Annalen der Physik. 7. Folge, Band 41, Heft 3, 1984, S. 172-178 J. A. Barth, Leipzig Cosmological Solutions of Bi-metric Theories of Gravitation By D.-E. LIEBSCHERand J. MUCKET Zentralinstitut f i i r Astrophysik der Akademie der Wissenschaften der DDR, Potadam-Babelsberg Abstract. The evidence in solar system tests of gravitation theories shows that in cosmological models the propagation velocities of gravitation and light may differ only by action of the local gravitational potential. The condition of approximately equal propagation velocities is shown to be consistent with cosmological models of some bi-metric tetrad theories. The way to analyze the situation is explained. We find a connection between the linear approximation of the theories with those properties of cosmological models, which are responsible for the local propagation of gravitation. Kosmologische Lbsungen Bi-metriseher (fravitationstheorien Inhaltsubersicht. Die Beobachtungen allgemeinrelativistischer Effekte im Sonnensystem zeigen, daB in kosmologischen Modellen sich die Ausbreitungsgeschwindigkeit der Gravitetion von der dea Lichts nur durch die Wirkung lokalen Gravitationspotentials unterscheiden kann. Es wird gezeigt, da.13 die Bedingung der angeniiherten Gleichheit der Ausbreitungsgeschwindigkeiten mit bimetrischen Gravitationstheorien vereinbar ist. Die Analyse der Theorien wird erliiutert. Es zeigt sich, da13 es eine enge Verbindung zwischen der linearen Niiherung der Theorien und den Eigenschaften der kosmologischen Liisungen gibt, die die lokale Ausbreitung der Gravitation bestimmen. Bi-metric Gravitation Theories The authors, remembering the long-term discussion of bi-metric theories with D. D. IVANENKO and the seminar of the department of theoretical physics a t the MGU, would like to present this paper in his honour. Bi-metric and tetrad theories have always been the means to discuss alternative ways to describe gravity, and to study real and supposed characteristics of EINSTEIN’S General Relativity [1- 221. They have been shown to be adequate alternatives t o test relativistic gravitation [20, 23, 241. Mainly they have been constructed ad hoc, and only later i t was shown that they all fit into a common scheme, which exhibits the main characteristics of bi-metric theories [25, 261. The main objective against the most widely discussed bimetric theory of ROSEN[21] is the behaviour of cosmological solutions [27], which yields wrong PPN-characteristics, if the effective metric of the Riemann space time does not turn out to be conformal to the Minkowski background metric. I n fact, if gik is not proportional to q i k in the cosmological solution, the propagation velocity of light (determined by g i k ) differs from that of gravitation (in general determined by l;lik). The PPN-parameter will show this straightforward [241 : D .-E. LIEBSCHER et al., Cosmological Solutions of Bi-metric Theories of Gravitation 1i 3 N. ROSENavoids this problem later by modifying his theory in such a way, that the cosmological model becomes itself the background metric, and is no more object of the field equations [28]. We will show that the general problem does not need such a complication in identifying a subclass of theories allowing for proportionality of g i k with qik. We look for cosmologies of bi-metric theories with respect to the condition in order to single out viable theories. We discuss the general class of homogeneous bimetric theories presented in [ 2 6 ] . In deriving the equations for Robertson-Walker models we find the desired subclass. Homogeneous Bi-metric Theories Bi-metric theories are in fact multi-metric. Given two symmetric tensor fields qik, and gik, we may introduce the root of their quotient by .the anholonomous transformation coefficients: g i k = qlnaabr* (3) If both tensors are metric tensors, we interpret these coefficients as the anholonomous transformation of a tetrad hp (orthonormal with respect to q i k ) into a tetrad hf (ortho0 normal to g i k ) : h% 