Energy-momentum tensor of fields in the standard cosmology

General Relativity and Gravitation, Vol. 20, No. 3, 1988

Energy-Momentum Tensor of Fields in the Standard Cosmology

Philip D. Mannheim 1 and Demosthenes Kazanas 2

Received March 27, 1987

With the use of the equations of motion of massless fields moving in a curved Friedmann-Robertson-Walker universe, we show, in some simple cases, that the energy-momentum tensor of a maximally 3-space symmetric distribution of the fields (i.e., an incoherent averaging over a complete set of modes of the field propagating in a Robertson-Walker background) has the standard perfect fluid form. As far as we know such an explicit demonstration, as well as the establishment of the compatibility of the equations of motion of the gravitational field with such an incoherently averaged source in the standard cosmology, has not previously been presented in the literature. Our results are found to hold for any value of the spatial curvature of the universe.

1. I N T R O D U C T I O N

In the s t a n d a r d F r i e d m a n n cosmology the universe obeys the cosmologica l pr inciple of being bo th i so t ropic and h o m o g e n e o u s in the 3-space associa ted with the three o rd ina ry spat ia l coordina tes . In such a cosmology the metr ic is then of the R o b e r t s o n - W a l k e r (R. W.) form and can be descr ibed by a scale p a r a m e t e r R ( t ) and a pure n u m b e r K (usual ly t aken to be 1, 0, or - 1) which serves to categor ize the curva ture of the 3-space. In this cosmology the Ricci tensor is also max ima l ly symmet r ic in the 3-space, so tha t the Einste in tensor, R~v - g ~ v R ~ / 2 , has the fo rm of a perfect fluid. The Einstein g rav i t a t iona l field equa t ions then oblige the e n e r g y - m o m e n - tum tensor to also be of the perfect fluid form and serve to fix the t ime dependence of R( t ) . They do not, however , de te rmine any value for K.

1 Department of Physics, University of Connecticut, Storrs, Connecticut 06268. 2 Laboratory for High Energy Astrophysics, NASA/Goddard Space Flight Center, Greenbelt,

Maryland 20771.

201

0001-7701/88/0300-0201506.00/0 �9 1988 Plenum Publishing Corporation

202 Mannheim and Kazanas

While the discussion of the Einstein equations is more or less analogous in the two main cases of interest, namely the matter-dominated and radiation-dominated eras, the actual a priori derivation of an explicit form for the energy-momentum tensor in the two cases is quite different. Specifically, for matter, and also for null rays, the covariant conservation of the energy-momentum tensor which follows from the Bianchi identities imposes an equation of motion on matter and on null rays which forces them both to move on the geodesics associated with a curved background metric. An incoherent maximally 3-space symmetric averaging over a complete set of noninteracting matter particles (i.e., dust) or null rays moving on these geodesics in the standard cosmology then gives the energy-momentum tensor its familiar generally covariant perfect fluid form.

However, if the source of the Einstein tensor is composed of fields rather than particles the above argument fails. Specifically, for fields the conservation of the energy-momentum tensor does not uniquely impose an equation of motion on the fields. Rather, this conservation will in general be satisfied by a whole family of field solutions. To force a unique wave equation on the fields it is necessary to also vary the actiori independently with respect to them. Thus for fields, as opposed to particles, the equations of motion of the fields are independent of the Einstein equations and thus contain additional information. This fact may be understood as follows. For particles on geodesics it is possible to go into a locally fiat coordinate frame at a position of the particle and apply the equivalence principle. For waves, however, which fill all space and are thus global in nature, we need to know the properties of the waves away from any point where the coordinate frame is made locally flat. Hence their motion is not fixed by the gravitational field equations alone, and is thus only constrained but not uniquely determined by the principle of equivalence. Thus, finally, in order to establish a perfect fluid form at all for an energy-momentum tensor composed of fields, we need to appeal to the equations of motion of the fields themselves and not just to the Einstein equations.

In this paper, therefore, we study and in fact derive the perfect fluid form for fields in the standard cosmology in some simple cases, namely scalar fields in an arbitrary R.W. metric, and then Maxwell fields in a spatially flat R. W. metric. The case of Maxwell fields in a spatially curved R.W. metric has-some technical difficulties which prevent our carrying through the complete calculation analytically. There do not appear to be any conceptual difficulties, however. Interestingly, and in a sense disappointingly, again we will find that even with the imposition of the field equations the spatial curvature K is still not determined.

