Volume 88B, number 3,4 PHYSICS LETTERS 17 December 1979

ANTIGRAVITY: A CRAZY IDEA? ¢r

J. SCHERK
Laboratoire de Physique Theortque de l Ecole Normale Supeneure , Paris C~dex 05, France

Received 11 September 1979

The theoretical aspect of antigravity is briefly discussed• It is shown that supergravity with N = 2, 3, ..., 8 fermionic gen-
erators leads naturally to antigravity.

Let us consider, between two particles, the tree The one-boson exchange graph is given by:
diagram due to the exchange of a massless graviton,
and of a massless vector field A ~ , which we shall call 16rrG r , g2 1
the antigraviton. The coupling of these two fields to
matter fields ¢i (scalars), or ×k (Dirac spinors) is In the static limit, setting k 2 = 4zrG, we find, for
given by: two particles of masses m, m', of charges g, g' the for-
12= -(1/4k2)VVUaVUbR uuab - 1 VgUpgVOF~ v F pa
~ mula:
.~ = (4k2mm'/q 2) [mm' - gg'[k2]. (6)

The first term is always positive (gravitational at-

traction). The sign of the second term is negative,
+ V ~J x k ( i T u ~ u - m k ) X It, (1) hence repulsive if we have two particles or two anti-
k
particles (assume that gi > 0 for all particles, gi = -gi
where we set: for an antiparticle, gi = 0 for a self-conjugate particle
F~uv = ~u A~v - avA~u ' (2) under C, such as the 7, the gluons, etc.), and positive,
hence attractive, between a particle and an antipar-
and ticle.
We shall call antigravity the lihenomenon which
% 4; = a - 'gi (3)
occurs if the net force (gravity + antigravity) is zero
%xk=~ xk-igkA~Xk. (4) between any two particles.
This implies the universal formula:
The vector A ~ is coupled to a conserved U(1) cur-
rent ]~ and the c~arges gi, gk are a priori unrelated. gi = kmi" (7)
Let us first see if this is ruled out. As two neutrons
Excerpts of talks given at the International Conference of attract each other, it seems that, immediately, anti-
Mathematical Physics (EPFL, Lausanne, August 20-25, gravity must be eliminated• However, the masses
1979) and at the 1 l~me Ecole d'Et~ de Physique des Par-
which appear in (1) are the quark and lepton masses,
ticules de Gif-sur-Yvette (September 3-7, 1979) (spon-
sored by the Institut National de Physique Nucl~aire et de not those of the proton and neutron. Indeed, the
Physique des Particules). antigraviton couples to e - , u, d, etc• and sees their
1 Laboratoire Propre du Centre National de la Recherche bare mechanical mass, since ]~ is conserved. In a com-
Scientifique, associ~ ~ l'Ecole Normale Sup~rieure et posite particle, such as p, it does not see the gluons
rUniversit~ de Paris Sud.
and its coupling to a proton is really given by:

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Volume 88B, n u m b e r 3,4 PHYSICS LETTERS 17 December 1979

j2 = k(m u ~u ?u Xu + md~dTUXd ), (8) be due to radiative corrections (of order g2t,ong,o~a or

k 2 , which can turn a potential having a minimum at
while the graviton is coupled to the real mass (it sees the origin into one which has a maximum, the true
the gluons). minimum being elsewhere [2]. To fix ideas, we can
As a result, the force between 2 atoms (Z, A), set m e ~ 1 GeV, (qS) ~ 1 GeV, which gives to the
(Z', A ' ) is given by: a tiny mass rn~ ~ 10 -19 GeV, or a Compton wave-
length R~ = 1/rn~ ~ 1 km ,2
F = (8nG/r 2) [MM'- 340340 '1, (9)
In this case antigravity is saved, since the poten-
where *~ tial is given by

M = Z(Mp + me) + (A - Z ) M n, (10) v = - ( a / r ) [ M M ' - M ° M ° ' exp -(r/RQ) f i R ,/R~)],

