may be of been mentioned. The problem of solution is made
use in trying to understand the internal consti- particularly difficult by the nonlinear character
tution of stars. ' of the equations which prevents the construction
of further solutions by the superposition of those
$2. THE GENERAL FoRM oF Sot.UTioN voR xN already obtained.
EQUILIBRIUM DISTRIBUTION OF FLUID In searching for solutions of the 6eld equations
(2.1) that would correspond to an equilibrium
If for. simplicity units are chosen which make
distribution of perfect Quid, considerable simplifi-
the velocity of light and the constant of gravi-
cation can be introduced at the start.
tation both equal to unity, Einstein's field
In the 6rst place, since the condition of
equations (connecting the distribution of matter
gravitational equilibrium for a fluid will on
and energy as described by the components of the
physical grounds be a static and spherically
energy-momentum tensor T„„with the resulting
symmetrical distribution of matter, we can begin
gravitational field as described by the potentials
by choosing space-like coordinates r, 8 and p, and
g„,) can be written in the form a time-like coordinate I, such that the solution will
Sir ,'Rg„.+kg„„, ——(2.1)
T„„=R„„— be described by the simple form of line element
where the cosmological constant is denoted by A, ds'= —s"dr' —r'd8' —r' sin' ed''+e"dt ', (2.2).
and where the contracted Riemann-Christoffel
with ) and u functions of r alone, as is known to
~
My own present interest in solutions of Einstein's field be possible in the case of any static and spherically
equations for static spheres of fluid is specially due to symmetrical distribution of matter. With the
conversations with Professor Zwicky of this Institute, and
with Professor Oppenheimer and Mr. VolkoA' of the Uni- simple expressions for the gravitational potentials
versity of California, who have been more directly con- appearing in (2.2), the application of the field
cerned with the possibility of applying such solutions to
problems of stellar structure. Professor Zwicky in a recent equations (2. 1) then leads to the following
note (Astrophys. J. 88, 522 (1938};see also Phys. Rev. 54, expressions for the only surviving components of
242 (1938}}has suggested the use of Schwarzschild's in-
terior solution for a sphere of fluid of constant density as
" the energy-momentum tensor
providing a model for a "collapsed neutron star. He is
making further calculations on the properties of such a
model, and it is hoped that the considerations given in this
article &gay be of assistance in throwing light on the ques-
tions that concern him. Professor Oppenheimer and Mr.
Volkoff have undertaken the specific problem of obtaining
numerical quadratures for Einstein's field equations applied Sm T'= 8aT',
to spheres of Quid obeying the equation of state for a
degenerate Fermi gas, with special reference to the particu-
lar case of neutron gas. Their results appear elsewhere in = —s "i —— +—+ )
—i1, (2.3)
this same issue. My own solutions of the field equations, as E2 4 4 2r J
given in the immediately following, can make only an
indirect contribution to the physically important case of a t'X'
Fermi gas, since it will be seen that they correspond to 87rT4=s "i ——1—.
y
(+—
1
—A,
equations of state which cannot be chosen arbitrarily. My
thinking on these matters has, however, been largely
influenced by discussions with Professor Oppenheimer and
Mr. VolkoE, and it is hoped that the explicit solutions ob- where differentiations with respect to r are
tained will at least assist in the general problem of develop- indicated by accents. '
ing a sound intuition for the kind of results that are to be
expected from the application of Einstein's field equations 'See for example, R. C. Tolman, Reluhvity, Thermo-
to static spheres of fluid. dynamics @md Cosmology (Oxford, 1934}, Eq. (95.3}.
366 RI CHARD C. TOL MAN
In the second place, since the matter involved problem will hence become determinate as soon
in the distribution is by hypothesis a perfect as we introduce one further independent equation
fluid, we can obtain a direct connection with the corresponding to some additional hypothesis as
properties of the fluid by making use of the to the nature of the fluid or of the distribution.
general expression From a physical point of view, it might seem
dx" dx" most natural to introduce this additional hy-
T""=(I +p) g""—
p, (2.4) pothesis in the form of an "equation of state"
dS d$ describing the relation between pressure p and
which by definition gives the components of the density p which could be expected to hold for the
energy-momentum tensor TI"" of a perfect fluid at fluid under consideration. Or since the properties
any point and time of interest in terms of the of the fluid could also depend on position, as
proper (macroscopic) density p, the proper pres- would be the case in treating a fluid of varying
sure p, and the components of "velocity" composition or in making an approximate appli-
dx"/ds and dx"/ds of the fluid at that point and cation to a Quid of varying temperature, it
time. With the simple form of line element (2.2), might seem natural to test the consequences of
this then leads to adding some equation connecting p and p with r.
From a mathematical point of view, however,
T'= T'= T'= —P, T~4= p, (2.5) the derivatives of X and v occur in our Eqs. (2.6)
as expressions for the only surviving components in such a complicated and nonlinear manner that
of the energy-momentum tensor in terms of the we cannot in general expect to obtain explicit
pressure and density of the fluid. analytic solutions when we complete the set by
Substituting (2.5) in (2.3), we now have adding a further equation connecting p with p or
p and p with r, and should usually have to resort
to a method of approximate quadrature to obtain
solutions in that way. In order to obtain explicit
—X') analytic solutions, it proves more advantageous
( v" —X'v' v"
+—
v'
8~p=s — y (2.6) to introduce the additional equation necessary to
)~yA,
~~
42 4 4 2r give a determinate problem in the form of some
relation, connecting ) or v or both with r, so
1q —
8~1 =s "~ ——— 1
I+ —~ chosen with reference to the occurrence of the
«') derivatives of ) and v in expressions (2.6) as to
as the desired expressions which make direct make the resulting set of equations readily
connection with the properties of the fluid in soluble. By adopting such a mathematically
terms of its pressure and density. In the case of rather than physically motivated procedure, we
gravitational equilibrium for a distribution of of course run the risk of obtaining solutions which
perfect Quid, we are thus provided with the may not be physically interesting or even physi-
general form of solution for the gravitational cally possible. Nevertheless, having once obtained
potentials g„„which is described by the line an explicit solution, it then becomes relatively
element (2.2), and with three differential equa- easy to examine the implied physics and see if
tions (2.6) which relate the unknown functions this has a character which affords insight into the
X and v appearing in that form of solution, along
equilibrium conditions that could be expected for
with the pressure p and density of the fluid p, to actual Quids.
coordinate position r within the distribution of To carry out the suggested method of attack,
fluid. it is desirable to re-express Eqs. (2.6) in a some-
what different form which will make it easier to
$3. METHOD OF OBTAINING EXPLICIT ascertain what conditions on X and v can be
ANALYTIC SOLUTIONS introduced to secure solubility. In the first place,
In accordance with the foregoing we now have it is helpful to equate the two different expres-
three differential equations for the four unknown sions for the pressure p given by (2.6) and thus
quantities X, v, p and p as functions of r. The obtain a single equation for the dependence of )
 SOL UT I 0 NS OF F I EL D EQUATIONS 367

