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19 66ApJ. . .145 . .544H PERTURBATIONS OF AN EXPANDING UNIVERSE S. W. Hawking Department of Applied Mathematics and Theoretical Physics, University of Cambridge Received September 14, 1965; revised February 7, 1966 ABSTRACT Perturbations of a spatially homogeneous isotropic universe are investigated in terms of small varia- tions of the curvature. It is found that rotational perturbations die away. Density perturbations grow relatively to the background, but galaxies cannot be formed by the growth of perturbations that were initially small. In the steady-state universe small rotational and density perturbations die away. The behavior of gravitational radiation in expanding universes is also investigated. Its “energy den- sity” decreases at the same rate as that of electromagnetic radiation, although its active gravitational effect is only half as great. If a small amount of viscosity is present, gravitational radiation will be com- pletely absorbed in the steady-state universe but not in an evolutionary universe. I. INTRODUCTION Perturbations of a spatially homogeneous and isotropic universe have been investi- gated in a Newtonian model by Bonnor (1957), in a Newtonian approximation to a relativistic model by Irvine (1965), and relativistically by Lifshitz (1946) and Lifshitz and Khalatnikov (1963). Lifshitz, method was to consider small variations of the metric tensor. This has the disadvantage that the metric tensor is not a physically significant quantity. That is, one cannot directly measure it but only its second deriva- tives. It is thus not always obvious what the physical interpretation of a given perturba- tion of the metric is. Indeed it need have no physical interpretation at all, but merely correspond to a coordinate transformation. Instead it seems preferable to employ a method which considers small variations of the physically significant quantity, the curvature. This method has an additional advantage in the discussion of the behavior of gravitational radiation in an expanding universe, since it includes the interaction be- tween the gravitational radiation and the matter. This interaction was not present in the approximations mentioned above. II. NOTATION Space time is represented as a four-dimensional Riemannian space with metric tensor gab of signature +2. Covariant differentiation is indicated by a semicolon, and covariant differentiation along a world line by a prime. Square brackets around indices indicate antisymmetrization; round brackets, symmetrization. The conventions for the Riemann and Ricci tensors are a;6c :=: a cbVp , -R-ab RaFbp Also Vabcd is the alternating tensor. Units are such that k, the gravitational constant, and c, the speed of light, are 1. III. THE FIELD EQUATIONS We assume the Einstein field equation Rah gabR- Ta5 , where Tab is the energy-momentum tensor of matter. We will assume that the matter consists of a perfect fluid. Then, Tab = fJLUaUb + phab 544 American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H EXPANDING UNIVERSE 545 where is the density, p is the pressure, ua is the velocity of the fluid, uaua = 1, and hab = gab + uaub is the projection operator into the hyperplane orthogonal to ua: habUb = 0 . We decompose the gradient of the velocity vector ua as a;& &ab 1” & ab 4“ sab % ab where ua = ua-bUh is the acceleration, 6 = ua,a is the expansion, aab = (cd)hcahdb %habO is the shear, and ab Uc;dhcahdb is the rotation of the flow lines ua. We define the rotation vector a as Oa = VabcdOcdUb . We may decompose the Riemann tensor Rabcd into the Ricci tensor Rab and the Weyl tensor Ca&c(i: Rabcd Cabed gadRcb gbcRda R/gacgdb Cabed =: Cabcd Cabca “ 0 Cabcd Cabed is that part of the curvature that is not determined locally by the matter. It may thus be taken as representing the free gravitational field (Jordan, Ehlers, and Kundt 1960). We may decompose it into its electric, and “magnetic” components. Rab =:= CapbqMM j Hab = CaP qryqrbsMpMS , Cabcd = %UaEb cud - UcaEd - 2rabpquHcu - 2vcdrsUrHsaub , Eab = E(ab) , Hab = H(ab) , -Ea = Haa = 0 , EabUb = HabUb 0 . Eab and Hab each have five independent components. We regard the Bianchi identities, Rabcd;e = 0 as field equations for the free gravitational field. Then Cab cdd = Rcb,a + igc&E;a (Kundt and Triimper 1962). Using the decompositions given above, we may write these in a form analogous to the Maxwell equations: habEbc,dhcd + 3HabO)b 7abCdUbaCeHde = ihabib, U) habHbc;dhcd 3EabO)b rabcdUbTceEde = (/X + p)a , (2) JLE ab + hb)cdeCHfd,e EabO EC(aOib)c EC(aOrb)c facdVbpqrC/PdqEer T 2Hd(arb) cdeMM e = 2 ( “4“ P) &ab y (3) - h (aVb) cdeCEfd,e “f HabQ Hc(aOib) c Hc(aGb) c VacdeVbpqrUaH61, + 2Hd(aVb) cdeMfM 6 0 , (4) American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H 546 S. W. HAWKING Vol. 145 where JL indicates projection by hob orthogonal to ua (cf. Trmper 1964). The contracted Bianchi identities give Rab - IgabR)* = - Tab* = 0, (5) / + (m + p)0 0 , (6) (M + P)ua + p-bhha = 0 . The definition of the Riemann tensor is Ua;bc RapbcM? Using the decompositions as above we may obtain what may be regarded as equations of motion., 0 = 2cu2 2a-2 J02 + wa; + $P) i U) -Leo ab = %&abQ “f“ 2ocoe*J6C H- p;qhPakqb ? 