A universe with torsion and cosmological constant
G. PLATANIAand R. ROSANIA
Dipartimento di Scienze Fisiche - Mostra d’Oltremare, Pad. 19, 80125 Napoli, Italy
(ricevuto il 19 Agosto 1996; approvato il 4 Febbraio 1997)
Summary. — The aim of this paper is to extend the solutions of the Einstein-Dirac equations in a Bianchi type-I Universe with torsion we have found(see PLATANIAG. and ROSANIA R., Nuovo Cimento B, 111 (1996) 1195)in the context of the ECSK theory of gravitation, to the case in which the cosmological constant L is non-zero. Since L does not appear in the off-diagonal field equations, we have again that the Universe has plane symmetry and that spin vector has only one non-zero space-component. The solutions of the Einstein-Dirac equations may be classified according to the value of a dimensionless parameter F. If F E1, we have the solutions, which extend the old ones when L is non-zero: they are singular and have a stationary-type asymptotical behaviour (t K6Q) to which, because of L, constant Hubble functions correspond.
PACS 04.50 – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.
PACS 98.80 – Cosmology.
1. – Introduction
According to the ECSK theory of gravitation (see, e.g., [1, 2]), also the spin of matter fields contribute to the space-time curvature, if the mass-energy density of the matter is of the order of 1071GeV/cm3(for electrons).
Since, according to standard cosmology, such density is attainable in the early universe, we may expect that spin plays a relevant role in the evolution of the early universe.
Moreover, since spin introduces naturally anisotropies even in the isotropic and homogeneous Fridmann-Robertson-Walker (FRW) models, studying the effect of spin in anisotropic cosmological models results to be more reasonable.
In particular, in this paper we consider a homogeneous and anisotropic cosmological model of Bianchi type-I with plane symmetry, in which the source of gravitational field is an unquantized Dirac field, the only non-zero component of the spin vector is oriented along the symmetry axis and the component of the current vector along the symmetry axis is zero.
Moreover, following Raychaudhuri [3], we assume that the cosmological constant Lis non-zero a priori.
The work is organized as follows: in sect. 2 we present briefly the formalism; in sect. 3 we present the Einstein-Dirac equations in a Bianchi type-I space-time; in sect. 4 we give the solutions; in sect. 5 we briefly analyze the solutions; in sect. 6 we give the conclusions.
2. – Formalism and definitions
In the ECSK theory of gravitation (see, e.g., [1]) the Einstein equations with cosmological constant L are modified to [1, 2]
Gab(] ()2Lhab4 K sab,
(1)
where Gab(] () is the Einstein tensor with respect to the anholonomic connection
gabsf 2Vabs1 Vbsa2 Vsab
(2)
of a V4space-time, valued on an orthonormal tetrad,
VabsfhmsAaiAb j Vij m , (3) Vijmf ¯[iBmj] (4)
is the anholonomy object [4], where Ba
i are the components of an orthonormal tetrads
field, valued on a given coordinate system, VAaiV f VBaiV21
sabfsab1 K
k
2 tasstbmm2 2 tbsrtars2 2 tbrstars1 tsrbtsra1 (5) 1habg
2tnmmtnss1 tnsrtnrs1 1 2 tnrst nrsh
z
,is the combined energy-momentum tensor in anholonomic coordinates, hab is the
Minkowski metric with signature (1, 2, 2, 2), sab4 1 k2g Aai dLm dBb i , (6)