While the energy-momentum tensor for fields has always been taken to have the perfect fluid form, as far as we know no explicit constructive

Energy-Momentum Tensor of Fields in the Standard Cosmology 203

derivation has previously been presented in the literature. The major g0al of this paper is to provide such a derivation. Our desire is to show that the coupled system of Einstein gravity with a massless field as its source possesses the standard cosmology as a self-consistent solution. (In principle there may also be other solutions to this coupled system, solutions with non-R. W. metr ics--our aim here, however, is not to find every possible solution to the coupled system but only to show that the standard R. W. one is in fact legitimate). To explore the full content of the coupled system of equations of motion we proceed as follows: We first assume a background R.W. metric and then solve for the propagation of the massless source in this background to obtain the appropriate modes of the source field. We next make an incoherent average over a bath of the entire set of these modes to see whether or not this averaging procedure yields a perfect fluid energy-momentum tensor. Should it yield such a fluid, insertion of this energy-momentum tensor into the Einstein equations would then recover the initially presupposed R. W. metric, and thus establish the self-consistency of the procedure and hence of the standard cosmology. With regard to the general structure of the R. W. metric, as noted above it is determined by symmetry considerations alone, except that R(t) and K are left as undetermined parameters to then be fixed dynamically. The dynamics of the Einstein equations then fixes R(t) but not K. It is thus of some interest to see whether the additional dynamics associated with the field equations of the source of the gravitational field then provides the additional information which selects out a specific value for K. As we shall see, this turns out not to be the case but rather we find that the field equations of gravity and its incoherently averaged source turn out to be compatible for any value of K in the standard cosmology.

In order to be able to discuss a tractable case we consider in detail a free massless scalar field in a curved R.W. metric (i.e., free except for gravity). Since our interest is in the radiation-dominated era we need an energy-momentum tensor that is generally covariantly traceless. With the canonical energy-momentum tensor for massless scalar fields not being traceless we instead use the traceless new improved energy-momentum tensor of Callan, Coleman, and Jackiw [ 1 ]. This tensor, if defined in a curved space simply as the general covariant generalization of the appropriate flat space form, is not, as we show below, in fact covariantly conserved in a curved space. However, this deficiency can readily be remedied by including some explicit curvature-dependent terms in the definition of what we call the generally covariant new improved energy-momentum tensor. With the inclusion of these additional curvature-dependent terms in the energy-momentum tensor we then find that they enter in just the right way to enable us to recover the standard cosmology at the end.

204 MannheimandKazanas

Our paper is organized as follows. In Section 2 we discuss some general properties of scalar fields in a curved R. W. background metric and analyze the covariant conservation properties of the new improved energy- momentum tensor. In Section 3 we show how an isotropic, homogeneous 3-space distribution of scalar fields and then of Maxwell fields leads to the perfect fluid form for the energy-momentum tensor in a flat space. This analysis is extended in Section IV to massless scalar fields in a curved R. W. space, and the Maxwell case in a curved R.W. space is discussed briefly in Section 5.

2. MASSLESS SCALAR FIELDS IN A CURVED R. W. B A C K G R O U N D

In order to describe the R.W. metric we follow the conventions and notations of [2]. The line element is given by

g"~ dxU dxV = -dt2 + R2(t) (1 - -~ r 2) -~- r2 dO2 q- r2 sin2 0 &b 2 (1)

and the nonvanishing elements of the Ricci tensor are given by (i, j = 1, 2, 3)

31~ Roo-R(t) Ro.= -[R(t) i~+2kZ+2K]y~ (2)

where ~v[ = giffR2(t)] is the 3-space metric. Thus the Ricci scalar is given by

6 R~= -R---~) [K + R(t) i~ + R2] (3)

For the generally covariant action of a massless scalar field S(x) coupled to gravity we take the conformally invariant action of [ 1 ]. (The utility of this particular conformal invariant action is that it yields a traceless energy- momentum tensor and hence a vanishing Ricci scalar, as required for the radiation-dominated era of the standard cosmology.) The required action is given by

1 R', (4)


The variation of this action with respect to g~,,(x) yields the Einstein equations

~g~,uR ~)

S~S,~ �89 ~' 1 2 2 ~: = - g ~ , , ( S ),~

- r ,v (5)

and serves to define an energy-momentum tensor T.v. The particular T~ defined by (5) is the naive general covariant generalization of the flat space energy-momentum tensor given in [ l ] . With the use of the Bianchi identities we see immediately that