4 -,,k - hAa41 ) (4) AhR gik = q A B h i k - q A B ~ f ~ ~ a = ! @q1rna~'r* Instead of using g i k as dynamical variable we may use ar. In doing this we get access to the matrix functions of a to construct Lagrangians. Furthermore, every function of a leads to a metric tensor, formally Fik = qlmfi(a) fk"(a) - (5) Of course these metrics have no physical meaning per se. They acquire such a physical meaning, if they are used to construct terms of the Lagrangian, and we will use such a construction to describe different propagation properties of gravitation. The matter Lagrangian is supposed to be constructed by force of the weak equivalence principle with g i k and the covariant derivative induced by it. We will not try to change here this concept of General Relativity. Only the Lagrangian for the gravitational field will be generalized. Its building blocks are - the flat-space derivatives of a in the combinations li i = qij(a )u ak,w(a')t, y/,vw - the scalar quotient of the determinants 1: = det a , ' - the metric G,, = qii(a")i(an)$, G,,G"" = SV,, to contract the covariant tensor indices. All the powers of a used in these construction elements could be substituted by other functions of a. In using only power functions we Ann. Physik Leipzig 41 (1984)3 174 get homogeneous theories. The Lagrangean considered reads L = I/-r (det a)m~ U V W ~ X Y Z [A,GuvGx~GWZ + A,,GuxGv~Gwz+ A,Gu~GVxGwz + A31GuWGxzGVU + A3,GVwGUZGux A 33 (GUWGUZGXV + GVWGXZGUV) + A 41(GUVGZWGYZ + GXUGVWGUZ1 f A (GUVGMWGVX + GxUGUWGuz ) 42 f + A5,G*GxWGYZ+ A,2GVzQUwGux f A,3(GUzG~WGVx + GVzGxWGu~ )I. We use the abbreviations A,, = A,, = iA5k9 + + f f = 2A33, A, = A, = 2A53, A, = A, A,. The coefficients A are numerical constants, which together with the exponents 1, m, n, and r define the theory. Special emphasis should be given to the exponent n. If n = 0, the propagation velocity of gravitation is determined by the flat background, and in weak and quasistatic fields gravitation is faster than light [29]. For n = 1, gravitation propagates by the same metric as other fields (Gt, = gik). For n > 1, gravitation is slower than light in weak quasistatic fields. Terms with indices contracted by cikZrn would destroy parity, and are not considered here. The class presented contains - elementary linear tetrad theories (m = n = 1 = r = 0), - linear theories for the metric tensor (m = n = r = 0, I = 1) [16], - the KOHLER class [8] (m = n = I = 1, r = 0), which itself contains - EINSTEIN'S General Relativity (m = n = 1 = 1, r = 0 , A , = -1, A , = A, = I , + 6 + + -21, TREDER'S equations [l5, 20, 251 (m = n = 1 = 0, r = -l), - ROSEN'S bi-metric theory [all (m = n = r = 0, 1 = -1). In the last two cases A , = -2A, = 1, the other coefficients vanish. The following properties are given without proof. Two cases are of special importance. First, if only A,, A,,, and A , do not vanish, then we get wave equations for all components of the tensor a straightforward (Poisson case). We are forced, otherwise, to define gauge-reduced quantities to exhibit the propagation properties. Secondly, we may postulate - in analogy t o EINSTEIN'S theory that the linear approximation allows for gauge transformations of the solutions. That means, together with a solution ui, also ui. Buh with - the special class of bi-metric tetrad theories developped from + Buk = + uj.,/tl (7) shall solve the field equations [16]. Gauge invariance in the linear approximation is equivalent to + A, A4 = 0, A,, f A63 = 0, -421 + A61 = 0, f + A4 = 0. -462 4 3 (8) D.