is the dynamical energy-momentum tensor,
tabs4 1 k2g AakAb j Bs i dLm dKijk , (7)
is the dynamical spin tensor, Lm(c , ¯c , Bai, ¯Bai, Sijk) is the material Lagrangian
density of the matter field c, minimally coupled to gravity,
Kijkf 2Sijk1 Sjki2 Skij
is the contorsion tensor valued on a coordinate basis, Sijkf 1 2(Gij k 2 Gjik) (9)
is the torsion tensor,
Gijkf ]k, ij( 2 Kijk
(10)
is the connection of a U4 space-time valued on a coordinate basis, being ]k, ij( the
Christoffel symbols, K 48pG, G is the universal gravitation constant, L is assumed to be positive. For a Dirac field c with mass m, minimally coupled to gravity, using units such that ˇ 4c41, one has that [5]
Lm4k2g 1 2 i(c g a ˜ac 2˜ac gac) 12imc c , (11) where c f c1g0
is the Dirac adjoint of c , ]ga( are the special-relativistic Dirac matrices, such that
]ga, gb( 4 2 habI, (12) ˜ac 4¯ac 2 1 8Gasb[g s, gb] c , ˜ ac 4¯ac 1 1 8Gasbc[g s, gb] , (13)
are, respectively, the covariant derivatives of c and c [5] with ¯a4 Aai¯i,
Gabs4 Ga[bs]4 2Kabs2 Vabs1 Vbsa2 Vsab
(14)
is the connection of a U4space-time valued on an orthonormal tetrad [5],
KabsfAaiAb j
AsmgmnKijn
(15)
is the contorsion tensor in anholonomic coordinates. From (11) and the definitions (6), (7) it follows that
sab4 S(ab)(] ()13 Khabtgtg
(16)
is the combined energy-momentum tensor of the Dirac field [6], tabsfhsgeabgntn
(17)
is the spin tensor of the Dirac field [6], where
tsf 1 4(c g5g
s
c) (18)
is the spin vector, with e01234 1 , g54 ig0g1g2g3,
Sab(] () f i 2
g
2 c gb˜ ]( ac 1˜ ]( ac gbch
(19)is the canonical energy-momentum tensor of the Dirac field, with ˜ ]( ac f ¯ac 2 1 8 gasb[g s, gb] c , ˜ ]( ac f ¯ac 1 1 8gasbc[g s, gb] . (20)
3. – The Einstein-Dirac equations
Now, we assume that the Universe is a Bianchi type-I space-time : its space-time separation is
ds2
4 dt22 (a1)2( dx1)22 (a2)2( dx2)22 (a3)2( dx3)2,
(21)
where a1, a2, a3depend on time t only.
The matter content of the universe is represented like un unquantized Dirac field c, minimally coupled to gravity.
Since such space-time is spatially homogeneous, it is reasonable to suppose that c does not depend on spatial coordinates:
c 4c(t) . (22)
Consistently with (21) we choose Ba ias
B0
04 1 , Bii4 ai, i 41, 2, 3,
(23)
being the other components of VBa
iVequal to 0.
The representation of Dirac matrices ]ga
( we choose is
.
`
`
/
`
`
´
g0 4C
`
`
`
D
1 0 0 0 0 1 0 0 0 0 21 0 0 0 0 21E
`
`
`
F
, g1 4C
`
`
`
D
0 0 0 21 0 0 21 0 0 1 0 0 1 0 0 0E
`
`
`
F
, g2 4C
`
`
`
D
0 0 0 2i 0 0 i 0 0 i 0 0 2i 0 0 0E
`
`
`
F
, g3 4C
`
`
`
D
0 0 21 0 0 0 0 1 1 0 0 0 0 21 0 0E
`
`
`
F
. (24)The Dirac equation in ECSK theory [5]
iga ˜ ]( ac 2 3 2 Kt dg 5gdc 2mc40
in a Bianchi type-I space-time, according to (22), (23) and (24), becomes
.
`
`
/
`
`
´
i c.11 i 2(H11 H21 H3) c11 3 2 K[ (it 2 2 t1) c22 t3c11 t0c3] 4mc1, i c.21 i 2 (H11 H21 H3) c21 3 2 K[2(it 2 1 t1) c11 t3c21 t0c4] 4mc2, 2i c . 32 i 2(H11 H21 H3) c31 3 2 K[t 3c 31 (t12 it2) c42 t0c1] 4mc3, 2i c . 42 i 2 (H11 H21 H3) c41 3 2 K[2t 3c 41 (t11 it2) c32 t0c2] 4mc4, (25) where c. fdc dt ,Hif aiN aiare the Hubble functions, a.if
dai dt and
.
`
/
`
´
t0 4 21 4 (c131 c241 c311 c42), t 1 4 21 4(c121 c341 c211 c43), t2 4 i1 4(c211 c432 c122 c34), t 3 4 1 4(c221 c442 c112 c33), (26)are the components of the spin vector tdf 1
4(c g5g
dc), being c
ADfcA(cD)* ,
A , D 41–4.
