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Generalised cosmological friedmann equations without gravitational singularity
Volume 80A, number 4
PHYSICS LETTERS
8 December 1980
GENERALISED COSMOLOGICAL FRIEDMANN EQUATIONS WITHOUT GRAVITATIONAL SINGULARITY A.V. MINKEVICH
Department of Physics, Byelorussian State University, Minsk, USSR Received 10 July 1980 Revised manuscript received 1 October 1980
Within the framework of the gauge approach to gravitation, including terms in the lagrangian quadratic in the curvature and torsion tensors, a generalised Friedmann equation for a homogeneous isotropic cosmology is obtained. This equation avoids the gravitational singularity with infinite mass density.
It is known that general relativity (GR) does not lead to any restrictions on admissible mass densities in gravitating systems. Various cosmological models and collapsing systems in GR have a singularity with infinite mass density. Since these states are physically unacceptable GR must be modified in the case of systems with extremely high density. In a number of papers (see e.g. refs. [ 1 - 4 ] ) this problem was discussed taking quantum effects into account. It should be noted, that the investigation o f quantum effects for highmass-density systems is a complicated problem, since a complete quantum gravitational theory does not exist. In the framework o f the gauge approach GR is modified by means of a space-time torsion [5]. In the simplest gauge theory of gravitation - the Einstein-Cartan theory - the influence of torsion connected with the spin of matter was investigated in refs. [6,7]. However, as was pointed out in refs. [ 8 10] the conclusions in this case essentially depend on the kind of spinning matter. In the case of spinless matter the Einstein-Cartan theory is identical to GR. The situation is different in a gauge theory of gravitation, in which the gravitational lagrangian contains both a scalar curvature and terms quadratic in the curvature and torsion tensors [11]. As is shown below this theory of gravitation essentially differs from GR for systems of high density and allows one to avoid the singularity with infinite mass density. In the framework o f the gauge approach, gravity is 232
described by two gauge fields: the tetrad field h(u and the rotating Ricci coefficients A!/~u * a. If we do not use the completely antisymmetric Levi-Civita symbol, then the most general expression for the gravitational lagrangian quadratic in the curvature and the torsion is
Lg = h[foF + (flFuv~a + f2Fuxvo + f3Fxa.v)FUVX~ + ( f 4 F v +fsFvta) F#v + f6 F2 + (alSuv x + a2Shvu)S uvx +a3SU.~vS'~ uh]
(Fur =Fa. uxv,F=FU..,h = det(h(.)),
(1)
where the curvature tensor is given by
Ff(uv = 20[uA(Jv] + 2A!![uA!IIIv],
(2)
the torsion tensor by
S x• uv = hiX(O[vh~.u] • - A i.[uv] )' f0 = 1/167rG (G is Newton's gravitational constant), a m (m = 1,2, 3) and fn (n = 1,2, ..., 6) are constants. We consider the case of a homogeneous isotropic space with metric
.1 i, j, ... are anholonomic Lorentz indices, #, v, ... are holonomic coordinate indices, the signature is equal to -2, the light velocity e = 1.
Volume 80A, number 4
PHYSICS LETTERS
R2(t)
guy = diag (1,
_R2(t)r2,_R2(t)r2sin2O)
8 December 1980
From the eqs. (6), using eq. (5), the conservation law follows:
b+ (k = 0,-+1), and with nonvanishing components of the torsion tensor (by conservation of the parity of the space)S.110 = $2 20 = S.330 - S(t) [ 12]. Then the curvature tensor F.UV.xa has two linearly independent components A -=F0!01
= F02
• .02
=
F03 • .03'
=
(3)
AR,
(4)
where a dot denotes differentiation with respect to time. There is a differential relation between eqs. (3) and (4), determined by the Bianchi identity, which in this case has the following form: k - 2(R/R)(A - B) + 4SA = 0.