-E. LIEBSCHER et al., Cosmological Solutions of Bi-metric Theories of Gravitation 17.5 TWO combinations of coefficients determine the behaviour of linear effects, C, = A , A,, and C , = A , A , A,. I n the gauge case just considered both C,, and C, vanish. The coefficient A , is chosen t o be 1, otherwise we would have to redefine the coupling constant x . The light deflection has the EINSTEIN value, if + + + + ( A , - A,) c, = (4 A,) c,. (9) I n the Poisson case we get 2A1 A , = 0 . I n the case of C, = 0 we find A, = A , = 0. The only nonlinear effect of the post-Newtonian approximation, the perihelion shift, is determined by the exponents of a in the Lagrangian. We get the EINSTEIN value, if m+n- 1 -r = 1. (10) + We will show in the following section that the desired subclass is given by 4C, + C, = 0, and 2n = m + 1. (11) Cosmological Models We want to inspect cosmological models i.e. the question how the Robertson-Walker metrics + + ds2 = H2(t)dt2 - A ( t ) ( d 9 d$ dz2) (12) fit into the theory. We do so only for the simplest case, in which t , x, y, and z are' Cartesian coordinates of the back-ground metric, i.e. qik shall take the usual numerical values. One can see by simple inspection that the field equations are reduced to two equations for H , and A , if we substitute into the field equations the form a!JH, a l - a 22 -- a 3 3- - A , a i = 0 , if i# k. (13) Therefore, we may substitute it already in the Lagrangian, and we may derive the field equations by variation with respect t o H and A . The action is reduced t o s = $ dt L + smatter, (14) with k = 2n - 1 - r , + A,, 01 = 3A1 B = 3c1, y=c,+c,. 6L and -contains no second derivatives. 6H For a n ideal fluid we get the Euler-Lagrange equations I n the gauge case, we find?!, = y = 0, (1.5) Ann. Physik Leipzig 41 (1984)3 176 I n addition, by the construction of Smatter we get the energy conservation equation in the form This relation is independent of the gravitational part of the Lagrangian. The structure of the gravitational part leads to another conservation law. L does not depend on t explicitely. Therefore, aL H- aL +A - L + %HA3@= const, aa aA which may be reduced by the homogeneity of L in A and H to + L xHA3e = M . This constant M means something like the energy content of the universe, including gravitation. We shall see, that a necessary condition to get the models we want to find is M = 0. The Euler-Lagrange equations read explicitely 2yD(H) + 2,9D(A)= H2n--nzfk-1A-3m ( ( m - 2n) L - %HA3@), (18) 2,9D(H) 2aD(A) = H2n-mA-3mfk-'(3mL 3xHA3p) . (19) with H (m - 2n) - + 3m H The first question to put is the question about the possibility of A = H . We have two equations for one variable A , if A = H . They are identical, if (a ,9) ((m - 2n) L - xHA3e) = (,9 y ) (3mL 3xHA3p). (21) Taking into account, that L = M - .HA", we bave to compare the coefficients of M , e, and p. Therefore, we find the following cases. 1. I n a radiation universe, e = 32, = NA-4, and eq. (21) is just a condition M = M ( N ) for the two integration constants M , and N . 2 . Any other equation of state p = p ( e ) requires ,6 y = 0 in order to allow for H = A. In t*hissecond case, we still have to ensure that (OL +j3) (m - 2 n + 1) = 0 . Again, we have to restrict the parameters by m - 2n 1 = 0. (22) The sum OL ,9 cannot be zero, or the equation of state is restricted again to radiation. If m - 2n 1 vanishes, the two Euler-Lagrange equations are equal modulo eq. (16). Their first integral is given by eq. (17), which represents the equivalent for the corresponding Friedmann equation. If we denote the cosmological time by + + + + + + + + + dz = H (t ) at, we may write this equation It coincides with the Friedmann expansion law, if 3m - 2k - 1 = 0. (23) D.-E. LXEBSCHER et al., Cosmological Solutions of Bi-metric Theories of Gravitation 177 After finding the condition for existence of solutions with H = A in ikl = 0, and 1 = 0, we have t o put the question about the behaviour of the other solutions to the solutions with H = A . We got one equation free of source. If = - y # 0, it reads rn - 2 n + D(H) = D ( A ) . The case o f @= y = 0 will be considered later. Setting H =A(l E), we get for the first approximation in E the equation (25) + For a n expanding universe 3m + 1 > 2k, c -> 0 ) we find damped oszillations in E , if and The case of @ = y = 0 represents theories, which are gaugeinvariant in the linear approximation. I n contrast to the case just considered, eq. (18) yields