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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 diagram due to the exchange of a massless graviton, and of a massless vector field A ~ , which we shall call the antigraviton. The coupling of these two fields to matter fields i (scalars), or k (Dirac spinors) is given by: 12= -(1/4k2)VVUaVUbR - F ~ uuab 1 VgUp gVO F~ v pa + V ~J xk ( iTu~ u - mk)X It, (1) k where we set: F~uv = ~ u A~v - avA~u ' (2) and % 4; = a - 'gi (3) %xk=~ xk-ig kA~Xk. (4) The vector A ~ is coupled to a conserved U(1) cur- rent ]~ and the c~arges gi, gk are a priori unrelated. Excerpts of talks given at the International Conference of Mathematical Physics (EPFL, Lausanne, August 20-25, 1979) and at the 1 l~me Ecole d'Et~ de Physique des Par- ticules de Gif-sur-Yvette (September 3-7, 1979) (spon- sored by the Institut National de Physique Nucl~aire et de Physique des Particules). 1 Laboratoire Propre du Centre National de la Recherche Scientifique, associ~ ~ l'Ecole Normale Sup~rieure et rUniversit~ de Paris Sud. The one-boson exchange graph is given by: 16rrG r , g2 1 In the static limit, setting k 2 = 4zrG, we find, for two particles of masses m, m', of charges g, g' the for- 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, hence repulsive if we have two particles or two anti- particles (assume that gi > 0 for all particles, gi = -gi for an antiparticle, gi = 0 for a self-conjugate particle under C, such as the 7, the gluons, etc.), and positive, hence attractive, between a particle and an antipar- ticle. We shall call antigravity the lihenomenon which occurs if the net force (gravity + antigravity) is zero between any two particles. This implies the universal formula: gi = kmi" (7) Let us first see if this is ruled out. As two neutrons attract each other, it seems that, immediately, anti- gravity must be eliminated However, the masses which appear in (1) are the quark and lepton masses, not those of the proton and neutron. Indeed, the antigraviton couples to e - , u, d, etc and sees their bare mechanical mass, since ]~ is conserved. In a com- posite particle, such as p, it does not see the gluons and its coupling to a proton is really given by: 265 Volume 88B, number 3,4 PHYSICS LETTERS 17 December 1979 j2 = k(m u ~u ?u Xu + md~dTUXd ), (8) while the graviton is coupled to the real mass (it sees the gluons). As a result, the force between 2 atoms (Z, A), (Z', A') is given by: F = (8nG/r 2) [MM'- 340340 '1, (9) where *~ M = Z(Mp + me) + (A - Z)M n, (10) M O=Z(2m u+m d+me)+(A-Z) (m u+2md) . (11) earth, we can replace M~earth/Mearth by For tile 3m. /M. Deviations from the equivalence principle u k' occur, and one finds that tile acceleration of two at- oms (Z, A), (Z', A') towards the earth differs by 6"//7 = (Z'A - ZA ') 3(m u/Mp) Ia2/MM ', (12) where /a 2 = me(M n - m u - 2rod) + 3(m u + rod)(M n - Mp) + (m u - rod)(M n + Mp). (13) In tile case of exact SU(2) symmetry (which is probably wrong anyhow), setting m u = md, we get: I~ 2 ~ meM n with M n - Mp ~ 1.7 MeV. Setting m u = I0 MeV (this is highly debatable), we get: /a 2 ~ 500 (MeV) 2, ~7/7 ~ 1.5 X IO-5(Z'/A ' - Z/A). (14) This is clearly bad news for antigravity, since the Ebtvos experiment gives 67/7 < 10 -9 and Dicke pushed tile limit down to 10 -11 [1]. Tile situation can be saved, however, if one of the scalar fields acquires a non-zero vacuum expectation value, as in the breaking of SU(2) X U(1) down to U(1). Then the ~ acquires a mass given by mQ = kmo (q3). (15) 2 > 0 can Tile reason why ~b acquires a v.e.v, while m e , IM M should really readM - hE M -z~Ewhere~LE p, n p , n is the nuclear binding energy per nucleon. This modifies slightly the proton and neutron masses which we shall, for the sake of simplicity, take to be equal both to 1000 MeV. Similarly, the ratio mu/M p will be small (10-2). be due to radiative corrections (of order g2t,ong,o~ a or k 2 , which can turn a potential having a minimum at the origin into one which has a maximum, the true minimum being elsewhere [2]. To fix ideas, we can set m e ~ 1 GeV, (qS) ~ 1 GeV, which gives to the a tiny mass rn~ ~ 10 -19 GeV, or a Compton wave- length R~ = 1/rn~ ~ 1 km ,2 In this case antigravity is saved, since the poten- tial is given by v = - (a / r ) [MM' - MM ' exp -(r/RQ) f iR ,/R~)], (16) where f (x) = (3/x3)[x chx - shx] , and R~ is the earth's radius. This factor arises because for the Yukawa potential of a spherical homogeneous object, all the sources cannot be considered to be at the cen- ter ("skin effect"). This sets an upper limit on RQ of the order of 2 m. To realize an antigravity device based on this idea would be cumbersome, but, in theory, possible: one would have to "heat up the vacuum" to reach the phase where (@ -- 0, but also the one where the quarks 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 this cost it might be possible if we know how to "heat up the vacuum" without destroying the engine. However, this clearly belongs either to