T ;v = ~( 2R~v- g ~ R % ) SS~ (6)

so that even while this particular 7"u~ is conserved in a flat space it is not in general conserved in an arbitrary background curved space. (As can be seen from (6), we note in passing that it is in fact conserved in Ricci flat spaces.) To remedy this deficiency we may instead consider as a candidate energy-momentum tensor the curvature dependent tensor

, i 2 i c~ Tu~ = T~u -- ~S (R~,~ -- ~g,,~R ~) (7)

a tensor which also reduces to the energy-momentum tensor of [1 ] in the flat space limit. This latter tensor obeys the Einstein equations

1 ct = -8~cGT'~ (8) R ~ - ~ g ~ R ~

and thus is covariantly conserved. As we see in Section 4, it will be T'~ rather than T,~ which will eventually produce the familiar perfect fluid tensor of the standard cosmology.

For the scalar field, variation of the action with respect to S ( x ) yields the equation of motion

1 S";. = - ~ S R ~ (9)

and enables us to reexpress the tensor Tu~ in the form

Ttlv-- 2 - x S . S v - ~g~,~S~S~ - 1 S S ~ ; v - ~- g , ,vS2R%

Taking the trace of (5) now yields

S 2 - - [ ~ R % = S2R%

(10)

(11)

206 Mannheinl and Kazanas

Hence, since G is not equal to infinity (i.e., since gravity is not induced here) we conclude that the Ricci scalar vanishes so that both T~ and T'~, v are traceless, as is required in the radiation-dominated era. With the vanishing of the Ricci scalar we see from (3) that the most general form for the time dependence of the scale parameter R(t ) is given by

R2(t) = a + bt - Kt 2 (12)

where a and b are constants of integration. Further, in this case the scalar field equation of motion reduces to

S~;~,= ( _ g ) 1/2 ~/z[-(_g)l/2 Gq#S ] = 0 (13)

In order to solve the scalar field equation of motion of (13), it is momentarily advantageous to begin with (9) and (3) and not yet require the Ricci scalar to vanish. In the R.W. case the scalar field equation is separable. It is convenient to set

f ( p ) g(r, O, qb) S (x ) - (14) R(t)

where we have introduced the so-called conformal time variable

P = r j , dt' (15) R(t ' )

The scalar field equation of motion is then found to reduce to (7 denotes the determinant of the spatial metric)

1 l- S+ ] 1 f ( p ) KS(p) =g(r , O, (~) ~--1/2 ~i[])l/2~ij ~jg(r, O, ~)]

= - 2 2 (16)

In (16) we have introduced a separation constant --2 2. Thus we see that our solutions are harmonic in the conformal time variable p. For a real scalar field we need all the cos cop and sin cop solutions to (16) where the frequencies have to satisfy

co2 = ,,],2 -{- K (17)

The spatial curvature thus acts somewhat like a mass term. For the spatial dependence of the real scalar field we can further set

g(r, 0, ~ ) = g~(r) Re Y~(O, ~) (18)


(together with an analogous Im Ffl(O, O) term) so that the radial field g~(r) obeys

#2 a l(l+ 1) [(1--Kr2)-~r2+(!--gKr) Or r 2 ~-221 gS~(r) = 0 (19)

For the K = 0 case the solutions to the radial equation are just the spherical Bessel functions jt(2r). For the K = - 1 case, i.e., the unit radius 0(3, 1) hypersphere, a complete set of solutions to the radial equation is given by the associated Legendre functions, which may conveniently be written as [3] (setting r = sinh )f and v2= 2 2 - 1)

~ + 1 ( v2 + 12) sinht Z cos v~ (20) gAr ) = vZ(v 2 2) ... d

With v taking on all real positive values I-3] we see that the allowed frequencies of (17) are still real even when K is negative. With the normalization of (20), the eigenfunctions of the covariant K = - 1 3-space Laplacian obey the addition theorem [3] (-l<~rn <~ l; l = 0 , 1, 2,..., c~)

gS.(rl) Yrff(O1, ~ 1 ) l m* V sin v O g;.(r2) Yt (02, ~b2) - 2rc2 sinh O

[,m (21)

where

cosh 0 = cosh X1 cosh 5~2

- s inh )~1 sinh Z2[-cos 01 cos 02 -}- sin 01 sin 02 cos(~ 1 - ~2)3

From (21) we obtain

(22)

so that

~ (2l+ 1) t l 2vsinv(zl--Z2) g~.(r2) (23) g~.(rl)

l ~ sinh(z1 - Z2)

~ ] g~,(r)' Y'/'(O, ~)1 2 = . _ ~ (21+ - (22--1) (24) l,m " 2:g2

We thus see that the summation on l in (24) removes the dependence on the radial coordinate r and yields a general covariant 3-space scalar without any need to sum on 2. This fact will prove crucial to our sub- sequent discussion.