no further condition on H and A. That means that we are free to choose any function H ( t ) , and we get again eq. (24), with LX @ = -2. This is an equation independent of H , because we use the cosmological time, eq. (23). It corresponds to the Friedmann equation for the Hubble parameter. H is now open t o be gauged t o H = A , and there is no question of stability of this choice. + Conclusion Bi-metric tetrad theories may well allow for cosmologies, which do not lead to preference in the local system of coordinates by different causal cones of the background and the effective metric gik. Therefore, bi-metric theories are not excluded by the evidence for the PPN-parameter a, to be zero. Furthermore, the condition of gauge invariance in the linear approximation just leads to these cosmologies. However, we get in these theories, for universes filled with matter, the equality H = A only in the case of eq. (22), 2n=mf1. If we now remind the reason for our effort, the evidence for a, = 0, we have also to take into account the perihelion condition, eq. (lo), k f m = n + 1. Combining both conditions, the Priedmann equation will be and the radiation universe expands like 2 - A oc zsn-4, the dust universe like A oc 2 - t81*-5. (28) 178 Ann. Physik Leipzig 41 (1984) 3 The conclusion a t hand would be, that n = 1 and by force of eqs. (22), and (28) also m = k = 1 would be strongly favoured by primordial nucleosynthesis data. However, if n = 1, there is no problem with the parameter 0 1 ~ . Now, the metric Gik of eq. (6), ruling the propagation of gravitation, does not differ from gik, ruling the propagation of non-gravitational fields. Therefore, the coefficient 0 1 ~vanishes independently of the relation between gik, and qin. The subclass n = 1 of hi-metric theories meets the a*condition without restriction on the other parameters k, m, and A . A c k n o w l e d g e m e n t s . The authors are obliged to H.-J. TREDER,and U. KASPER for valuable and helpful discussions. References [l] EINSTEIN,A. : SBer. PreuB. Akad. Wiss. 1928, 217. [2] EINSTEIN,A.: SBer. PreuB. Akad. Wiss. 1929, 1, 156. [3] EINSTEIN,A.; MAYER,W.: SBer. PreuB. h a d . Wiss. 1931, 257. [4] IVANENHO, D. D.; FOCK,V. A.: Phys. Z. 30 (1929) 648. D. D.; FOCK, V. A.: Z. Phys. 54 (1929) 798. [5] IVANENKO, R.: SBer. PreuB. Akad. Wiss. 1938, 466. [6] WEITZENBOECH, N.: Phys. Rev. 57 (1940) 147, 150. [7] ROSEN, [8] KOEUER,M. : Z. Phys. 134 (1953) 306. A.: Z. Phys. 139 (1954) 518. [9] PAPAPETROU, [lo] BELINFANTE, F. J.; SWMART,J. C.: Ann. Phys. NY 1 (1957) 168. [ll]YILMAZ, H.: Phys. Rev. 111 (1958) 1417. D. D.; BRODSPI,A. M.; SOPOLIK,G. A.: JETP 41 (1961) 1307. [12] IVANENKO, [13] IVANENHO, D. D.: Einstein-Symposium Berlin 1966, p. 300. [14] IVANENPO, D. D.; in: Relativity, Quanta, and Cosmology, New York: Johnsson Reprint Corp., 1980, p. 295. [15] TREDER, H.-J.: Ann. Physik Lpz. 20 (1967) 194. P.; BARBOUR, J. B.: Z. Phys. PO3 (1967) 82. [16] MITTELSTAEDT, [17] MITTELSTAEDT, P. : Z. Phys. 811 (1968) 271. [18] LIEBSCHER, D.-E.; TREDER, H.-J.: GRG Journal l(1970) 117. [19] NI, WEI-TOU:Astrophys. J. 169 (1971) 141. [20] TREDER, H.-J. (ed.): Gravitationstheorie und Aquivalenzprinzip, Berlin: Akademie-Verlag1971. [21] ROSEN,N.: GRG Journal 4 (1973) 435. A. A.; FOLOMESEKIN,V. N.: TMPH 32 (1977) 147, 167, 291. [22] LOCUNOV, [23] NORDTVEDT, K.; WILL, C. M.: Astrophys. J. 185 (1973) 31. D.-E.: Astron. Nachr. 295 (1974) 11. [24] KASPER,U.; LIEBSCHER, [%I KASPER,U.; LIEBSCHER, D.-E.: Ann. Physik Lpz. 30 (1973) 129. D.-E.: Ann. Physik Lpz. 34 (1977) 402. [26] LIEBSCHER, [27] EARDLEY, D. M.; WILL,C. M.: Astrophys. J. 812 (1977) L 91. [28] ROSEN,N.: Astrophys. J. 211 (1977) 357. D.-E.: Ann. Physik Lpz. 34 (1977) 295. [29] LIEBSCHER, Bei der Redaktion eingegangen am 24. Mai 1984. Anschr. d. Verf.: Prof. Dr. sc. nat. DIERCK-EKKEHARD LIEBSCHER Dr. rer. nat. JAN MUCHET ZI f. Astrophysik d. AdW der DDR DDR-1502 Potsdam-Babelsberg Rosa-Luxemburg-Str. 17a

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