Class.Quanrum Crav. 11 (1994) 23752382.
Rinted in the UK
Spinning fluid cosmologies in EinsteinCartan theory Larry L Smalleyt and J P Krisch$ t Depamnent of physics, University of Alabama in Huntsville. Huntsville, AL 35899, USA I Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Received 6 June 1994 Abstraft We consider selfconsistent spinning fluid cosmologies in bath general relativity theory in RiemanoCW spacetimes. First we in Riemannh spacetimes and Einstein
extend slightly the cosmological calculation of Marlin eral for general relativistic selfconsistent spinning Buids. The existence of spinsquared terms in the field equations in the Einsteincutan theory shows, however, that an expanded class of meaningful cosmologies is possible. Under certain assumptions on the arbikariness of the cosmological shear and expansion the results for the ad hoc Weyssenhofl spin fluid in a spherically symmehic spacetime can be reproduced. PACS numben: 0440,0450,9880
1. Introduction
Selfconsistent gravitational theories are generalizations of the RayHilbert variational principle for perfect fiuids in general relativity [l] (GR) in the holonomic framework in which certain matter variables such as specific entropy and volume are also treated as thermodymical variables. The selfconsistent formulation can easily be extended, for perfect fluids, to more complex spacetimes such as RiemannCartan (RC) spacetimes (which is the spacetime arena for the Einsteinx3artan theory (EC) [2]) or in general, a metricaffine spacetime (Poincar.6 gauge theory) [3]. Some additional matter variables have been included such as spin density [4]; and for charged fluids in EC, the electromagnetic field [571. Charged fluids have also been investigated and the theory extended to include the electromagnetic field as a thermodynamical variable [SI. Selfconsistent spinning fluids have also been investigated in the holonomic framework in Ec by de Ritis et al [9] in the anholonomic framework in EC by Obukhov and Korotky [IO] and in OR by Obukhov and Piskareva [I I]. Other variational principles are known but not considered here; for example, the Lagrange variational principle of Kopczyhski [ 121 in EC which selfconsistently yields both the matter currents and the equation of motion of the currents. On the other hand, the phenomenological approach does not rigorously yield the equations of motion since the currents are not defined in term of the actual dynamical variables of the fluid (see, for example, discussions in [13]). However earlier cosmological models based upon the EC theory [14] using ad hoc prescriptions for torsion are known. See for example the discussion on a nonsingular universe by Kopczyhski [15] using the Cartan relation between spin and torsion [16] or the discussion of early cosmological times [17], using the Weyssenhoff spin fluid model [18]. A general feature of the selfconsistent spinning fluid theories in the holonomic frame is consistency conditions. They were first discussed by A ” [SI for selfconsistent, charged spinning fluids in Ec, with later corrections by Ray and Smalley [7]. Within this framework 02649381/94/092375+08$1950 @ 1594 IOP Publishing Ltd
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L L Smalley and J P Krisch
Gasperini has discussed spindominated inflation in EC theory [19]. More recently Martin et a1 [20] have discussed consistency conditions for a spinning fluid in GR 1211. However, Martin et a1 have indicated that withiin the cosmological context for a spinning fluid in GR [20], the consistency relation may limit useful solutions to the case of purely stationary spinning fluids. These consistency conditions seem to occur in all known solutions of selfconsistent theories in the holonomic framework [2224] but do not seem to be a feature of the anholonomic framework because of a different formulation of the thermodynamics [ 10,l I]. In principle these two distinct types of formulations should be equivalent. This seems to be the general importance of the consistency conditions. .We make the following observations. (1) One must be careful in taking EC solutions to the GR limit. The limiting procedure in which one zeroes the spin contributions from the antisymmetric connections while keeping any explicit spin contributions 6om the stress energy content is incorrect. Given a set of selfconsistent Ec field equations, the correct limiting procedure will zero any spin squared terms in the EC equations, leaving all others as contributions to the selfconsistent GR system. In EC, the torsion field equation is given by [4]
s^;
i r =K psjju (1) 2 where the torsion is the antisymmetric p m of the connection, Sjjk = r[jjlk,the caret signifies the @acefreeproper torsion, K = 8 z G , where G is the gravitational constant, p is the density, sij is the spin density, and ut is the four velocity. Thus the limit S$ + 0 gives GR. But by equation (1) in this limit, si, f 0 as well, and one arrives at GR with a perfect fluid. A selfconsistent spinning fluid in GR is not the simple limit of a selfconsistent spinning fluid in an EC [20]. (2) The appropriate spacetime arena for spinning fluids is in EC in which a natural geometric objectthe proper torsionis directly related to the spin density as in equation (1). No such geometric object occurs in GR. We will see that the consequence of this observation is that no spinsquared terms, complementary to angular momentum squared terms can arise in the description of energy density of the spinning fluid. However spinsquared terms arise naturally for spinning fluids described in ~ c . A selfconsistent spinning fluid in GR is a very different theory than a spinning fluid in ECt. Because of this we will show in this work that within Ec, nontrivial cosmologies occur. In order to demonstrate this feature, we write the EC metric field equation in an ) by expanding the connection into Christoffel plus torsion effectiveEmteinian ( e ~ form parts and then replace the torsion using the torsion field equation (1). In this form the Bianchi identities then give the conservation laws. In section 2, we revisit the GR calculation by Martin et a[ [20] and extend their results slightly as background to the EC case which is written in the EB form in section 3. Application to spherically symmetric cosmologies for EC and results are discussed in section 4, and we summarize our results in the last section.