The solutions to the radial wave equation for the K-- + 1 case, i.e., the unit radius 0(4) hypersphere, were first given by Fock in his study of the


hydrogen atom [4] and may conveniently be written (setting r = sin )~ and Y2=22+ 1)

... sln/Z ~ ) cos v Z (25)

In the K = + 1 case the spatial geometry is compact, so the separation constant has to take on discrete values, viz. 22 -- n(n + 2), where n = 0, 1, 2,..., with the allowed values for v then being of the form v = n + 1. Further, the angular momentum is bounded in this case with its allowed values being given by l = 0, 1 ..... v - 1. (With v and l being bounded an alternative description of the basis for the radial functions can be given as a complete set of orthogonal Gegenbauer polynomials). With the normalization of (25) the eigenfunctions of the covariant K = +1 3-space Laplacian obey an addition theorem [4]

~ ( 2 l + 1 ) g~(ra) g~(r2)= 2v sin v(za--Z2) (26) l n sin(zl--Z2 )

so that

_ x - , 2 ( ' ~ 2 + 1 ) 1 ~ (2 l+ 1)[g~(r)] = 2n 2 (27)

4n

in complete analogy to (24). Armed with these solutions to the radial equations for each value of K, we can now proceed to study perfect fluids composed of these fields.

3. P E R F E C T FLUID FIELDS IN A FLAT SPACE

Before actually embarking on a discussion of perfect fluids in a curved space it is useful to see first how things work in a flat space, and so we shall study classical massless scalar fields and then briefly Maxwell fields in a flat background with Cartesian metric t/,v = ( - 1, 1, 1, 1,). For a free flat space massless scalar field the energy-momentum tensors of both (10) and (7) reduce to

_ 2 1 ~ 1SO.OvS T,v - g O u S ~ v S - gq~O S c ~ S - (28)

while the wave equation of (9) simplifies to

d~0,S = 0 (29)


In a plane wave basis there are two sets of real solutions

) S = \ sin k

where k~k~, = 0. For a given k~, the energy-momentum tensor is then given by

T.v = \ k•k,, sin 2

in the two sets of solutions. Adding these two contributions incoherently thc,'n gives, for the total contribution to T~v due to a given k u

Tuv =k~,kv (32)

Therefore, for a lightlike mode moving in the z direction with frequency ~o, i.e,., for k~=co(1, 0, 0, 1), we find that T~v takes the form

T.V = co 2 0 0 0 0

0 0

(33)

Consequently, adding incoherently to this a lightlike mode with the same the negative z direction then brings T~,v to the

( 000 1 'r'/tv = (D2 0 0 0 (34)

0 0 0

0 0 2

frequency moving in diagonal form

Proceeding in the same way, we now add incoherently two further pairs of modes moving in the positive and negative y directions and in the positive and negative x directions, but all with the same frequency to obtain

0 o Tuv=~2 2 0

0 2 0 0

(35)

Thus we see that we can bring T~ to the general form of (35) by averaging


over momenta directions. Hence, now doing the calculation completely by averaging the original k~k~ tensor of (32) over all directions yields

(T~v)= f 1 0 (36) 3 0 1

0 0

Introducing the 4-vector U" = (1, 0, 0, 0) enables us to rewrite (36) in the form

< r.v> = pn.v + (p + p) u.uv= 0 p 0

0 0 p

0 0 0

(37)

SO that we recognize our energy-momentum tensor to be of the standard perfect fluid form. Also we see that the relation p = 3p is automatically enforced for massless scalar fields since T,v is traceless (i.e., no appeal need be made to the statistical mechanical properties of black body radiation). Thus we see that an incoherent superposition of the contributions of all modes with a given frequency yields the perfect fluid form for T,v in a flat space.