and v on r. In the second place, it is then helpful In this case the assumed equation makes (3.1)
to rewrite this result in a form which is already immediately integrable since the second two
nearly integrable so that the desired kind of terms drop out. The resulting solution is the
condition on X and v can be more easily seen. well-known one for a static Einstein universe
Introducing such a. re-expression, Eqs. (2.6) are with uniform density and pressure throughout.
then found to be equivalent to the set The solution could correspond to a distribution
d "—1) d (e "v')
(e
I+
dr( r' ) dry 2r )
I
dr42r)
—
d (e"r'q
— I+e-"- — ]
of actual fluid, with p and p non-negative, only
with the cosmological constant satisfying the
condition 3/R') h. & 1/R'.
(3 1)
Solution IL (Schwarzschild-de Sitter solution)
(v'
8~p=e-"( —y — ——+ ~,
r') (
r'
(3.2) Assumed equation
4 r
"=const.
—
~
e
(X' 1) —
— 1
8~p=e-"( I+ —~.
r') r'
(3.3) Resulting solution
q r -'
2m r'y
where the first of these equations (3.1) has been
r R') (4.2)
obtained in the manner just described. The one
term in (3.1) which is not immediately integrable r'y
contains X and v in a sufficiently simple manner e"=c'] 1-2m
rR')
so that conditions on those quantities can be
readily found which will make it easy to obtain a 8n. p = 3/R' —A 8+p = —3/R'+A
first integral of that equation.
In this case the assumed equation makes it
immediately possible to obtain a first integral of
$4. SPEcIFIc SQLUTIQNs
(3.1), since it ma, kes the third term at once
This method of attack can now be used to integrable, The necessary second integration then
obtain a number of specific solutions. We shall also proves to be possible and leads to the well-
summarize the results thus provided by first known combined Schwarzschild-de Sitter solution
stating the additional assumed equation expressing for a de Sitter universe with a spherically
the condition on ) or v that was taken to secure symmetrical body at the origin of coordinates.
the integrability of (3.1), and then giving the With p and p both non-negative the space around
resulting solution for e", e", p and p as functions of this body has to be empty with 3/R'=A and
r which can be obtained by combining the new p = p=0. With R= ~ the solution goes over into
equation with (3.1), (3.2) and (3.3). New the usual form for the Schwarzschild solution
symbols such as A, 8, R, c, m, n, etc. will be used surrounding an attracting particle of mass m,
in these expressions to denote constants of and with m=0 it goes over into the usual form
integration; they are to be regarded as adjustable for the de Sitter universe.
parameters having real, not necessarily integral,
values.
Solution III. (Schwarzschild interior solution)
Assumed equation
Solution L (Einstein universe)
e—"= 1 r'/R'—
Assumed eguati on