The gravitational-field equations (1) are G00(] () 4 K s00: (27) H1H21 H1H31 H2H32 L 4 3 K2habtatb1 K i 2 ]2 c . 1(c1)* 2c . 2(c2)* 2 2 c . 3(c3)* 2c . 4(c4)* 1c1(c1* ) . 1 c2(c2* ) . 1 c3(c3* ) . 1 c4(c4* ) . ( , G01(] () 4G10(] () 4 K s014 K s10: (28) c.1(c4)* 1c . 2(c3)* 1c . 3(c2)* 1c . 4(c1)* 2c4(c* )1 .2 c3(c* )2 .2 2c2(c3* ).2 c1(c4* ).4 0 , G02(] () 4G20(] () 4 Hs024 H s20: (29) c.1(c4)* 2c . 2(c3)* 1c . 3(c2)* 2c . 4(c1)* 1c4(c* )1 . 2 c3(c* )2 . 1 1c2(c3* ).2 c1(c4* ).4 0 , G03(] () 4G30(] () 4 K s034 K s30: (30) c.1(c3* ) 2c . 2(c4)* 1c . 3(c1)* 2c . 4(c2)* 2c3(c1* ) . 1 c4(c2* ) . 1 2c1(c3* ) . 1 c2(c4* ) . 4 0 ,
G11(] () 4 K s11: H . 21 H . 31 (H2)21 (H3)21 H2H32 L 4 3 K2habtatb, (31) G22(] () 4 K s22: H . 11 H . 31 (H1)21 (H3)21 H1H32 L 4 3 K2habtatb, (32) G33(] () 4 K s33: H . 11 H . 21 (H1)21 (H2)21 H1H22 L 4 3 K2habtatb, (33) G12(] () 4G21(] () 4 K s124 K s21: (H22 H1) t34 0 , (34) G13(] () 4G31(] () 4 K s134 K s31: (H12 H3) t24 0 , (35) G23(] () 4G32(] () 4 K s234 K s32: (H22 H3) t14 0 . (36)
Among the various cases which can occur according to eqs. (34)-(36), we consider the one already examined in our previous paper [7]: we choose
t1
4 t24 0 D c211 c431 c121 c344 0 ,
(37)
H24 H1.
(38)
Therefore it is reasonable to set
a14 a2.
(39)
Furthermore, in order to have
t04 0 D c311 c421 c131 c244 0 ,
(40)
from the relation jmt
m4 0, where jmfc gmc is the current vector of the Dirac field c,
we set
j3
4 c g3c 40 D c312 c421 c132 c244 0 .
(41)
From the eqs.(40), (41) we have that
c131 c314 0 ,
(42)
c241 c424 0 .
(43)
Now, derived c.iand (c
.
i)* from the Dirac equation (25) and replaced them in (27), (28),
(29) and (30), we obtain that eqs. (28), (29) and (30) are identically satisfied and H1H21 H1H31 H2H32 L 4 23 K2habtatb1 m K (c331 c442 c112 c22) .
(44)
Taking into account (37), (38) and (40), eqs. (44), (31), (32), (33) become
G00(] () 4 K s00: (H1)212 H1H32L43 K2(t3)21m K (c331c442c112c22) , (45) G11(] () 4 K s11: H . 11 H . 31 (H1)21 (H3)21 H1H32 L 4 23 K2(t3)2, (46) G33(] () 4 K s33: 2 H . 11 3(H1)22 L 4 23 K2(t3)2, (47)
and the Dirac equation becomes
.
`
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/
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`
´
i c.11 i 2( 2 H11 H3) c12 3 2 Kt 3 c14 mc1, i c.21 i 2( 2 H11 H3) c21 3 2 Kt 3c 24 mc2, 2i c . 32 i 2( 2 H11 H3) c31 3 2 Kt 3c 34 mc3, 2i c . 42 i 2( 2 H11 H3) c42 3 2 Kt 3c 44 mc4. (48)4. – The solutions of the Einstein-Dirac equations
The solutions of eqs. (45)-(48) are classified according to the value of the dimension-less parameter F 4 L(g 16k 2 ) 2 b2 (49)
(see below the definitions of g and b).
A) If F E1, one has that
.
`
/
`
´
a1(t) 4a2(t) 4a10Nk1 2F cosh (x)21N 1 /3N
ex2 z1 ex 2 z2N
p1 , a3(t) 4a30Nk1 2F cosh (x)21N1 /3N
ex 2 z1 ex 2 z2N
22 p1 , (50).