(5)
The gravitational equations corresponding to the lagrangian (1) have the form ]using eq. (5)]
a(R [R - S)S + 12f(A 2 - B 2) - 6foB = -p, ~ a ¢ + 2(R/R)S -
S 2) -
Eqs. (3) and (4) with the functions S, A, B defined according to (7) are the generalization for a space-time with torsion of the cosmological Friedmann equations, into which eqs. (3) and (4) transform when f ~ 0. Together with the conservation law (8), and the equation of statep = P(O), eq. (3) defines the functions R(t), Ifa = 0 the generalised Friedmann equation (3) is a first order differential equation:
k + (R - 2RS) 2 = BR 2, 2(RS)"
4f(A 2
(8)
p(t).
B = F! 2.12 = F ! "313 = F23 • • 23' and according to the definition (2)
J~ -
+p)= 0.
-
B 2)
(6) - 2f0(ZA + B) = p, f [ ( . , / + k ) + 4 S ( A +B)] + b S = 0 ,
1 +f(-o - ~P)/3f 2 R]
(9)
1 p +f(P - 3p)2/12f2 R 2. 6 f 0 1 +f(p - 3p)/3f~
Integrating eq. (9) gives that for f < 0 for systems with p < ±P 3 the mass density cannot exceed the limiting value Pmax defined by the condition Pmax - 3P(Omax) = 3f2/Ifl ,2. This is because the right hand side of eq. (9) would become negative while the left hand side is a square. The existence of Pmax prevents the singularity of the metric. If p ~ #max, eq. (9) transforms into the cosmological Friedmann equation of GR. Let us consider a cosmological model with noninteracting dust matter (Pd= 0) and radiation (Pr = ~Pr): P =~d + Pr,P =Pd +Pr [13]. In this case: Pd max = 3fo/Ifl, and the minimal value of the function R(t) is
[ PdO 1[3
where P is the mass density, p is the pressure,
Rmin=RO~Pdmax]
a = 3(2a 1 + a 2 + 3a3),
' Prmax
=P [0.4,~0~ r 0,Rmin]
1
P(Pmax) ~ 5Pmax'
f = f l +"~f2
+f3
+f4
+f5 + 3f6' b=fo +~'44a"
Eqs. (6), taking into account eqs. (3), (4) with b 4: 0, give F = 6(A +B) =
S =-
ft~ 6b(1 + 2fF/3b)'
a 2k + (R2) "" ÷ 8b R2 '
B = p + l f F 2 + la(k +/~2)/R2 66(1 + 2fF/3b) (7)
where the index zero denotes the values of the modern epoch. The state with Pr max has the limiting temperature Tmax = T ORo/Rmi n. Integrating eq. (9) we obtain for the time t, which corresponds to the change of R(t) from Rmi n to some value R, the expression
l . 2 In the case of equations of state for which p > Tp, at ultra-
high densities a limiting mass density appears, iff > O.
233
Volume 80A, number 4
R
PHYSICS LETTERS
,qb 1 ,1/2
.o, R rain
1
dp1 = (Pdmax x3 + 7C1
)2
alP2 = (rid max x 3 _ C1 ) {_k(p d max x 3 _ C1 ) + ( 6 f 0 ) - I [(C1/x)(P d max x3 - 1 C I )
+ PdmaxProR4xl )
(CI = PdoR30).
For small t, when R(t) - R m i n "~ Rmi n we have
R(t) = R m i n + r I t 2 + r 2 t 4 + . . .
(r 1 > 0 ) ,
by t ~ 0: R -+ Rmin, I~/R -+ O, R/R > 0. The obtained solution (10) means the absence of a big bang and the presence o f a regular transition from compression to expansion. It takes place because o f the gravitational repulsion effect by mass.densities near Pmax" Let us note, that in the case a = 0 at the point R = Rmi n the functions S, A and B diverge, while the metric and the scalar curvature F are regular. I f a 4= 0 then eq. (3) is a third order differential equation. It is easy to see, that neglecting the torsion in eqs. (3), (4) if the conditions 2 l / S I ~ [/~1,
2 l ( R S ) ' l ' ~ I/~1,
are satisfied corresponds to small values o f the mass density * a :
p + p ~12bfo/f(1 - 3~p/b#)l,
(11)
and leads to the cosmological Friedmann equations of , a Condition (11) does no t restrict the value of p ff the equation of state is exactly p = -~#. In the case p = ~# the singular solutions of GR also satisfy eqs. (3), (4).