Resulting solution
e" = const.
e" =,.
Resulting solutiori

1
e" [A
2/R2'—
= 3/R' —A,
= —B(1 —r'/R') &j', (4.3)

87rp
e~=
1—r'/R'
= 3/R' A,
ev c2 I

8~P = —1/R'+—A
(4. 1)
8m. p

8 p= —1

R'
(3B(1
i
— EA
r'/R') I
B(1 —r'/R')&
Ay—
)
—
368 RI CHARD C. TOL MAN
In this case the assumed equation immediately nating the third term. The solution for ) and v
simplifies (3.1) by making the first term drop out, then becomes easy. With suitable values for A
and a first integral can at once be obtained, after and R the solution represents a sphere of com-
multiplying through by e"v'/2r. The second pressible fluid with the pressure dropping to zero
integration then also proves to be easily possible at the boundary as will be discussed more fully
and leads at once to the well-known Schwarzschild later.
interior solution for a sphere of fluid of constant
Solution V
density p and a pressure p which decreases with r.
With the constant = 0 the solution degenerates
8 Assumed equation
into the Einstein universe and with A =0 into e»= const. r'~
the de Sitter universe.
Resulting solution
It should be noted that Schwarzschild's interior
solution as given by (4.3) is not the most general 1+2n —n2
solution for a sphere of constant density, since a ex ev +2r2n
1 —(1+2n —n') (r/R) ~
more general assumed equation, e "=1—r'/R'
+C/r with C an arbitrary constant, would also 2(1+2n —n')
correspond to p = const. 4 The consequences of not where N=
setting C=0 in this expression have been studied
by Mr. VolkoS and will be communicated 2n —n' 1 3+Sn —2n' ( r &
elsewhere. Exp= —
—+
1+2n n' r' (1+n)R' 0 R)
Solution IV
n2 1 1+2n(r) ~
Assumed eguati on
e"v'/2r

Resulting soluti on
1+2r2/A2
= const.
where M=
' R' ( RJ
1+2n n' r—
2m(1 I)—
ex In this case the substitution of the values of e"
(1 r'/R') (1+ r'/A —
') and ~' which are given by the assumed equation
e" = B'(1+r'/A'), makes (3.1) immediately integrable if we multi-

1+3A'/R'+3r'/R'
(4 4)
ply through by r™
/(n+1) The so.lution is a
8m. p =—
1 natural one to use in investigating spheres of
A' 1+2r'/A' fluid with infinite density and pressure at the
center, as will be discussed in more detail later.
2 1 r'/R'—
—
+A' (1+2r'/A')'
A, Solution VI
Assumed equation
—A'/R' —3r'/R'
8s.P = —
1 1