`
/
`
´
c14 b1 (2g)1 /4 (e 2imt 2 iG(t)) , c34 b3 (2g)1 /4 (e imt 2iG(t)) , c24 b2 (2g)1 /4(e 2imt 1 iG(t)) , c44 b4 (2g)1 /4(e imt 1iG(t)) , (51) where x fk3 L (t 1b) , (52) z14 1 1 kF k1 2F , z24 1 2kF k1 2F , (53) p14 k 3 sign (b)k
k2 1 ( 1 /6 ) g , (54)k2g 4 b L[k1 2F cosh (x)21] , (55) G(t) f 3 2 K
t3 dt 4 1 2o
g g 16k2 lnN
ex2 z1 ex 2 z2N
, (56) being t3 4 m k2g , g f 6 K2m2, (57) m f 1 4(Nb2N 2 1 Nb4N22 Nb1N22 Nb3N2) , (58) b f 1 2mn K , (59) n f Nb3N21 Nb4N22 Nb1N22 Nb2N2. (60)k , b , a01, a03 are real integration constants, bi (i 41– 4) are complex integration
constants.
The constants k and b can be fixed by requiring that, if L 40, the solutions (50), (51) reduce to the ones corresponding to L 40 as in our previous paper [7]: we have that
k 4 1 3(l12 l3) , (61) b 4 2 l11 l3 3 b . (62)
From the constraints t0
4 t14 t24 0 , j34 0 one has that
b1b2* 1b3b4* 40 , b1b3* 1c.c.40 , b2b4* 1c.c.40 .
(63)
Moreover, we have that
a102a304 b
Lsign
(
k1 2F cosh (x)21)
. (64)The equation k2g 4 0 admits two (one if and only if k 4 0 and g 4 0) distinct real
roots t14 2b 1 1 k3 L ln
u
1 1kF k1 2Fv
, t24 2b 1 1 k3 L lnu
1 2kF k1 2Fv
. (65)They are singularities of metric and spin vector. Such singularities are physical because the curvature scalar
R 424 H.12 2 H
.
32 6 H122 2 H322 4 H1H32 6 K2(t3)2
(66)
Finally, sincek2g D 0 (t, we conclude that a) t ] 2Q , t2[U] t1, 1Q[ if bD0, b) t ] t2, t1[ if b E0.
B) If F D1, one has that
.
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/
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´
a1(t) 4a2(t) 4a10N[6kF 21 sinh (x)21N 1 /3m
N
ex2 z1 ex 2 z2N
n
6p1 , a3(t) 4a30N[6kF 21 sinh (x)21]N1 /3m
N
ex 2 z1 ex 2 z2N
n
Z2 p1 , (67).
`
/
`
´
c14 b1 (2g)1 /4 (e 2imt 2 iG(t)) , c34 b3 (2g)1 /4 (e imt 2iG(t)) , c24 b2 (2g)1 /4(e 2imt 1 iG(t)) , c44 b4 (2g)1 /4(e imt 1iG(t)) , (68)where p1 is given by (54), bi and a0 i are subjected to the constraints (63) and a102a304
b
Lsign
(
kF 21 sinh (x)21)
, respectively,k2g 4 b L[6kF 21 sinh (x)21] , (69) z14 1 1 kF kF 21 , z24 1 2 kF kF 21 , (70) G(t) 4 3 2 K
t 3 dt 4 1 2o
g g 16k2 lnN
ex 2 z1 ex2 z2N
(71) ifk2g 4 b L[1kF 21 sinh (x)21], G(t) 4 3 2 K t3 dt 4 1 2o
g g 16k2 lnN
ex2 z2 ex 2 z1N
(72) ifk2g 4 b L[2kF 21 sinh (x)21].z1is the only real singularity since, being z2E 0, the equation ex4 z2 has no real
solutions.
In this case, since L is inferiorly limited by a positive constant, it does make no sense to consider the limit L K0.
C) If F 41, we have that: C1) if
k2g 4 a(ex2 d) ,
where a is a real arbitrary constant, then . / ´ a14 a24 a10Na(ex2 d) N1 /3N1 2 de2xNp1, a34 a30Na(ex2 d) N1 /3N1 2 de2xN22 p1, (74)
.
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/
`
´
c14 b1 (2g)1 /4(e 2imt 2 iG(t)) , c34 b3 (2g)1 /4(e imt 2iG(t)) , c24 b2 (2g)1 /4(e 2imt 1 iG(t)) , c44 b4 (2g)1 /4(e imt 1iG(t)) , (75) where G(t) 4 1 2o
g g 16k2 ln N12de 2x N . (76)We notice that if a 41, b4g40, k40, so that p14 0 and a104 a30so that a104 1, from
(74) we obtain a stationary type solution [3]. C2) if
k2g 4 a(e2x
2 d) , (77)
where a is a real arbitrary constant, then
.