234
8 December 1980
GR. The problem o f obtaining regular solutions of eq. (3) with Friedmann behaviour for small values o f the density requires a more careful consideration. In conclusion we note the following. Usually one connects the torsion with the spin o f matter. As was shown above, in the quadratic gauge theory of gravitation the torsion is important in the case of spinless matter with a high density and can prevent a gravitational singularity with infinite mass density. Besides in order that the correspondence principle be satisfied, i.e. in order that the conclusion of this theory should agree with GR for systems with rather small mass densities by neglect of the torsion, it is necessary to take into account a term linear in the curvature in the gravi. tational lagrangian.
References [1] Ya.B. Zel'dovich and I.D. Novikov, Structure and evolution of the universe (Nauka, Moscow, 1975). [2] V.L. Ginzburg, D.A. Kirzhnits and A.A. Lyubushin, Zh. Eksp. Teor. Fiz. 60 (1971) 451. [3] V.Ts. Gurovich and A.A. Starobinski, Zh. Eksp. Teor. Fiz. 77 (1979) 1683. [4] V.P. Frolov and G.A. Vilkovisky, preprint IC/79/69, Trieste (1979). [5] F.W. Hehl, P. vonder Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. [6] A. Trautman, Nature (Phys. Sci.) 242 (1973) 7. [7] W. Kopczynski, Phys. Lett. 43A (1973) 63. [8] G.D. Kerlick, Ann. Phys. (NY) 99 (1976) 127. [9] J. Tafel, Bull. Acad. Pol. Sci. ser. math. astron, phys. 25 (1977) 593. [10] A. Inomata, Phys. Rev. D18 (1978) 3552. [11] A.V. Minkevich, Izv. Akad. Nauk BSSR Ser. Fiz.-Mat. no. 2 (1980) 87. [12] V.I. Kudin, A.V. Minkevich and F.I. Fedorov, preprint N 3794-79, VINITI Akad. Nauk SSSR (1979). [13] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Vol. 2 (Moscow, 1977) p. 398.
PHYSICS LETTERS
8 December 1980
GENERALISED COSMOLOGICAL FRIEDMANN EQUATIONS WITHOUT GRAVITATIONAL SINGULARITY A.V. MINKEVICH
Department of Physics, Byelorussian State University, Minsk, USSR Received 10 July 1980 Revised manuscript received 1 October 1980
Within the framework of the gauge approach to gravitation, including terms in the lagrangian quadratic in the curvature and torsion tensors, a generalised Friedmann equation for a homogeneous isotropic cosmology is obtained. This equation avoids the gravitational singularity with infinite mass density.
It is known that general relativity (GR) does not lead to any restrictions on admissible mass densities in gravitating systems. Various cosmological models and collapsing systems in GR have a singularity with infinite mass density. Since these states are physically unacceptable GR must be modified in the case of systems with extremely high density. In a number of papers (see e.g. refs. [ 1 - 4 ] ) this problem was discussed taking quantum effects into account. It should be noted, that the investigation o f quantum effects for highmass-density systems is a complicated problem, since a complete quantum gravitational theory does not exist. In the framework o f the gauge approach GR is modified by means of a space-time torsion [5]. In the simplest gauge theory of gravitation - the Einstein-Cartan theory - the influence of torsion connected with the spin of matter was investigated in refs. [6,7]. However, as was pointed out in refs. [ 8 10] the conclusions in this case essentially depend on the kind of spinning matter. In the case of spinless matter the Einstein-Cartan theory is identical to GR. The situation is different in a gauge theory of gravitation, in which the gravitational lagrangian contains both a scalar curvature and terms quadratic in the curvature and torsion tensors [11]. As is shown below this theory of gravitation essentially differs from GR for systems of high density and allows one to avoid the singularity with infinite mass density. In the framework o f the gauge approach, gravity is 232
described by two gauge fields: the tetrad field h(u and the rotating Ricci coefficients A!/~u * a. If we do not use the completely antisymmetric Levi-Civita symbol, then the most general expression for the gravitational lagrangian quadratic in the curvature and the torsion is
Lg = h[foF + (flFuv~a + f2Fuxvo + f3Fxa.v)FUVX~ + ( f 4 F v +fsFvta) F#v + f6 F2 + (alSuv x + a2Shvu)S uvx +a3SU.~vS'~ uh]
(Fur =Fa. uxv,F=FU..,h = det(h(.)),
(1)
where the curvature tensor is given by
Ff(uv = 20[uA(Jv] + 2A!![uA!IIIv],
(2)
the torsion tensor by
S x• uv = hiX(O[vh~.u] • - A i.[uv] )' f0 = 1/167rG (G is Newton's gravitational constant), a m (m = 1,2, 3) and fn (n = 1,2, ..., 6) are constants. We consider the case of a homogeneous isotropic space with metric
.1 i, j, ... are anholonomic Lorentz indices, #, v, ... are holonomic coordinate indices, the signature is equal to -2, the light velocity e = 1.