A' 1+2r'/A'
+A. e—"= const.
Resulting solution
In this case, we obtain the first of the new sx —2 ~& sv —(Arl —n Br1+n) 2 (4.6)
solutions to be considered. The assumed equation
makes (3.1) immediately integrable by elimi-
87rp= ——A,
4See R. C. Tolman, Relativity, Thermodynamics and 2—n2 r2
(Oxford, 1934), $ 96. In connection with the —(1+m)'Br'"
(1 —n)'A
Cosmology
treatment of Schwarzschild's interior solution given in that 1
place, it should be mentioned that the precise upper limits 8m. p = +A.
for rP and 2m should really be taken only eight-ninths as 2 —n' r'
large as given in (96.14), since the solution ceases to have
physical significance when the pressure p becomes infinite In this case, it is convenient to substitute the
at r =0, i, e., when A becomes equal to 8 rather than when
A becomes imaginary. assumed equation in the form e "= (2 — n') ', and
 SOLUTIONS OF F I ELD EQUATIONS 369

perform the indicated differentiations in (3.1). A

first integral of the equation can then be obtained 8mp=
after multiplying through by r"+'e"", and the (a —b) (a+ 2b —1)
'+"-'
second integral after multiplying the result by
r-' '". The solution gives a very simple expression - (a+2b-2) (2m'
Er) I

for the dependence of p on r, and is again a

natural one to use in investigating spheres of
fluid with infinite density and pressure at center,
)rq' b-
(4.8)
as will be discussed in more detail later. ER) r'
Solution VII 87rP = 2 be "/r —8~ p,
Assumed equation with (a+b)(p —1) —2b —2=0.

e—"=1— — +
4r4
—.
R' A4
In this case, it is convenient to substitute the
assumed expression for e " in (3.1) and perform
the indicated differentiations. A first integral of
Resulting solution Eq. (3.1) can then be obtained after multiplying
through by r where u is connected with b by
e" = the last of Eqs. (4.8); and the second integral
1 r'/R'+ 4—
r4/A ' can then be obtained after multiplying through
by r ' '. With a=2, b=0, the solution degener-
pe
""+—
2r'/A' 4R'~: '
A'/— ates into the Schwarzschild-de Sitter solution as
e" = B' sin log
! already treated under .(4.2).
c ) This is the last of the new solutions which we
3 20r' shall present.
8xp= ——
R' A4
$5. CONNECTION OF INTERIOR AND
ExTERIQR SoLUTIQNs
8vrp= —
— + +
4r2
R' A4 A'
(B'e " 1)'*+A. —
With an appropriate choice of parameters,
some of the foregoing solutions would correspond
In this case, the substitution of the assumed to a distribution of fluid in which the pressure p
equation into the first term of Eq. (3.1) makes drops from its central value at r =0 to the value
it reduce to a constant times r, and the equa- zero at some particular radius r=rb where the
tion then becomes integrable after multipli- density p still remains finite. Such a solution
cation by e" v'/2r The depend. ence of p on r, with could then be taken as describing the condition
e-"" and e " explicitly expressed in terms of r, is inside a limited sphere of fluid with a definite
so complicated that the solution is not a con- boundary at r=r&, and in the empty space
venient one for physical considerations. outside this boundary would be taken as re-
placed by the Schwarzschild-de Sitter solution, in
Solution VIII accordance with Birkhoff's theorem as to the
Assumed equation most general spherically symmetrical solution in
empty space. It will now be of interest to
e "=const. r 'be". consider the interconnection of the two forms of
solution at the surface of discontinuity at r = rb.
Resu/ting solution
Inside the sphere of fluid, we may take the
solution as described by a line element of the
2 (2m)'+'b ' (r)o— b
—
general form (2.2) on which we have based
(a —b)(a+2b —1) ( r ) ER) our considerations
eR +2r2be —x ds'= e"dr' r'de' r' ' Hdqrs+— e"dt',
sin— —(5.1)
370 RICHARD C. TOL MAN
where ) and v are functions of r. Outside the spheres surrounded by empty space. In using
sphere, we may take the solution as described these equations, it is convenient to regard Eq.
by a line element of the simple Schwarzschild (5.3) as determining the radius rq of the sphere
form of Quid in terms of the parameters appearing in
the expressions for t! and e" in the line element
df
ds —r'd82 (5. 1). Eq. (5.4) may then be regarded as imposing
1—2m/r a condition which connects the parameters
appearing in e" with those in e~ and with rb, a
r' sin' —
8dp'+
(1 —2m/ ~dt', (5.2) condition which can always be readily satisfied
r i since it will be noted that e" is originally always
I