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/
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´
a14 a24 a10Na(e2x2 d) N1 /3N
1 d 2 e xN
p1, a34 a30Na(e2x2 d) N1 /3N
1 d 2 e xN
22 p1, (78).
`
/
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´
c14 b1 (2g)1 /4(e 2imt 2 iG(t)) , c34 b3 (2g)1 /4(e imt 2iG(t)) , c24 b2 (2g)1 /4(e 2imt 1 iG(t)) , c44 b4 (2g)1 /4 (e imt 1iG(t)) , (79) where G(t) 4 1 2o
g g 16k2 lnN
1 d 2 e xN
. (80)In both cases p1 has the usual value (54), bi are subjected to the usual constraints
and d 4 b aL , (81) (a10)2a304 1 , (82)
b 46
o
L(g 16k 2) 2 , (83) x 4k3 L t . (84)To the class of solutions with F 41 as well the static solution belongs: a14 a24 a10, a34 a30,
(85)
with g 42L, NbN4L, k40, bE0.
We notice that a static solution exists even in a spatially flat space-time [8]. That is possible since the spin term acts as a curvature term in the field equations as we can see looking at the expression of the curvature scalar which in such case is R 426 K2(t3)2. Differently from the Einstein static cosmological model, the scale
factors aido not depend on L and, so, on the mass-energy density of the matter, but on
arbitrary constants.
Moreover, c must depend on time in order to avoid contradictions which would result from the Dirac equation if c were constant (this is not a problem since c itself is not measurable : what is important is that every measurable quantity depending of c is constant):
.
/
´
c14 b1(e2imt 2 i( 3 /2 ) K t 3t ) , c34 b3(eimt 2i(3/2) K t 3t ) , c24 b2(e2imt 1 i( 3 /2 ) K t 3t ) , c44 b4(eimt 1i(3/2) K t 3t ) , (86) where t3 4 m 4 6o
L 3 K2 .Equation (86) guarantees that every measurable quantity depending on c is constant, being bilinear in c or power of such bilinear forms.
5. – Some remarks about the solutions
Among these solutions, only those of (50), (51) may be regarded as an extension of those corresponding to L 40 as found by us [1] because, with the settings (61) and (62), they reduce to the old ones if L 40. In the other cases we have that either, if FD1, it is not possible to consider the limit L K0 or, if F41, then L40 implies that b40, i.e. n 40 or m40 or both.
So, we limit ourselves to analyze briefly the solutions (50).
We see that L rules the evolution of the scale factor ai through the time constant
1
ck3 L, so that for any instant t of time such that t b 1
ck3 L, particularly in a
neighbourhood of the singularities t1and t2, the cosmological constant has no effect on
the evolution of the universe which is well approximated in this case by the solutions given in [7].
So, as t Kt1, t2, the evolution of the universe is strongly anisotropic, because of spin
and initial anisotropy (difference l12 l3) (see [7]).
Subsequently, the mass-energy damps increasingly the anisotropies of the universe whose metric, at infinity, is isotropic and acquires the stationary form aiP ek3 LNtNsince
it is ruled by L. Correspondently, the Hubble functions Hi become constants with the
Moreover, we notice that, given L, the constraint F E1 gives an upper limit to the anisotropy (due to spin and initial conditions); vice versa, if g c 6 k2, then F Bl2L, indipendently of initial conditions, where l 4ˇ/mc is the Compton length of a particle with mass m : in such case, for any given m, F E1 sets an upper limit to L.
Now, let us describe the evolution of the Universe as ruled by (50).
For any sign of p1, a14 a2is zero at t 4t1and t 4t2and does not diverge at any finite
value of t.
Along the x12 x2directions one has that:
a) if t ] 2Q, t2[ the universe contracts up to t 4t2;
b) if t ] t2, t1[ the universe expands up to an instant at which a14 a2 reaches a
maximum and then collapses up to t 4t1;
c) if t ] t1, 1Q[ the universe expands up to infinity.
Along the x3-direction we can have the following cases:
1) p1] 21/6, 1/6[, a3is zero at t 4t1and t 4t2and does not diverge at any finite
value of t. We have that:
a) if t ] 2Q, t2[ the universe contracts up to t 4t2;
b) if t ] t2, t1[ the universe expands up to an instant at which a14 a2reaches a
maximum and then collapses up to t 4t1;
c) if t ] t1, 1Q[ the universe expands up to infinity.