2. Spinning fluids in general relativity
The selfconsistent spinning fluid formalism [20] gives the metric field equation G’j = K(T,!’
+ r:’)
(2)
t By ignoring the torsion held equalon, an improper GR limit can be obtained from the Rc “IC equalon by pulling the torsion to zem on t k gsomehic ‘fields’ side and replacing the RC COMFZ~O~ in the ‘ m e r ’ side with the CbWoffel connection.
Spinning fluid cosmologies in EinsteinCartm theory
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where Gij is the Einstein tensor (in GR), the perfect fluid energymomentum tensor is
T,' = [p(I + E )
+ plu'uj + pg'j
(3)
where p is pressure, E is energy density, and the spinning fluid energymomentum tensor is

~,= ' j @ ~ j ) t i , . + ~ : l [ ~ ( k ~ i ) kw] k ( i s j ) k
(4)
where the spin tensor S'j = ps'j, the overdot on the four velocity ut = U L @ U k represents the covariant derivative along fluid flow lines, represents the covariant derivative in GR with Chistoffel connection ( f j ) , the spin angular velocity is 0.'rj
.*
 a iawj
(5)
where aej are a set of tetrads such that the mehic is given by gi,
= qflva*iav,
(6)
qLLV= (1, 1,1, 1) is the Lorentz mehic, and a* e ui.
One finds that the spin density FermiWalker transports [25] S;j
+ 2UliSjlXli' = 0.
(7)
In the OR selfconsistent formalism, the spin density is also treated as a thermodynamic variable so that the first law is given by 141
where T is the temperature and s is the specific entropy. Dividing equation ( 8 ) by d x k and contracting with 'U gives the convective derivative of the thermodynamic law. In the selfconsistent theories, entropy is conserved in the comoving frame, S = 0. We then have €+P(b)
1
..
 2wijs'J = 0.
(9)
The last term on the lefthandside vanishes because of the definitions of the angular velocity and the spin density si, = 2k0a1p2,1 where ko is the constant magnitude of the spin vector in the comoving frame [26]. The expression given by equation (9) is also true in both OR and EC because of the FermiWalker transport of the spin density given by equation (7). Thus €
+ p (;)*
= 0.
The conservation of particle number given by ~ ' ( P U ' = ) 0 can be rewritten using the definition of the cosmological expansion parameter [U],
0 = @lui
(11)
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L L Smnliey and J P Krisch
as
p+pe=o.
(12)
The consistency conditions are obtained from equation (2) using the Bianchi identity and contracting with uj so that 0 =u
j v p ; j
+T3.
(13)
Using equations (3). (10) and (12), the perfect fluid part is conserved automatically. Before we give the consistency relation which follows from the spin part, we prove an identity. In E,an identity for the antisymmetric components of the Ricci tensor [28] is ~Iij= l ~ l i j= l oksijk
(14)
Taking the EC covariant derivative 0, of equation (14) and contracting with U( gives the EC equation
But in Riemannian spacetime, GtiiJ=_ 0, so that setting EC connection to Christoffel (see previous footnote) on the righthandside of equation (15) gives the GR equation V‘i’[S”Ui]
= 0.