In order to be able to generalize this procedure to a curved R.W. space where plane waves do not provide a good basis, we reformulate the above flat space discussion in terms of a spherical basis. In polar coordinates a complete set of solutions to the flat space scalar wave equation with metric ( - 1, 1, r 2, r a sin 2 0) is given by

/cos o~t'~ (COS mr S(x) = NT' (,sin on) j,(cnr) P~n(cos 0) \ s in rnC ] (38)

where

[ (2l + l )(l-m)!ll/2 N f ' = ( - 1 ) m [ _ 4n(l+m)[ J (39)

To illustrate the nature of the incoherent averaging which is required in a spherical basis we consider the specific quantity O~SO~S which appears in


(28). Averaging incoherently over the two frequency components and also over the azimuthal dependence yields

t , -- ~ j~(cor) PT(cos 0) 2 r 2 sin 2 0

[A ]2 t + [j;(cor)] 2 PT'(cos 0) 2 + r2 [_dO Pf'(cos 0) (40)

We now sum this average over m. Manipulation of the addition theorem for spherical harmonics

E m*

m

(2I + 1 - 4~ ) P+ [cos 01 cos 02 + sin 0~ sin 02 cos(~bx - ~b2)] (41)

yields the required m sums

(Nlm)2 pT,(co s 0 ) 2 (2l+ t) 4~z m

(N+m)2 m2pT(cos 0) 2 _ (2l+ 1) 1(l+ 1) sin 2 0 ,n 8re

d ]2 ( 2 / + l ) l ( l + l ) (N~) z ~ PT'(cos 0) - 87r

m

so that the rn summation yields

(42)

(43)

The summation on m thus eliminates the dependence on 0 and not just that o n q~.

This summation procedure can now be applied to the entire Tuv. With regard first to the off-diagonal elements of T,, , we find that those which involve one angular coordinate average to zero because of relations such as

d Z (NT') 2 P7'( c~ 0) d-0 P~(cos 0) = 0 (44) m

and an analogous one for qt. Further, the averaging over the two frequency components makes (Tot ) vanish. Finally, the full averaging is then found

212 Mannheimand Kazanas

to be sufficient to make the entire ( T ~ ) completely diagonal. As for the diagonal elements of ( T . v ) , use of the relation

d 2 ,,, ( 2 1 + l ) l ( l + l ) (45) Y. (NZm) 2 PT'(COS 0) ~ Pz (cos 0) = 8r~ m

which also follows from the addition theorem, yields

(2l+ 1) I5co2j ~ + j~2 + l ( l + 1) .2] (Too>= 24= - - ~ - - a' J

[ .2 .... 1(1+ 1)j ] (2l+1) ~o2j2+3j ' - 2j, jz r2 ( Zrr ) = 24rc

r2(21+ 1)[-~.,2-2 ;,2 2l(l+ 1)j~] ( Too ) = 24rc r L J, _ , , _2j , j; ,2

[ 2l(l+1) j21 (46) r2sin20(2l+1) o92j2_j~2.2j, j~_~ r2 ( T# ) = 24rc r

To complete the calculation we next sum on l. By manipulating

e i . . . . . 0= ~ (2l+ 1) jl(cor) ilPz(cos 0) (47) 1

we find the following summation formulae for the spherical Bessel functions

(2l+ 1) j 2 = 1 l

(2l+ 1)jzj ~=0 l

(2l + 1 )(j;)2 = a~2/3 l

(2l + 1 ) Jz J~' = -co2/3 l

(48)

while the use of the spherical Bessel differential equation yields

~2~ (2l+ 1)l( l+ 1) j2= E (2l+ 1)jt(co2jz+j7 +! j ; ) l

= 2m2/3 (49)

In all of these formulae the summation on l is seen to remove the depen-


dence on r. Finally, then, for the energy-momentum tensor itself the summation on l yields

3 0 0 o) 2 0 1 0

/ , - r k

~1"v2=1-5~ 0 0 r 2

0 0 0

which is precisely of the perfect fluid form the prescription for obtaining a perfect fluid sum incoherently over the contributions to the basis with a given prescribed frequency. tion below in the curved R.W. space.

o (5o) 0

r 2 sin 2

in spherical coordinates. Thus in a spherical basis is again to T.v due to all of the modes of We therefore use this prescrip-

To complete this section we discuss very briefly the corresponding situation for electromagnetism. For the free Maxwell field in a flat space the energy-momentum tensor is given by

- - ~ , l ,uv L ~fl ~t

with the electromagnetic field tensor being given by

0 -E l -E2 ) - - E 3

Fur= E1 0 B 3 - B 2

E 2 - - B 3 0 B 1

o E3 B2 --B 1

(51)

(52)

For either a left or a right circularly polarized plane wave moving along the z axis, i.e., for

E/R'L = [cos co(z - t), T-sin co(z- t), 0]

B~ ,L = [T-sin co(z- t), cos co(z- t), 0]

kU = co(l, 0, 0, 1)

(53)

we find explicitly that T.v is given by

(~~ 0 0 1 t TU~=co 2 0 0 0 0 0 0 =k"k ~ (54)

0 0 1

Adding incoherently left or right circularly polarized plane waves moving along the negative z axis then makes (T.v) diagonal since the average


Poynting vector must vanish for a system whose total momentum is zero. From this point on the discussion then follows the scalar field case with the perfect fluid form for Maxwell fields then emerging in flat space after an incoherent averaging over all momentum directions. (In this case the right circularly polarized and the left circularly polarized plane waves separately form perfect fluids.) Having set up the problem in flat space we now turn to an analogous discussion in curved space.

