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The singularity of homogeneous cosmological models with electromagnetic fields
Volume 81A, number 9
PHYSICS LETTERS
23 February 1981
THE SINGULARITY OF HOMOGENEOUS COSMOLOGICAL MODELS WITH ELECTROMAGNETIC FIELDS B.L. SPOKOINY L.D. Landau Institute for TheoreticalPhysics of the USSR Academy of Sciences, 117334 Moscow, USSR Received 10 December 1980
The regime of approaching the singularity in the presence of electromagnetic fields has been found. The rotation angles of the Kasner axes are calculated.
Belinsky and Khalatnikov [l}, Bogoyavlensky [2] and Ruban [3] investigated the effect of a sourceless electromagnetic field (EMF) on the evolution of homogeneous cosmological models near the singularity. But they restricted themselves to the simplest case of a diagonal metric and a simple type of parallel electric and magnetic fields. In the present note we show how their results are generalized for the case of an arbitrary (non-diagonal) metric and an arbitrary six-component sourceless EMF. Such a generalization results in the appearance of some new properties, as compared to the diagonal case: rotation of the Kasner axes takes place under successive changes of the Kasner epochs. We have calculated the angles of this rotation. In our paper we give the fmal results of the work and discuss physical derivation these resultstheir will be given meaning. in a more The detailed paper.of The Einstein equations have the form R~ç 1~R6~ç = —FkIF’1 + t~6~FimFlm
a
~ b tP2, c ~ <~
1’~,
t
Passing to the new basis vectors without changing the structure constants [4,5] we can get the components of the vectors L, M, N in the new basis in the following form: L =(1, 0,0), M=(cos 0m’ ~ 0m’ 0), (2) N (cos O~,sin °n cos ~n’ sin O~sin ~Pn). All three-dimensional vectors and tensors are projected onto three basis vectors. The components of such projections are functions of time only. Let us introduce the vectors of the “reduced” electric and magnetic fields B’1 = _ 71/2çOa D’~= X FbcI2, where a, b, c run through the values 1, 2,123 3, detII7abII, e’~~ is the antisymmetric symbol (eare == +1). Here and elsewhere all tensorial operations performed with the help of metric (1).
where ElkKasner is the EMF In the epochtensor. [5], when the energy—momentum tensor of matter and the three-dimensional Ricci tensor can be neglected in the Einstein equations, the three-dimensional metric has in the general case the form [4]
Since a(t) b(t), c(t); b(t) 0, c(t) -÷0fort —*0, it may be shown that for Bianchi types V1 1)2 + (D1)2 const. [the 0, V110, VIII, IX, I, II we have metric coefficient a2(t)(Bincreases in the direction 1]. All the remaining quantities: B2, B3 and D2, D3 are almost constant (up to terms -~-t~, where F> 0). The existence of the Belinsky—Khalatnikov—Lifshitz (BKL) oscillation regime depends only on whether
c2(t)N~N 1, where L, M, N are constant vectors,
1)2 + (D1)2 are zero[5]simultaneously. the “main curvatures” and the quantity U0 = (B
—
Yab
,
a2 (t)LaLb + b2 (t)MaMb
~‘
+
,
(1)
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
—~
493
Volume 81A, number 9
PHYSICS LETTERS
When analysing the Einstein equations, we have found that in the metrics of Bianchi types VI0, VII0, VIII and IX there occurs an oscillatory regime, whereas the other types of metric have the Kasner asymptotics. caseV1the constant quantity U0 in the modelsInofthis types 0 and Vu0 is assumed to be nonzero (in other words, there is an electric or a magnetic field in the direction 1); otherwise, these models have a Kasner asymptotics too. For types VIII and IX such a requirement (U0 > 0) is absent. The above-mentioned results generalize the results of refs. [1—3]. The spaces of type III, IV, VIa, Vila do not permit the existence of a nonzero electric or magnetic field component in the direction of