arbitrary as to a multiplicative factor. Finally,

which arises from the full expression for the Eq. (5.5) may then be used to calculate a value
Schwarzschild-de Sitter solution (4.2), when we of ihe gravitational mass of the sphere m in
set A. =3/R'=0 in agreement with the known terms of rb and the parameters mentioned. It
fact that the cosmological constant is too small wi)1 be noted from the form of (5.5) that the
to produce appreciable effects within a moderate gravitational mass of the sphere m can in any
spatial range, and where for convenience we set case not be larger than rq/2, and hence can go to
c'=1 in order that m shall be the mass of the
—
sphere as measured by its external gravitational
infinity only as the size of the sphere goes to
—
field expressed in the usual units which make
infinity. This, however, does not necessarily
imply that the mass 3f of the Quid before it
the velocity of light and the gravitational con- has been condensed into the sphere could not
stant equal to unity. At the boundary of the go to infinity with rq finite.
sphere at r = rq both forms of the solution must
then give the same results for physical measure-
$6. DETAILED CGNSIDERATIoN QF SoLUTIoN IV
ments made at that radius.
Since the pressure p of the fluid falls to zero We shall take Solution IV as providing the
at the boundary, we may calculate the radius of first of the examples of a sphere of Quid sur-
the boundary r~ by setting the general expression rounded by empty space, which we wish to
for p given by (3.2) equal to zero. With A=O, consider in more detail.
this then gives us The line elt!rnent describing this solution has
the form
(5.3) 1+2r'/A 2
ds'=— dr' —r'd0'
(1 —r'/I ') (|1+r'/A')
as the equation which determines the radius of
r' sin' Hdrlr'+B'(1+—r'/A')dt'. (6.1)
the boundary rq At this. radius (employing for
simplicity a unified system of coordinates r, 8, Inside the sPhere of fluid, the density p and
t applicable both inside and outside the
Pressure P (with A=0) are given by expressions
sphere) we must then demand the equalities of the form

in order that the two forms of line element

shall lead to the same results for measurements
made at the boundary with stationary meter
sticks and clocks.
We shall make use of the three equations (5.3),
(5.4)
(5.5)
Sm

and
1
p=—
A'
1+3A'/R'+3r'/R'
1+2r'/A'

Swp
1 I
=—
A'
—,
r'/R'—
2
+A'

1+2r'/A '
1

(1+2r'/A')'
—A2 / R2 —3r2 / g2
62

(6.3)
(5.4) and (5.5) in the following three sections
whef Q wc consider spec1fic examples of Quid The central density p, and central pressure p,
 SOLUTIONS OF F I ELD EQUATIONS 37i