0-a6 = Eab C0acC0C6 (Tac(T% %TabO (9) ” hab(22 2a2 + uc;c) -f ufaufb + up;q)hpahqb , where 2co2 = coa&coa& , 2a-2 = o-a-6 . We also obtain what may be regarded as equations of constraint. 0;bhba = f(wbc;6 + Tbcfi)hCa “ ub()ab + = A = QfihPa . If we assume an equation of state of the form p = (ju), then by equations (6) and (1A), PfihPa = A = a . This implies that the universe is spatially homogeneous and isotropic since there is no direction defined in the 3-space orthogonal to ua. In this universe we consider small perturbations of the motion of the fluid and of the Weyl tensor. We neglect products of small quantities and perform derivatives with respect to the undisturbed metric. Since all the quantities we are interested in, with the exception of the scalars ju, p, and 0, have unperturbed value zero, we avoid perturbations that merely represent coordinate transformation and have no physical significance. To the first order, equations (l)-(4) and (7)-(9) are Eab* skaflib y (13) Hab;b = (M + , (14) American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H No. 2, 1966 EXPANDING UNIVERSE 547 Eab + + hf (aVb) cdeCHfde “ K/* “t“ Hab + HabO hf (aVb) cdeUcEfd,e = 0 , (16) 0 = - ie2 + - Km + 3p), where r measures the proper time along the world lines. As the surfaces r = constant are homogeneous and isotropic they must be 3-surfaces of constant curvature. Therefore the metric can be written, ds2 = - dr2 + Wdy2 where 2 = (r), and dy2 is the line element of a space of zero or unit positive or negative cuivature. We define t by cU = l dr I Then ds2 = 22(- dt2 + dy2) . In this metric, ua = (,0,0,0), 3 _ 3 d ti2 dt Then, by equations (5) and (7), ( + p)3, i ( M + . (20) (21) If we know the relation between /x and p, we may determine . We will consider the two extreme cases, p = 0 (dust) and p = p/3 (radiation). Any physical situation should lie between these. The Case for p 0 By equation (20), /x = f/3, M = const. Therefore, 3 M E = const. a) For E 0: 2 = cosh VEM/3 ) / 1 , t=-L V(3/EM)sinh/(EM/3)i . JOj American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H 548 b) For E = 0: c) Fot 0: b) For E = 0: c) For E = 0, For p = ti/3, V. ROTATIONAL PERTURBATIONS wc:d hachhd = Mgbp P co = Therefore w = coo/2. Thus rotation dies away as the universe expands. This is in fact a statement of the conservation of angular momentum in an expanding universe. VI. PERTURBATIONS OF DENSITY For p = Owe have the equations, , 0 = _ 102 _ . American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H No. 2, 1966 EXPANDING UNIVERSE 549 These involve no spatial derivatives. Thus the behavior of one region is unaffected by the behavior of another. Perturbations will consist in some regions having slightly higher or lower values of E than the average. If the universe as a whole has a value of E greater than zero, a small perturbation will still have E greater than zero and will continue to expand. It will not contract to form a galaxy. If the universe has a value of E less than zero, a small perturbation can contract. However, it will only begin contract- ing at a time hr earlier than the whole universe begins contracting, where jr_8E To Eo Here t0 is the time at which the whole universe begins contracting. There is only any real instability when E = 0. This case is of measure zero relative to all the possible values E can have. However, this cannot really be used as an argument to dismiss it, as there might be some reason why the universe should have E = 0. For a region with energy 5E in a universe with E = 0, !14i(_T2+ ) TT215l(*-+ ) For E = 0, n = fr-2. Thus the perturbation grows only as r2/3. This is not fast enough to produce galaxies from statistical fluctuations even if these could occur. However, since an evolutionary universe has a particle horizon (Rindler 1956; Penrose 1964), different parts do not communicate in the early stages. This makes it even more difficult for statistical fluctua- tions to occur over a region until light had time to cross the region. For p = n/3, 6= -0- + uaa, 0=-p. As before, a perturbation cannot contract unless it has a negative value of E. The action of the pressure forces makes it still more difficult for it to contract. Eliminating 6, u MM-(M)2-fM3 + fMVo = 0, = ua.bh + uall = - 1 to our approximation; hacVchhaVb is the Laplacian in the hypersurface r = constant. We represent the perturbation as a sum of eigenfunctions Sn) of this operator, where S(n);cuc = 0 , hac(hbaSW.b) ;C = - Hi (w) * These eigenfunctions will be hyperspherical and pseudohyperspherical harmonics in cases (c) and (a), respectively