Volume 80A, number 4
PHYSICS LETTERS
R2(t)
guy = diag (1,
_R2(t)r2,_R2(t)r2sin2O)
8 December 1980
From the eqs. (6), using eq. (5), the conservation law follows:
b+ (k = 0,-+1), and with nonvanishing components of the torsion tensor (by conservation of the parity of the space)S.110 = $2 20 = S.330 - S(t) [ 12]. Then the curvature tensor F.UV.xa has two linearly independent components A -=F0!01
= F02
• .02
=
F03 • .03'
=
(3)
AR,
(4)
where a dot denotes differentiation with respect to time. There is a differential relation between eqs. (3) and (4), determined by the Bianchi identity, which in this case has the following form: k - 2(R/R)(A - B) + 4SA = 0.
(5)
The gravitational equations corresponding to the lagrangian (1) have the form ]using eq. (5)]
a(R [R - S)S + 12f(A 2 - B 2) - 6foB = -p, ~ a ¢ + 2(R/R)S -
S 2) -
Eqs. (3) and (4) with the functions S, A, B defined according to (7) are the generalization for a space-time with torsion of the cosmological Friedmann equations, into which eqs. (3) and (4) transform when f ~ 0. Together with the conservation law (8), and the equation of statep = P(O), eq. (3) defines the functions R(t), Ifa = 0 the generalised Friedmann equation (3) is a first order differential equation:
k + (R - 2RS) 2 = BR 2, 2(RS)"
4f(A 2
(8)
p(t).
B = F! 2.12 = F ! "313 = F23 • • 23' and according to the definition (2)
J~ -
+p)= 0.
-
B 2)
(6) - 2f0(ZA + B) = p, f [ ( . , / + k ) + 4 S ( A +B)] + b S = 0 ,
1 +f(-o - ~P)/3f 2 R]
(9)
1 p +f(P - 3p)2/12f2 R 2. 6 f 0 1 +f(p - 3p)/3f~
Integrating eq. (9) gives that for f < 0 for systems with p < ±P 3 the mass density cannot exceed the limiting value Pmax defined by the condition Pmax - 3P(Omax) = 3f2/Ifl ,2. This is because the right hand side of eq. (9) would become negative while the left hand side is a square. The existence of Pmax prevents the singularity of the metric. If p ~ #max, eq. (9) transforms into the cosmological Friedmann equation of GR. Let us consider a cosmological model with noninteracting dust matter (Pd= 0) and radiation (Pr = ~Pr): P =~d + Pr,P =Pd +Pr [13]. In this case: Pd max = 3fo/Ifl, and the minimal value of the function R(t) is
[ PdO 1[3
where P is the mass density, p is the pressure,
Rmin=RO~Pdmax]
a = 3(2a 1 + a 2 + 3a3),
' Prmax
=P [0.4,~0~ r 0,Rmin]
1
P(Pmax) ~ 5Pmax'
f = f l +"~f2
+f3
+f4
+f5 + 3f6' b=fo +~'44a"
Eqs. (6), taking into account eqs. (3), (4) with b 4: 0, give F = 6(A +B) =
S =-
ft~ 6b(1 + 2fF/3b)'
a 2k + (R2) "" ÷ 8b R2 '
B = p + l f F 2 + la(k +/~2)/R2 66(1 + 2fF/3b) (7)
where the index zero denotes the values of the modern epoch. The state with Pr max has the limiting temperature Tmax = T ORo/Rmi n. Integrating eq. (9) we obtain for the time t, which corresponds to the change of R(t) from Rmi n to some value R, the expression
l . 2 In the case of equations of state for which p > Tp, at ultra-
high densities a limiting mass density appears, iff > O.