(16)
MO = Z ( 2 m u + m d + m e ) + ( A - Z ) ( m u+2md). (11)
where f ( x ) = (3/x3)[x c h x - s h x ] , and R~ is the
For tile earth, we can replace M~earth/Mearth by earth's radius. This factor arises because for the
3 m .u/ M .k' Deviations from the equivalence principle Yukawa potential o f a spherical homogeneous object,
occur, and one finds that tile acceleration of two at- all the sources cannot be considered to be at the cen-
oms (Z, A), (Z', A') towards the earth differs by ter ("skin effect"). This sets an upper limit on RQ of
the order o f 2 m.
6"//7 = (Z'A - ZA ') 3(m u/Mp) Ia2/MM ', (12)
To realize an antigravity device based on this idea
where would be cumbersome, but, in theory, possible: one
would have to "heat up the vacuum" to reach the
/a2 = me(M n - m u - 2rod) + 3(m u + r o d ) ( M n - Mp)
phase where (@ -- 0, but also the one where the quarks
+ ½(m u - r o d ) ( M n + Mp). (13) are free so that Mp --- 2m u + m d , M n = 2m d + m u,
which may be a real disaster for the space ship. At
In tile case of exact SU(2) symmetry (which is this cost it might be possible if we know how to
probably wrong anyhow), setting m u = md, we get: "heat up the vacuum" without destroying the engine.
I~2 ~ m e M n with M n - Mp ~ 1.7 MeV. Setting m u = However, this clearly belongs either to UFOlogy [3]
I0 MeV (this is highly debatable), we get: or Science Fiction [4] not yet to Technology.
Now what about the strange relation gi = krni?
/a2 ~ 500 (MeV) 2,
This universal formula clearly comes from the sky
~7/7 ~ 1.5 X IO-5(Z'/A ' - Z/A). (14) in the previous discussion. However, in extended N =
2 supergravity, it was shown that the spin-1 partner
This is clearly bad news for antigravity, since the of the graviton couples to a massive matter scalar
Ebtvos experiment gives 67/7 < 10 - 9 and Dicke supermultiplet with precisely this strength. This was
pushed tile limit down to 10 -11 [1]. deduced from the algebra of supersymmetry trans-
Tile situation can be saved, however, if one of the formations [5] and it was guessed [6] (but unproved)
scalar fields acquires a non-zero vacuum expectation that N = 2 supergravity led to antigravity. In 1977
value, as in the breaking of SU(2) X U(1) down to K. Zachos worked out the coupling o f N = 2 super-
U(1). Then the ~ acquires a mass given by gravity ( V r , qJ't (i = 1,2), A Q) to a massive matter
multiplet (~i, q~') and discovered the phenomenon
mQ = kmo (q3). (15)
of antigravity [7] within the framework of super-
Tile reason why ~b acquires a v.e.v, while m e2 > 0 can gravity. In the case N = 8 it is now known that anti-
gravity also occurs [8,9]. In the spontaneously broken

, I M p, Mn should really readM p - hE , M n -z~Ewhere~LE

is the nuclear binding energy per nucleon. This modifies ,2 This can also occur classically if (@ 4: 0, provided that in
slightly the proton and neutron masses which we shall, for the original lagrangian, A ~ couple with strength +-2khul
the sake of simplicity, take to be equal both to 1000 MeV. even when ~2 ~ 0. We thank E. C r e m m e r for pointing
Similarly, the ratio mu/M p will be small (10-2). this out.

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Volume 88B, number 3,4 PHYSICS LETTERS 17 December 1979

N-- 8 supergravity theory with masses, g = +-2km ture unified theory of gravity + matter, and predicts
holds (note the factor of 2) but there is a massless antigravity. If the SU(8) local invariance discovered
Brans-Dicke scalar, coupled with strength x/~ km by Cremmer and Julia [13] in the massless case leads
so that there is a net cancellation of gravitational to the appearance of SU(8) gauge fields through
forces, which holds for all 256 states of the model. bound states, it might even be a realistic grand uni-
In addition, the formula lgl = 2km in the N = 8 fied theory [14].
theory is not so mysterious if one realizes that the
N = 8 theory with 4 mass parameters is obtained by The author acknowledges the hospitality of the
dimensional reduction from 5 dimensions to 4. If EPFL of Lausanne and of the Gif-sur-Yvette Summer
we exchange a massless graviton in 5 dimensions be- School where these ideas were developed. Useful and
tween 2 massless particles of momenta p~, p~ in challenging conversations with W. Thirring, P.
the static limit we find the amplitude given by Choquard, D. Olive, A. Morel and L. Maiani (who
suggested to "heat up the vacuum") are also acknowl-
sff= (8rrG/q2) (Pl "P2 )2" (17)
edged. W. Nahm is thanked for his deciphering of
Setting the letter £ which reads in Aegyptian hieroglyphs
A
"shen" and signifies "binding" or "rope". E. Crem-
p~ = m l ( a , 0 , 0 , 0 , e l ) , (18)
mer is also warmly thanked for reading and correct-
A
ing the manuscript. J.H. Schwarz is also thanked for
p~ = m 2 ( l , 0 , 0 , 0 , e 2 ) , (19)
correcting an error in the evaluation of the upper
where e I = +1 for a particle, - 1 for an antiparticle limit on R~ set by the EbtviSs experiment.
(by convention), we get:
References
.~= (8rrG/q2)m~ m2(1 - el e2)2. (20)

The cancellation of forces is obvious for 2 par- [1] See S. Weinberg, Gravitation and cosmology (Wiley,
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[2] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973)
D = 5 translates in D = 4 into
1888.
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(Jaymac-Hollmann, 1973) p. 82; La Recherche no. 102
This phenomenon of cancellation of forces is (June 1979) p. 752.
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p. 64.
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[5] S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121
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it has an SU(3) × (U(1)) 2 invariance. At zero mass, Caltech (April 1979).
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[13] E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48;
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M, a c quark of mass M + 2m, as well as a triplet of [14] T.L. Curtright and P.G.O. Freund, Chicago preprint
a-type (Q = - 1 / 3 ) gravitinos of mass m, and a singlet EFI 79[125; /

gravitino of charge 0 and of mass M. Although poor J. Ellis, M.K. GaiUard, L. Maiani and B. Z'umino, un-
published.
in leptons and quarks, this might be a model for a fu-

267