and plane waves in case (b). In case (c) n will take only discrete values but in a) and b) it will take all positive values. American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H Vol. 145 550 S. W. HAWKING where no is the undisturbed density. Therefore Bmo - iBWno - BW (j|Mo2 - Mo) = 0 As long as /o n2/tl2, (n) will grow. For /o)w2/42, Ct + Dt1 . These perturbations grow for as long as light has not had time to travel a significant distance compared to the scale of the perturbation Until that time, pressure forces cannot act to even out perturbations. When n2/ )$ no, d2B(n) dt2 Therefore B(n) = CQrll2ei(n,vVt. We obtain sound waves whose amplitude decreases with time. These results confirm those obtained by Lifshitz and Khalatnikov (1963). From the foregoing we see that galaxies cannot form as the result of the growth of small perturbations. We may expect that other non-gravitational forces will have an effect smaller than pressure equal to one-third of the density and so will not cause rela- tive perturbations to grow faster than r. To account for galaxies in an evolutionary universe, we must assume there were finite, non-statistical, initial inhomogeneities. VII. THE STEADY-STATE UNIVERSE To obtain the steady-state universe we must add extra terms to the energy-momen- tum tensor. Hoyle and Narlikar (1964a) use Tab = nUaUb + phab CaCb + hgabCdP* , where Ca = U;o , U;aa := jaa , ja = (M F pMa Since Tah = 0 , / + (ju + )0 + uaCaCbh = 0 and (n p)ufa *F P;bhha hhaCbCdd = 0 . (22) There is a difficulty here, if we require that the “C” field should not produce accelera- tion or, in other words, that the matter created should have the same velocity as the matter already in existence. We must then have hhaCb = 0 . (23) However, since C is a scalar, this implies that the rotation of the medium is zero. On the other hand, if equation (23) does not hold, the equations are indeterminate (cf. Ray- chaudhuri and Bannerjee 1964). In order to have a determinate set of equations we will adopt equation (23) but drop the requirement that Ca is the gradient of a scalar. The condition (23) is not very satisfactory, but it is difficult to think of one that is more satisfactory. Hoyle and Narlikar (19646) seek to avoid this difficulty by taking a par- ticle rather than a fluid picture. However, this has a serious drawback since it leads to infinite fields (Hawking 1965). From equation (17), ca = - 1 - American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H No. 2, 1966 EXPANDING UNIVERSE 551 Therefore, + p)-n+p)d=-e P p+p+in+p) 6 Pl_ + P + ( + P)e For p, p p = 61 (ju + p). Therefore /* + /!. Thus, small perturbations of density die away. Moreover, equation (18) still holds, and therefore rotational perturbations also die away. Equation (19) now becomes 0 = J02 K/x + ?p) + 1 . Therefore 0 * /3(i p)- These results confirm those obtained by Hoyle and Narlikar (1963). We see therefore that galaxies cannot be formed in the steady-state universe by the growth of small per- turbations. However, this does not exclude the possibility that there might by a self- perpetuating system of finite perturbations which could produce galaxies (Sciama 1955 ; Roxburgh and Saffman 1965). VIII. GRAVITATIONAL WAVES We now consider perturbations of the Weyl tensor that do not arise from rotational or density perturbations, that is, = Hob* = 0 . Multiplying expression (15) by ucyc and expression (16) by ha(p7iq)rbsUrV8, we obtain, after reduction, Eab - (Ecdehcfhdgheic) ;ihk Wah0b + lEabB + EabBf + |02 + K/x 3) + (TabQ(p + ) + I(m/ + p) 0 . (24) In empty space with a non-expanding congruence ixa, this reduces to the usual form of the linearized theory, nEab = 0 . The second term in equation (24) is the Laplacian in the hypersurface r = constant, acting on Eab. We will write Eab as a sum of eigenfunctions of this operator: Eab = VAVab , where VabW = 0 , (V(ncdtehcfhdghek)-ihkihfahdh ri1 2 Vab, Then m,b;6 = o, m* = o. F -iyV -dA(n) oI 2 2Fa! dt Similarly, E ab ti2 SFai,() ( dP i o dt dt ) (Tab = XDVabM . Then, by expression (19), American Astronomical Society Provided by the NASA Astrophysics Data System 19 66ApJ. . .145 . .544H 552 dD dt S. W. HAWKING Vol. 145 Substitution in equation (24) yields + /lwr,+6.i av dt dt L 2 dt2 2dt/ 3 J + (M + /)+i(/ + ,)o2 = o. We may differentiate again and substitute for dDin)/dt. For and 2 1/w2, J (w) _rt_ J_ int A We so the gravitational field Eab decreases as 2“1 and the “energy” %(EabEab + HajbHah) as 26. We might expect this, as the Bianchi identities may be written, to the linear ap- proximation, gei JSCaU) Jabc Therefore, if the interaction with the matter could be neglected Cabcd would be propor- tional to 2 and Eab, Hab to 2_1. In the steady-state universe when n and 6 have reached their equilibrium values, Eab = (I + p)gab Thus Jabc = -Kca;& 6C;&1 = Thus the interaction of the “C” field with gravitational radiation is equ