233
Volume 80A, number 4
R
PHYSICS LETTERS
,qb 1 ,1/2
.o, R rain
1
dp1 = (Pdmax x3 + 7C1
)2
alP2 = (rid max x 3 _ C1 ) {_k(p d max x 3 _ C1 ) + ( 6 f 0 ) - I [(C1/x)(P d max x3 - 1 C I )
+ PdmaxProR4xl )
(CI = PdoR30).
For small t, when R(t) - R m i n "~ Rmi n we have
R(t) = R m i n + r I t 2 + r 2 t 4 + . . .
(r 1 > 0 ) ,
by t ~ 0: R -+ Rmin, I~/R -+ O, R/R > 0. The obtained solution (10) means the absence of a big bang and the presence o f a regular transition from compression to expansion. It takes place because o f the gravitational repulsion effect by mass.densities near Pmax" Let us note, that in the case a = 0 at the point R = Rmi n the functions S, A and B diverge, while the metric and the scalar curvature F are regular. I f a 4= 0 then eq. (3) is a third order differential equation. It is easy to see, that neglecting the torsion in eqs. (3), (4) if the conditions 2 l / S I ~ [/~1,
2 l ( R S ) ' l ' ~ I/~1,
are satisfied corresponds to small values o f the mass density * a :
p + p ~12bfo/f(1 - 3~p/b#)l,
(11)
and leads to the cosmological Friedmann equations of , a Condition (11) does no t restrict the value of p ff the equation of state is exactly p = -~#. In the case p = ~# the singular solutions of GR also satisfy eqs. (3), (4).
234
8 December 1980
GR. The problem o f obtaining regular solutions of eq. (3) with Friedmann behaviour for small values o f the density requires a more careful consideration. In conclusion we note the following. Usually one connects the torsion with the spin o f matter. As was shown above, in the quadratic gauge theory of gravitation the torsion is important in the case of spinless matter with a high density and can prevent a gravitational singularity with infinite mass density. Besides in order that the correspondence principle be satisfied, i.e. in order that the conclusion of this theory should agree with GR for systems with rather small mass densities by neglect of the torsion, it is necessary to take into account a term linear in the curvature in the gravi. tational lagrangian.
References [1] Ya.B. Zel'dovich and I.D. Novikov, Structure and evolution of the universe (Nauka, Moscow, 1975). [2] V.L. Ginzburg, D.A. Kirzhnits and A.A. Lyubushin, Zh. Eksp. Teor. Fiz. 60 (1971) 451. [3] V.Ts. Gurovich and A.A. Starobinski, Zh. Eksp. Teor. Fiz. 77 (1979) 1683. [4] V.P. Frolov and G.A. Vilkovisky, preprint IC/79/69, Trieste (1979). [5] F.W. Hehl, P. vonder Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393. [6] A. Trautman, Nature (Phys. Sci.) 242 (1973) 7. [7] W. Kopczynski, Phys. Lett. 43A (1973) 63. [8] G.D. Kerlick, Ann. Phys. (NY) 99 (1976) 127. [9] J. Tafel, Bull. Acad. Pol. Sci. ser. math. astron, phys. 25 (1977) 593. [10] A. Inomata, Phys. Rev. D18 (1978) 3552. [11] A.V. Minkevich, Izv. Akad. Nauk BSSR Ser. Fiz.-Mat. no. 2 (1980) 87. [12] V.I. Kudin, A.V. Minkevich and F.I. Fedorov, preprint N 3794-79, VINITI Akad. Nauk SSSR (1979). [13] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Vol. 2 (Moscow, 1977) p. 398.