have the values The solution may be a useful one in studying

the properties of spheres of compressible fluid
Sxp. = —
3
+
3
—,
A' R'
and Sx-p, = ———. (6.4)
1
A' R'
since the equation of state (6.5) which connects
the density of the fluid with its pressure is
relatively simple. This equation of state, how-
In terms of these central values, the efluation of
ever„ is of course a very special one, since the
state, connecting the density and pressure of the coefficients in the terms which give the linear
fluid inside the sphere, can then be written in
and quadratic dependence of density on pressure
the convenient simple form
are not arbitrary but are of the form which has
(p. p)'- arisen from the particular assumption that was
p= p. 5(P. —P)+S— (6.5) introduced to secure integrability. Nevertheless,
pc+ pc it has been shown by Professor Oppenheimer and
The boundary of the sphere occurs at Mr. Volkoff, in the article mentioned, that this
equation of state leads to results in some respects
R similar to those which would be obtained from
rc (1 —A '—R—
/— ') ', (6.6)
the equation of state for a Fermi gas in cases of
intermediate central densities.
where the pressure has dropped to the value
zero, and the boundary density has the value $7. DETAILED CONSIDERATION OF A SPECIAL
= p. 5P. + SP'—/(p. +P.). CASE OF SOLUTION V
pb (6.7)
We shall take Solution V as providing the
From the connection with Schmo, rsschi ld's
second of the examples of a sphere of fluid
exterior solution (5.2), holding outside the bound-
surrounded by empty space which we wish to
ary, we have consider in more detail. In carrying this out we
B'= (1 r~'/R')/(1+— 2rc'/A') (6.8) shall make the specific choice n = —, ' for the
parameter n which appears in the description of
as a condition on the parameter B' appearing in the solution (4.5), since this will give a ratio of
e". And we have central density to pressure of special physical
rc'/R') —
(1+rg'/A ') interest.
m= —
rt,
1—(1 (6.9) The line element describing the solution (with
2 1+2rP/A' n= —,') has the form
as an expression for the mass of the sphere as
measured by its external gravitational field. ds dr' —r'de'
4 —7 (r/R)'~'
It will be seen from the foregoing that all the
properties of the sphere and its surrounding field r' sin' +B'rdt'.
ed&'— (7.1)
can be regarded as determined by the choice of
the two independent parameters A and R, Inside the sphere of Quid the density p and
although it is not necessarily most convenient pressure p (with A=O) are given by expressions
to express those properties solely in terms of of the form
these parameters. With R')A', it will be found 3 10 try&
that the pressure and density of the fluid would Sap= + (7.2)
both fall continuously from their central to their 7r' 3R' ER )
boundary values where the density would still
be 'positive. And with R'&11.5A', it will be and (7.3)
found that the ratio of pressure to density would 7r' R' ER &
nowhere exceed one-third. As R' and A' go to
infinity the central density and pressure ap-
The central density p, and central pressure p.
then become infinite with the ratio
proach zero, and the sphere becomes larger
without limit. p./p. = 3 (7 4)
~
372 R I CHARD C. TOL MAN
The equation of state connecting the density and fluid which have infinite central pressures and
pressure of the Quid inside the sphere can be densities. In accordance with the equation of
written in the form state (7.5), however, the ratio of pressure to
I/6
density drops too fast with decreasing density to
p=3p+ (7.5) correspond very closely with what would be
6xR'i3 43 (3p+5P) expected in the case of a Fermi gas having
infinite central pressure and density.
The boundary of the sphere occurs at
r3 —R/143'', (7 6)
$8. DETAII. ED CONSIDERATION OI& A SPECIAL
CASE OF SOLUTION VI
where the pressure has dropped to zero and the
We shall now take Solution VI as providing
boundary densify has the value
the last of the examples of a sphere of fluid
83.p3 = 28/(3 )& 14"'R') (7.7) surrounded by empty space which we wish to
consider in more detail. In carrying this out we
From the connection with Schmarzschild's shall make the specific choice n = —, ' for the
exterior solution (5.2), holding outside the bound- parameter n which appears in the description of
ary, we have the solution (4.6), which will again give us the
B' = 14~'/2R, value one-third for the ratio of central pressure
(7.8)
to density.
as a condition on the parameter B' appearing in The line element describing the solution (with
e". And we have n= —,') has the form
m=r, /4=R/(4X l43i') (7.9)
ds' = —(7/4) dr' r'de' r' —
sin' tld —
p'
+(Art BrI)3dt3. (—
8. 1)
as an expression for the mass of the sphere, as
Inside the sphere of fluid the density p and
measured by its external gravitational field.
pressure p (with A=0) are given by expressions
It will be noted in this case that the solution of the form
corresponds to a sphere of fluid of infinite density
8xp = 3/7r', (8.2)
and pressure at the center, having at that point
and
the ratio which would hold for radiation, or for
particles of such high kinetic energy that their
1 1 9(B/A)r-
87t p= —(B/A)r (8.3)
rest mass may be neglected in comparison with 7r' 1
their total mass. Other ratios could, of course,
be obtained with a different choice for the The central density p, and centra/ pressure p,
parameter n in (4.5). then become infinite with the ratio
With the adjustable parameter R lying any- p&/p& 3' (8.4)
where in the range 0(R( ~, it will be seen
The ectuation of state connecting the density and
that the solution satisfies the necessary physical
conditions for a sphere of Quid, with a pressure pressure of the fluid inside the sphere can be
which drops continuously from an infinite value written in the form
at the center to zero at the boundary, with a p 1
—9(3/56m)'*(B/A) p '
density which drops continuously from an infinite (8.5)
value at the center to a value which is still 3 1 —(3/56m)&(B/A) p:
positive at the boundary, and with a ratio of
The boundary of the sphere occurs at
pressure to density which never exceeds one-
third. As R approaches infinity, the ratio of pres- r3 = A/9B, (8.6)
sure to density approaches one-third throughout
where the pressure has dropped to zero and the
and the sphere becomes larger without limit.
boundary density has the value
The solution may in some cases be a useful
one in studying the properties of finite spheres of 8xp3 = 3'B'/7A'. (8.7)
 SOLUTIONS OF F I EL 0 EQUATIONS