4. PERFECT SCALAR F L U I D S IN THE STANDARD C O S M O L O G Y

The adaptation of the analysis of the previous section to a general R.W. curved space is essentially straightforward. In the radiation- dominated era the scalar field energy-momentum tensor of (10) (which we take as a convenient starting point for the curved space analysis) takes the form

T # v __ ~O,uS~vS2 __ gguv01 ~ SO~S - 7SOuOvS1 + ~SF~,vO~ ~ (55)

while a complete set of solutions to the scalar field equation of motion, (9), may be written as (for any value of the spatial curvature K)

(cos e)p~ g~(r) PT'(cos 0 ) ( c o s m~b~ S(x) = R@t) \ s in e)p/ \ s in m~b]

(56)

where again p is the conformal time. The incoherent averaging over the two frequency components and over the azimuthal dependence, as well as the summation on m, proceed exactly as in the fiat space case to again yield a diagonal tensor whose diagonal elements are given this time by (for any value of K)

(2l+1) f t . 2 (Too) = 24rtR4(t ) ~L)CO + 2R(t)/~ -- k 2 ] [g~(r)] 2

+ (1 - Kr2)[g~'(r)] z +---7--l(l + 1 ) [g~(r)]2 }

T "~ (2 /+ 1) {[o)2_k2][g~,(r)]Z+3(l_KrZ)[g~,(r)] 2 ( r~ /= 24~rR2(t)(1 L Kr 2)

- 2(1 - Kr 2) g~(r) t,, g~. (r) + 2Krg~(r) g~'(r) - l(l + 1) [g~(r)]2 r 2 3


( 2 l + 1 ) r 2 { [ t r 2 ( T o o ) - 24rcR2(t) [ ( ' 0 2 - / ~ 2 ] g;~( )] --( 1 -Kr2)[g~'(r)] 2

-- 21(l+ 1) } 2(1 r Kr2) g~(r) g~'(r) -t r2 [g~(r)] 2

(21+l)r2sin20{ R ][g).(r)] ( 1 - K r 2) l, 2 247zR2(t) [002 _ "2 , 2 _ [g;~ (r)]

2(1 --Kr 2) 2l(l+ 1) } g~.(r) "'r" + g; , ~ ) r e [g~(r)] 2 r

(57)

To effect the final summation on l we note that manipulation of (23), (26), and (19) yields (for K equal to plus or minus one)

(2l + 1 t 2 )rg~j(r)] = 20)2/rc l

s (2l+ 1) gS.(r) g~'(r)=0 l

2 0 ) 2 ( 0 ) 2 - - K) Z (2/+ 1)[g~'(r)] z - ~-~-7~.rT~

l

2 0 ) 2 ( 0 ) 2 - - K) (2l+ 1) g~(r) g~"(r)= 3re(1 - K r 2)

l

~ (2l+ 1) l(1+ 1)[g~(r)]2=40)2(0)2-K)" 3=

(58)

The general structure of these relations may in fact be anticipated since, for instance, the quantity

~, g~(r) Y~ (0, r k , m * -/~gc3~)[g;.(r) Y'/'(O, r l,m

is a generally covariant rank two tensor and therefore has to be propor- tional to the only 3-space tensor available, namely the metric tensor 7,~ itself. (Apart from an irrelevant overall normalization, (58) also reduces to (48) and (49) when K = 0 , as it should. In order to be able to treat all values of K simultaneously in the following, we change the overall normalization of (48) and (49) to bring them in line with (58).) With the use of (3) when R~, = 0, the final summation on l then yields for K positive, zero, or negative

216 Mannheimand Kazanas

(02(6602 -- 3 K - 3R 2) (Too) = 12n2Rn(t )

~o2(2~o 2 - K - k 2)

r2co:(2co 2 -- K - k 2) .(Too) = 12rcZRZ(t)

r 2 sin 2 0 o92(2co 2 - K - k 2) (Tr162 = 12nZR2(t )

(59)

Thus we see that (T~v) is exactly of the standard general covariant perfect fluid form, i.e.