UFOlogy [3] or Science Fiction [4] not yet to Technology. Now what about the strange relation gi = krni? This universal formula clearly comes from the sky in the previous discussion. However, in extended N = 2 supergravity, it was shown that the spin-1 partner of the graviton couples to a massive matter scalar supermultiplet with precisely this strength. This was deduced from the algebra of supersymmetry trans- formations [5] and it was guessed [6] (but unproved) that N = 2 supergravity led to antigravity. In 1977 K. Zachos worked out the coupling o fN = 2 super- r t Q gravity (V , qJ' (i = 1,2), A ) to a massive matter multiplet (~i, q~') and discovered the phenomenon of antigravity [7] within the framework of super- gravity. In the case N = 8 it is now known that anti- gravity also occurs [8,9]. In the spontaneously broken ,2 This can also occur classically if (@ 4: 0, provided that in the original lagrangian, A~ couple with strength +-2khul even when ~2 ~ 0. We thank E. Cremmer for pointing this out. 266 Volume 88B, number 3,4 PHYSICS LETTERS 17 December 1979 N-- 8 supergravity theory with masses, g = +-2km holds (note the factor of 2) but there is a massless Brans-Dicke scalar, coupled with strength x/~ km so that there is a net cancellation of gravitational forces, which holds for all 256 states of the model. In addition, the formula lgl = 2km in the N = 8 theory is not so mysterious if one realizes that the N = 8 theory with 4 mass parameters is obtained by dimensional reduction from 5 dimensions to 4. If we exchange a massless graviton in 5 dimensions be- tween 2 massless particles of momenta p~, p~ in the static limit we find the amplitude given by sff= (8rrG/q2) (Pl "P2 )2" (17) Setting A p~ = ml (a ,0 ,0 ,0 ,e l ) , (18) A p~ = m2( l ,0 ,0 ,0 ,e2) , (19) where e I = +1 for a particle, -1 for an antiparticle (by convention), we get: .~= (8rrG/q2)m~ m2(1 - el e2)2. (20) The cancellation of forces is obvious for 2 par- ticles or 2 antiparticles. The relation (p~7) 2 = 0 in D = 5 translates in D = 4 into p2_ m 2 = 0. (21) This phenomenon of cancellation of forces is known to occur also in vector scalar systems, both for classical fields [10] and magnetic monopoles [11] and has the same interpretation [12]. The N = 8 theory with 4 mass parameters [9] is of particular interest. In the limit where ml,2, 3 = m and m 4 =M, it has an SU(3) (U(1)) 2 invariance. At zero mass, it has a graviton, 8 gluons of electric charge Q = 0, and 2 vectors, one which is identified with the pho- ton (7), and one which is the antigraviton (~) and gauges the charge g = +2krn for all states of the theo- ry and thus can hardly be a Z . In addition, the mod- el contains a a quark of mass 2m, an electron of mass 3m, a u quark of mass M - 2m, 8 gluinos of mass M, a c quark of mass M + 2m, as well as a triplet of a-type (Q = -1 /3) gravitinos of mass m, and a singlet gravitino of charge 0 and of mass M. Although poor in leptons and quarks, this might be a model for a fu- ture unified theory of gravity + matter, and predicts antigravity. If the SU(8) local invariance discovered by Cremmer and Julia [13] in the massless case leads to the appearance of SU(8) gauge fields through bound states, it might even be a realistic grand uni- fied theory [14]. The author acknowledges the hospitality of the EPFL of Lausanne and of the Gif-sur-Yvette Summer School where these ideas were developed. Useful and challenging conversations with W. Thirring, P. Choquard, D. Olive, A. Morel and L. Maiani (who suggested to "heat up the vacuum") are also acknowl- edged. W. Nahm is thanked for his deciphering of the letter which reads in Aegyptian hieroglyphs "shen" and signifies "binding" or "rope". E. Crem- mer is also warmly thanked for reading and correct- ing the manuscript. J.H. Schwarz is also thanked for correcting an error in the evaluation of the upper limit on R~ set by the EbtviSs experiment. References [1] See S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972)p. 11. [2] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888. [3] J. McCampbell, UFOlogy, L.C.C.C.N. 73-93488 (Jaymac-Hollmann, 1973) p. 82; La Recherche no. 102 (June 1979) p. 752. [4] A.C. Clarke, Profiles of the future (Pan Books, 1962) p. 64. [5] S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393. [6] J. Scherk, La Recherche (October 1977) p. 878. [7] K. Zachos, Phys. Lett. 76B (1978) 329; Ph.D. Thesis, Caltech (April 1979). [8] J. Scherk and J.H. Schwarz, Phys. Lett. 82B (1979) 60; Nucl. Phys. B153 (1979) 61. [9] E. Cremmer, J. Scherk and J.H. Schwarz, Phys. Lett. 84B (1979) 83. [10] C. Montonen and D. Olive, Phys. Lett. 72B (1977) 117. [11] N.S. Manton, Nucl. Phys. B126 (1977) 135. [12] D. Olive, Imperial College preprint (1979). [13] E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48; LPTENS preprint 796, to be published in Nucl. Phys. B. [14] T.L. Curtright and P.G.O. Freund, Chicago preprint EFI 79[125; / J. Ellis, M.K. GaiUard, L. Maiani and B. Z'umino, un- published. 267