2) p1D 1 /6, a3diverges at t 4t1and is zero at t 4t2.
We have that:
a) if t ] 2Q, t2[ the universe contracts up to t 4t2; b) if t ] t2, t1[ the universe expands from t 4t2up to t 4t1;
c) if t ] t1, 1Q[ the universe contracts from t4t1up to an instant at which a3
reaches a mininum and then expands up to infinity.
3) p1E 21 /6, a3is zero at t 4t2and diverges at t 4t1.
We have that:
a) if t ] 2Q, t2[ the universe contracts up to an instant at which a3reaches a
mininum and then expands up to t 4t2;
b) if t ] t2, t1[ the universe contracts from t 4t2up to t 4t1; c) if t ] t1, 1Q[ the universe expands up to infinity.
4) if p14 11 /6, then a3P e2x/3Nex2 z2N2 /3: the universe contracts up to t 4t2,
where a3takes over a finite value, and then expands.
5) if p14 21 /6, then a3P e2x/3Nex2 z1N2 /3: the universe contracts up to t 4t1,
where a3takes over a finite value, and then expands.
We notice that if b E0, so that t]t2, t1[, and p1] 21/6, 1/6[, we have an
oscillating solution [9].
In such case, t12 t24 1 /k3 L ln
(
( 1 1kF) /( 1 2kF))
is the lifetime of theIf g c 6 k2
, then F BLl2
. Assuming that L B10256cm22[8], then F b 1 for any
reasonable value of m, so we have that, developing the logarithm, t12 t2B l/c 4 ˇ/mc2,
Compton time [9].
In particular, if we set k 40 and a104 a30, we get the isotropic solution R(t) 4a1(t) 4a2(t) 4a3(t) 4N b L
(
k1 2F cosh (x)21)
N 1 /3, (87) where x 4k3 Lt.Assuming again that F b 1, i.e. l2b1 /L, we find that spin is effective for time intervals Nt2t1N 4 Nt 2 Nt2V B l/c [9], after which the mass-energy rules the evolution
of the universe up to t F1/ck3 L ; finally, when t c 1 /ck3 L , we get an empty universe which evolves as a stationary-type one.
Finally, if in (87) we set g 40, one has F40 and we get the solution of Isham-Nelson with cosmological constant in a spatially flat FRW universe [10].
6. – Conclusions
In a Bianchi type-I universe with plane symmetry and cosmological constant and torsion characterized by the only non-zero space-component t3 of the spin vector, we have found the solutions of the Einstein-Dirac equations.
If F E1, such solutions extend our previous ones [1], as Lc0.
In such case, if b D0, so that t]2Q, t2[N]1 , t1, 1Q[, the evolution of the
universe goes through three stages: one, strongly anisotropic, ruled by spin as t K t1, t2; a second one ruled by the mass-energy which damps anisotropies; finally, a third
one, altogether isotropic, ruled by L, which shows a stationary-type empty universe. If p1] 21/6, 1/6[ and bE0, we obtain an oscillating solution even in a spatially flat
space-time: in a particular case, g c 6 k2, such Universe has a lifetime t
12 t24 l/c
independent of initial conditions.
* * *
One of the authors (RR) would like to thank with all his heart Mrs. M. C. TINO for inspiring the present paper.
R E F E R E N C E S
[1] HEHL F. W., VON DERHEYDE P., KERLICK D. G. and NESTER J. M., Rev. Mod. Phys., 48 (1976) 393.
[2] DE SABBATA V. and SIVARAM C., Spin and Torsion in Cosmology (World Scientific, Singapore) 1994.
[3] RAYCHAUDHURIA. K., Theoretical Cosmology (Oxford University Press, London) 1979. [4] SCHOUTENJ. A., Ricci Calculus, 2nd edition (Springer-Verlag, Berlin) 1954.
[5] HEHLF. W. and DATTAB. K., J. Math. Phys., 12 (1971) 1334. [6] KERLICKD. G., Phys. Rev. D, 12 (1975) 3004.
[7] PLATANIAG. and ROSANIAR., Nuovo Cimento B, 111 (1996) 1195.
[8] ZELDOVICH J. and NOVIKOV I., Struttura ed evoluzione dell’universo, 1st edition (Editori Riuniti, Roma) 1982.
[9] PLATANIAG. and ROSANIAR., unpublished.