( T ~ ) = p(t) g,v + [p(t) + p(t)] U, Uv (60)

Hence we see that with R(t) given by (12) and with K allowed to be positive, zero, or negative, the averaging over all of the angular momentum states of a given frequency removes the dependence of (T~v) on the coordinates r, 0, and ~b in just the right way to yield finally a maximally 3-space symmetric tensor as is desired.

We note that while we have obtained a perfect fluid form for (Tuv) in (59), and while we even obtain p(t) = 3p(t) since (Tuv) is traceless, we find that in general p(t) does not have the familiar R-n(t) behavior since, as we discussed earlier, the tensor Tu~ introduced in (5) is not covariantly conserved. To obtain, finally, a perfect fluid density which does have this requisite familiar form, we instead consider the curvature-dependent tensor T~,v introduced in (7). To obtain its incoherent average we must also average the S 2 dependent term present in (7). With the use of (58) this then yields a perfect fluid form for (T'~v) so that the Einstein equations simplify to (we leave out the frequency summation)

R~ ooZRuv (-o4 [4Uu Uv + gu,-I (61) 8nG = (T'~v) = ( T ~ ) 12zt2R2(t) = 67r2R4(t )

From the time-time component of (61) (the space-space components conta in the same information since R~ is traceless) we find that

3J~(t) 094 8nGR(t) = 2~2R4(t) = (T~o) (62)

so that we see that the density (T~o) has just the right R-a(t) behavior, as required. Thus we see that the extra nonpure R-4(t) terms in (Tuv) precisely cancel the averaging of the additional S 2 dependent term which is


present in (7), so that (T',v), the source of the Einstein tensor, emerges as the familiar perfect fluid tensor of the standard cosmology. Finally, the insertion into (62) of the specific time dependence obtained earlier for R(t) in (12) then yields the familiar

3(4aK + b 2) 0) 4 32zcGR4(I) 2rc2R4(t)

(63)

so that the Einstein equations (with initial condition R ( 0 ) - 0) then express the parameter b in terms of the total energy density while leaving K undetermined. Thus we see that it is (T',~) which is to be identified with the perfect fluid tensor of the standard cosmology, with all of the standard cosmology associated with a scale parameter R(t) in an expanding universe then being recovered at the end. We thus demonstrate the complete consistency of the Einstein and the scalar field equations in the standard cosmology for any value of the spatial curvature.

5. MAXWELL FIELDS IN A CURVED R.W. BACKGROUND

For a free classical Maxwell field in a curved background metric the Maxwell equations take the form

JFlZV;v '~" 0

F~v;.~ + F;,~;v + Fv;.;. = 0

while the energy-momentum tensor is given by

- - cr Tu_F~,~F~ la, 17 l~afl

(64)

(65)

(66)

The Maxwell energy-momentum tensor is covariantly conserved for field configurations which obey the equations of motion and is generally covariantly traceless (in fact, it is even traceless in nonstationary paths). Since F~,. is antisymmetric we may replace the covariant derivatives by ordinary derivatives in (65) so that in a curved space the Ei and Bi fields, as conveniently defined via F~v in (52), then obey the following usual flat space Maxwell equations (written immediately in spherical coordinates)

~?B r OEr OEo OB o OE r OEr - - = 0 k 0

0t 00 &b 0t &b Or

OB~ kOE~ OEr=o OBr+ OB~ OBr 0t ~r O~ Or --g-d-+--~ - = ~ (67)

842/20/3-2


The presence of the metric is only felt in (64), i.e., in

(__g) 1/2 OvE(__g)l/2 g~,~gV~F~] = 0 (68)

Thus in the curved R.W. background metric of (1) the Ei and B~ fields obey

(1--Kr2)l /2sinO~r[r2(1--Kr2)U2Er]+ (sinOEo)-~ sinO 0 r - = 0

1 8Bo R(t) r 2 sin 0 ~t [R(t) E~] - [sin OBo] q sin 0 0--r = 0

R( t ) r 2 sin 0 --~ c~t JR(t) Eo]

1 OB r g sin 0 8~b ~- r2(1 - Kr2)1/2 sin 0 ~r [(1 - Kr2) m B~] = 0

20 R(t) r -~ [R(t) E~]

- r2 (1 -Kr2) l /2 -~r[ (1 -Kr2) l /2Bo]+s inO ~ = 0

(69)

The time dependence can be greatly simplified in (67) and (69) by introducing the eonformal time of (15), which obeys

d d ap-7- = R(t) Z (70)