(16)
The spin part of equation (13) w1 be reduced to the equation 0 = U i V ‘ i l T i j = ~VJ!+Sj$i()
 U i V I1j [ut( isj ) ! 1.
(17)
Using equation (16) therefore gives the OR consistency relation u ; v j ~ [ , t ( ~ s j )= ~ ]0.
(18)
The cosmological shear parameter is given by [28] ajj
= +&Uj)
+ U ( j U j )  +9(UIUj +g j j )
(1%
Expanding the covariant derivatives in equation (18), using the identity equation (16) plus the definitions for the cosmological parameters given by equations (1 1) and (19). the consistency relation becomes
guijsije +
~
;
s
=j 0.~
~
~
~
(20)
This is a slight generalization of the results of Martin er al in [20] where they leave unresolved the term given in the GR equation in equation (16). In the next section, we investigate spinning fluid cosmologies in EC.
Spinning fluid cosmologies in EinsteinCartan theory
2379
3. Spinning fluids in EinsteinCartan theory In EC the metric field equation now takes the form (assuming the torsion vector vanishes) G!;)
 2vk[,yk(ij)]= K [ T ; j + ~ ; j ]
(21)
where the subscript U: refers to the Einstein tensor in EC,and where now T j j has the same 'form' as equation (4) except that covariant derivatives are in terms of the EC connection
r..k { ki j } + Sijk  Sjki +si:.
(22)
,J
We can rewrite equation (21) in a EE form by noting that the lefthand side becomes ~ $ 0 2vk[sk(i~3] = GU + 2 + Snk(iS I) nk
s ( i t n p + S(iL,Sj)nt
 128i j [sLmn &mn
+
 @ j  K 2 [Ts 1 iIS''
+ 2 SCmnStnml
:~,ts"'U'Uj
 i g i j S h s C m ] K V ( : [ S k ( i U j ) ]
(23) where in the last two lines, we have used the torsion field equation (1). Similarly the righthandside of equation (21) becomes K[$+$]
= K ( [ P ( 1 + E ) + p ] U i U j + p g i j + 2 U ( i S j ) k ' ut
+ @ [ u ( i s j ) k ]  mk(isj)k K s i k s j k  swskLuiuj K 2
I
where we have used the identity for the spin angular velocity in an Ec [29] &i$) = U k ( i s j ) k  ski$k. K 2
(24)
(25)
The effective Einsteinian form now becomes G'j
K
( [ P ( l + E ) fP ] U i U i
+ Pg"
f 2u('SnkUk
 K 2 ( $ s i k s j i + iSkeSkeu'Ui + isk,s'g'j} .
 6Jk(isj)k} (26)
Using the GR results in the last section, the EC consistency relation is
0
uiq)Gij=
f
=
K ~ '. $j~ [ U ~ ( ~ SJ ?K2uiVj1[+SikSjp ~]
+ Q&SkLg"]
i~ktskLU'Uj
 K U i $ !I
s + &SikSjk
[ U k (i j)k
1
where we have used the identity [s.1.I S j k y = 28s.Jk Sjk.
(27)
(28)
The last line in equation (27) can be rewritten in U:form [7] as UiVj[Ok"s"']
= 0.
(29)
Using equation (19). we can put equation (27) in the final form
which demonstrates that nontrivial cosmologies with both expansion and shear are possible in EC. In the next section, hy way of example, we investigate this particular consequence.