From the connection with Schwarzschi ld's p— 3p = const. p' which is that for a highly
exterior solution (5.2), holding outside the bound- compressed Fermi gas.
ary, we have
$9. CONCLUDING REMARKS
A' = (3'/2')& 7) (B/A) (8.8) In conclusion we must call attention to two
as a condition on the parameter A' appearing in points which are more completely considered in
e", where B/A may still be taken as arbitrary. the article of Professor Oppenheimer and Mr.
And we have Volkoff. It is necessary to mention these points
also here in order to guard against misconcep-
m = 3rI,/14 = A/42B, (8 9) tions as to the nature of the static solutions of
Einstein's field equations for spheres of fluid
as an expression for the mass of the sphere as
which have been presented in this paper.
measured by its external gravitational field.
In the first place, it should be remarked that
Again it will be noted as in the previous case
the static character of any such solution is in
discussed in II7, that the solution corresponds to
itself only sufficient to assure us that the solution
a sphere of fluid of infinite density and pressure describes a possible state of equilibrium for a
at the center, having at that point the ratio fluid, but is not sufficient to tell us whether or
which would hold for radiation or for particles
not that state of equilibrium would be stable
of such high kinetic energy that their rest mass
towards disturbances. Further investigation is
may be neglected in comparison with their
necessary to settle the question of stability under
total mass. Other ratios could be obtained with
any given set of circumstances. The question is
a different choice for n in (4.6).
an important one, since we cannot regard a
With the adjustable parameter B/A lying
static solution as representing a physically per-
anywhere in the range 0&B/A & ~, it will be
manent state of a fluid if the equilibrium turns
. seen that the solution satisfies the necessary
out to be unstable towards small disturbances, as
physical conditions for a sphere of Quid, with a
for example in the well-known case of the
pressure which drops continuously from an
Einstein static universe.
infinite value at the center to zero at the bound-
In the second place, it should be emphasized
ary, with a density which drops continuously that the imposition of static character on the
from an infinite value at the center to a value
solutions to be considered is from a physical
which is still positive at the boundary, and with
point of view a severe restriction. It is, of course,
a ratio of pressure to density which never exceeds
immediately evident that solutions having a
one-third. As B/A approaches zero, the ratio
strictly static character could in any case be
of pressure to density approaches one-third
applicable only in first approximation to spheres
throughout, the sphere becomes larger without of fluid where slow changes are actually taking
limit, and the solution as given by (8.1) ap-
place. In addition it is to be noted that there
proaches the same form as is approached by the might be a possibility for an important class of
solution given by (7.1) as R goes to infinity. quasi-static solutions with e"=g44 going to zero
This limiting form, which agrees with that given at the center of the sphere, as in certain of static
in the article of Professor Oppenheimer and solutions considered above. Such solutions could
Mr. Volkoff by Eq. (22), might be called the be said to have quasi-static character, since
blackbody radiation solution. changes taking place at the center would exhibit
The solution in its more general form proves a very slow rate when measured by an external
to be a somewhat helpful one for use in connection observer. Further discussion of the possible
with the results of Professor Oppenheimer and existence and importance of such solutions will
Mr. Volko6, since with p large the equation of be found in the article by Professor Oppenheimer
state (8.5) goes over into the approximate form and Mr. Volkoff.