With the rescaling

F.i = R(t) E~ (71)

we find that (67) and (69) simplify to

8Br OEo ~Eo + = 0

0p a0 ar

OB~ -~ = 0 81) Or 80

OB o OE r OEcj ~ - - = 0

0p 0r Or

8B~ OB o ~?Br or +-Sg =~

(72)


and

(1 - - Kr2) 1/2 sin 0 ~rr It2(1 - gr2) l /2 F'r] + [sin 0E0] -t sin 0 a~ - 0

1 c3B o 0Er ~3 [sin0B~] ~ - - = 0 r2 sin 0 --@-p - ~3"--d sin0 c~r (73)

r 2 sin 0 - - OE o 1 ~ B r

@ sin 0 0r

0 - - - - + r2(1 - K r 2 ) ~/2 sin 0~r [(1 - K r 2 ) 1/2 B~] = 0

- ~ p - r 2 ( 1 - K r 2 ) ~ / 2 - ~ r [ ( 1 - K r 2 ) ~ / 2 B o ] + s i n O ~ = 0

respectively, since the Maxwell equations are only first-order and not second-order differential equations.

ff the 3-space geometry is flat, i.e., if K is zero, we see immediately that the Maxwell equations reduce to those associated with a flat 4-space with coordinates p, r, 0, r so that in this space the fields Bi and R( t )E i are simply plane waves with Cartesian cos (o~p-# .2) and s i n ( e ) p - # . 2 ) behavior. Averaging incoherently over all these modes then completely duplicates the discussion leading to (54) for the rescaled fields, so that for the original fields we find that the averaging of the energy-momentum tensor of (66) yields

<Lv 3 0 0 0 01 = (-.04 0 R2(t) 0 0 " (74) 6~2R4(t) 0 0 R2(t) r 2 0

0 0 0 R2(t) r 2 sin 2

which is precisely of the standard perfect fluid form in a flat 3-space. Inser- tion of (74) into the Einstein equations then recovers the familiar relation R(t) = bl/2t 1/2 (on setting a = 0 in (12)). With this form for R(t) the confor- real time is then given p = 2ta/2/b 1/2 so that the various components of the electric field vector E i propagate as cos(2tmco/b m - # . 2)It m in a spatially flat universe. We thus establish the consistency of the coupled Einstein-Maxwell system in the standard cosmology with a flat spatial metric.

Unlike the spatially flat .case, we have not been able to decouple the various components of Fuv in the equations of motion in the spatially curved case, and so we are currently unable to make a similar analysis in this case. However, in passing we note that we have found that the radial


components Er and B r do decouple with the quantities R(t) r(1 -Kr2) 1/2 E r

and Br/r sin 0 (written collectively as Cr) obeying the equation of motion

02Cr 202Cr ~0 (sin 0 - "--072 + ( 1 - Kr ) -~rf-rZ + ( ! - 3Kr ) OCt+Or r 2 sinl 0 --frO- ~

1 ~2C~ -t r2 sin2 0 O~ 2 - - K f r : 0 (75)

which is just the scalar wave equation discussed previously. While we are unable to carry through the complete calculation in the spatially curved case, we are not aware of any conceptual difficulties which would prevent the establishing of a perfect fluid form in this case.

N o t e Added in Proof

We have now suceeded in establishing the existence of the perfect fluid form in the Maxwell case, and our results are reported by Y. Deng and P. D. Mannheim, Perfect Maxwell Fluids in the Standard Cosmology, to be published.

ACKNOWLEDGMENTS

We would like to thank Professor S. Nussinov for his enthusiastic sup- port and for many fruitful discussions. One of us (P.D.M.) would like to thank the National Aeronautics and Space Administration for the award of a NASA/ASEE Summer Faculty Fellowship, which enabled him to spend the summer of 1985 at the Goddard Space Flight Center, and in particular to thank Professor F. P. Emad for his kind invitation to participate in the NASA program. He would also like to thank Professors R. Ramaty and F. Stecker for their interest and the kind hospitality of the high-energy astrophysics theory group at the Goddard Space Flight Center. This work has also been supported in part (P.D.M.) by the U.S. Department of Energy under Grant No. DE-AC02-79ER10336.A.

R E F E R E N C E S

1. Callan, C. G., Coleman, S., and Jackiw, R. (1970). Anns. Phys. (N.Y.), 59, 42. 2. Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General

Theory of Relativity (Wiley New York). 3. Bander, M., and Itzykson, C. (1966). Revs. Mod. Phys., 38, 346. 4. Fock, V. (1935). Z. Phys., 98, 145.

Energy-momentum tensor of fields in the standard cosmology

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