L L Smalley and J P Krisch
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4. EinsteinCartan cosmologies
The consistency relation in Ec equation (30)indicates that there are at least three distinct cosmologies possible. It is interesting that there is at most only a weak condition on the vorticity depending on both arbitrary expansion and shear; otherwise the vorticity is arbitrary. In this case, arbitrary expansion and shear lead to the conditions 2 ~k
Sit
 Ks.I t s j k = 0
(31)
respectively. Note that the trace of equation (32) contains equation (31). Consider the example of a spherically symmetric cosmology in EC spactime [30].In this case, equation (21) can be written using equations (23) and (24): =K[T;j
Gij
where
T'j
Tij
(33)
KTjj
is the effective spherically symmetric energymomentum tensor with
+
=[p(l+ E ) T:' = $[4&Sfk Tdj
+ T i j +d j ]
plu'uj +sKs2
+
(34)
pg'j
 WckSu]U'U'
+ [&KS2]g'j
 !$[&su
+ 2s' 
Otksfk]g'j
=[~Ks2]uiUj
(35) (36)
where the OR vorticity is defined as 1271
and  T i ] is the spinsquaredpart of G$$ given in equation (23). and S;,Sij = 2s'. The form of the average stressenergy tensor depends on assumptions about the fields and the various cosmological models that one can have in a spherically symmetric Ec. If the consistency conditions are included in the analysis of the average value of equation (33), we can make definitive predictions. The average energy momentum tensor will depend on the particular assumptions about the type of cosmology being investigated. In taking the average of a spherically symmetric isotropic system of randomly oriented spins, the average of the spin itself is assumed to vanish, ( S i ] ) = 0, but for the spinsquared terms, ( S j j S ' j ) # 0. Since SijjS'j = 2Sz, where S is the spin fourvector, then we set (9) S2 for convenience. For the case of a cosmology with arbitrary shear and expansion developed above, then the definition of the GR vorticity equation (37). defined in terms of the fourvelocity, and the spin density, described after equation (9),shows that there is no correlation between the randomly oriented spins and the congruences associated with the fourvelocity. We therefore assume that the average of the vorticityspin term vanishes (ajjtSik)
=0
138)
but on the other hand, equation (32) gives a non zero average for the angular velocity (Ojksik)
= KS'.
(39)
Spinning fluid cosmologies in EinsteinCartan theory
2381
Using these results, the average of the energy momentum tensor equation (33) becomes (Ti’), = [ p ( l f€ f p / p )  4 K S2 ] U i U j f [ p  Z1 K sZlgij
(40)
where the subscript I refers to the improved energy momentum tensor of the selfconsistent spinning fluid in w. This is identical to the results of Hehl eta1 [31J for the earlier EinsteinCartan theory for a Wessenhoff spinning fluid. In addition, the cases of no expansion but arbitrary shear, or vice versa, will give the same results. Note that equation (39) is equivalent to the condition that the average value of the EC angular velocity spin term vanishes. Other cases are possible. For example, the case of no expansion or shear, equation (38) still holds, but there is no specific prediction for the term (oj&. However it could be chosen to give the same results as equation (40). amongst other possibilities. On the other hand, both the vorticity and the angular velocitiy terms could satisfiy weak conditions similar to equation (38). Thus in the absence of any further information, one couZd assume that on the average, there are no correlations between the GR vorticity or the spin angular velocity with the spin. Then the average value of the effective spherically symmetric energy momentum becomes
found in an earlier work [ 3 1 ] .
5. Conclusious
We have Seen through the investigation of selfconsistent spinning fluids in a cosmological setting that the most active arena for spinning fluids is in a RiemannCartan spactime. Spinning fluids within the general relativistic context are very limited in the types of cosmologies that seem to exist [PI. In fact for certain assumptions on the cosmological model, the restrictive relation between the spin angular velocity and the spin given in equation (20) must be satisfied everywhere (not just on the average). Including the assumption of the vorticityspin average given by equation (38) yields the effective energymomentum tensor of a perfectfluid given by equation (30). When the spinning fluids reside in a EC, a larger class of spacetimes are allowed. The less restrictive relationships given in equation (30) between the spin density and the spin angular velocity, due to the addition of spinsquared terms in the field equation, further enriches the character of the types of cosmologies that can exist. This leads to very general relationships amongst the spin density and the spin angular velocity, such as shown by equations (31) and (32) for arbitrary shear and expansion. If we include the vorticity assumption given by equation (38), then we obtain the effective energymomentum tensor given by equation (40). It is in the EC case that one begins to see the importance of the consistency relations. Without using the consistency relations of the selfconsistent theories, further assmptions on the averages of the spin angular velocityspin terms must be made. In such cases, the average value of the improved energymomentum tensor can be very different, as seen in equation (41), from the results for the classical Weyssenhoff fluid given in equation (40). However by including the consistency condition in the analysis, we find that the classical Weyssenhoff fluid is just one of several different possible spherically symmetric cosmologies with spin density in EC. This does not happen for spinning fluids in GR.
L L Smalley and 3 P Krisch
2382
Addendum Special recognition is given to the referee who noticed that the angular velocity equation (3, which occurs in the consistency condition equation (27), is not as arbibary as one might at first guess. Taking the traces of equations (5) and (32) and using the defintion of spin, he derives the relation 201'az~= ~ p between k the tetrad spin angular velocity and the spin density in the fluid frame which must be satisfied in order to have a solution in the self consistent model used in this work. This observation solves a long time misconception on the arbitrariness of the angular velocity tensor in the selfconsistent theories with extended thermodynamics. An alternative approach to this problem used by the authors (and perhaps others) is to assume a metric form which can be as arbitrary as necessary. Consistent with this, a set of tetrads can be generated which then gives a dynamical basis for the spin angular velocity. This technique is very general within the limits of the metric and the spin density subsumed by the tetrads (see, for example, 132,331). References [I] Ray J R 1972 3. Mdh Phys. 13 1451 p] Ray J R and Smalley L L 1982 Phys. Rev. D 26 2615 L L I993 Class. Qua" Grav. 10 1179 Ray J R and Smalley L L 1982 P e s . Rev. Lcn. 49 1059; 1983 50 6268; 1982 Phys. Rev. D 26 2619 Amorim R 1984 Phys. Len. 40A 259; 1985 Pkys. Rev. D 31 3099 de Ritis R, Lavorgna M,Platonia G and Stomaiola C 1985 Phys. Rev. D 31 1854 [71 Ray J R and Smalley L L 1986 Phys. Rev. D 34 3268 [8] SmaUey L L and Krisch I P 1991 Clms. Qunnhun Gmv. 8 1889; 1992 3. Mnfh Phys. 33 1073 [9] de Ritis R, Lavagna M,Platania G and Stomaiolo C 1983 Phys. R e . MS 713 [IO] Obukhov Yu N and Komtky V A 1987 Class. Quunrum Gmv. 4 1633 [ I l l Obukhov Yu N and F ' i s b v a 0 B 1989 C h s . Q u a " Gmv. 6 L15 [12] Kopcyhski W 19% Phys. Rev. D34 352; 1990 Ann Phys. 203 308 [I31 O b W v Yu N and Tresguem R 1993 Hyperfluida model of classical malter wilh hypennomenmm Preprint University of Cologne [I41 Kibble T W B 1961 3. M a i h Phys. 2 212 Sciama D W 1962 Recen! developments in General Relativity (F&schrij+flir Infeld) (Oxford:Perpamon) p415 [I51 Kopcqhski W 1972 Phys. Len. 39A 219 [I61 CarIan E 1922 C.R. Acad Sci., Paris 174 593 [I71 Nurgaliev I S and Ponomariev W N 1983 Phys. Len. 130B 378 [I81 Weyssenhoff J and Raabe A 1947 Acta Phys. Pol. 9 7 [I91 Gaswrini M 1986 Pkys. Rev. kfr.56 2873 [a] Martin M A P. VasconcellosVnidya E P and Som M M 1991 Class. Qumhun Gmv. 8 2225 [21] Ray J R, Smalley L Land Krisch J P 1987 Phys. Rev. D 35 3261 [221 Smalley L L 1985 Pkys. Rev. D 32 3124 [23] Krisch J P and Smalley L L 1988 3. Math Phys. 29 1640 [24] Kiisch J P and SmaUey L L 1990 Class. Quantum Grav. 7 481 [W] Misner C W. Thome K S, and Wheeler J A 1973 Gmvirntion (San Pmcisco, C A Freeman) section 6.5 [26] Smalley L L and Ray J R 1988 Phys. Lea 134A 87 [271 Ehlers J and Kundf W 1962 Cravitnrion: M Introduction lo Current Research ed L Wllten (New York Wdey) p 57 [28] Schouten J A 1954 R i d Culculur (Berlin. Springer) [29] Fennelly A J. Krisch 1 P. Ray J R and S d l e y L L 1991 3. Moth Phys. 32 485 [30] Smalley L L and Krisch I P On s p h e r i d y symmeLric cosmologies in Riemann4ktan spacetime with spin density Repon 8.7.92 Univenity of Alabama in Huntsville 1311 Hehl F W. von der Hey& P and Kerlick G D 1974 Pkys. Rev. D 10 1066 [32] Krisch J P and Smalley L L 1990 Class. Q u a " Gmv. 7 481; 1988 3. Maih Phys. 29 1620 [33] S d e y L L 1985 Phys. Rev. D 32 3124 [3l [41 [SI [6]
S d e y