the “vector” a [5]. This direction is physically separated for these models. In the direction of the “vector” a the “main curvature” equals zero. And, hence, the spaces of types III, IV, VIa, VIIa have the Kasner asymptotics with a negative index in the above-mentioned direction. A nonzero quantity U0 which “plays the role” of the “main curvature” in the direction of the “vector” a is absent in this case. It should be noted that in the spaces of type v, in the presence an EMF therein appears a Kasner asymptotics with a of negative index the direction of the “vector” a. For the sake of definiteness, suppose that the “vector” a lies in the direction 1. The negative index P 2D1 B1D2. 1 equals (1 +exists q)/3, for where q =q B< —1. This asymptotics —2 ~ The law of rotation of the Kasner axes of successive Kasner epochs for the “oscillating” models of types V1 0, V110, VIII and IX has the form 0m + a cotg 0’m + a1 2u 1 cotg 1 2u + (3) —
—
u+2 cotg o’ = cotg o ~ u—2 —~---~
(a1 cos
+ a2
sin
)
where u is the BKL parameter [4]. If the main curvature in the direction 1 of the
494
increasing Kasner axis equals zero, which may be the case for the models of types V10 and V110, then we have in eq. (3) 2/2B1 = D212D1, a 3/2B1 = D3/2D1. (4) a1 = B 2 =B If the main curvature in the direction 1 of the increasing Kasner axis is not zero, we have in eq. (3) 1B2 + D1D2)/u 1B3 +D1D3)/u a1 = (B 0, a2 = (B 0. (5) The quantities a1 and a2 in eq. (4) are constants near the singularity up to terms ~ where F> 0. In eq. (5) the quantitiesa1 and a2 are constant for a finite number of epochs near the singularity, and they are twice as large as in eq. (4). The quantitiesa1 and a2 have the meaning of the relative “non-parallelity” of the EMF to the increasing axis. When the electric and magnetic fields lie in the direction 1 we have a1 = a2 = 0 and expression (3) co~ incides with the known BKL law [4] for models of type IX without an EMF. The law (3) has a simple physical meaning. The theas“non-parallel” 2, B3, D2greater D3 are, compared to EMF paralcomponents B lel B1 and D1 the “greater” are the angles of rotation of the axes. ,
,
I am very grateful to Professor I.M. Khalatnikov for constant interest in this work and for useful discussions of the results of the paper. References [1] V.A. Belinsky and I.M. Khaiatnikov, preprint, Landau Institute of Theoretical Physics, Chernogolovka, USSR (1976). [2] 0.!. Bogoyavlensky, Teor. Mat. Fjz. 27(1976)184 (in [3] Russian). V.A. Ruban, preprint, Leningrad Institute of Nuclear Physics, USSR, N411 (1978).
—
+
23 February 1981
[4] V.A. Lifshitz and I.M. Zh. Eksp. Belinsky, Teor. Fiz.E.M. 60 (1970)1969 [SoY.Khalatnikov, Phys. JETP 33 (1971) 1061].
[5] L.D. Landau and E.M. Lifshitz, Field theory (Nauka, Moscow, 1973) section 116.
PHYSICS LETTERS
23 February 1981
THE SINGULARITY OF HOMOGENEOUS COSMOLOGICAL MODELS WITH ELECTROMAGNETIC FIELDS B.L. SPOKOINY L.D. Landau Institute for TheoreticalPhysics of the USSR Academy of Sciences, 117334 Moscow, USSR Received 10 December 1980
The regime of approaching the singularity in the presence of electromagnetic fields has been found. The rotation angles of the Kasner axes are calculated.
Belinsky and Khalatnikov [l}, Bogoyavlensky [2] and Ruban [3] investigated the effect of a sourceless electromagnetic field (EMF) on the evolution of homogeneous cosmological models near the singularity. But they restricted themselves to the simplest case of a diagonal metric and a simple type of parallel electric and magnetic fields. In the present note we show how their results are generalized for the case of an arbitrary (non-diagonal) metric and an arbitrary six-component sourceless EMF. Such a generalization results in the appearance of some new properties, as compared to the diagonal case: rotation of the Kasner axes takes place under successive changes of the Kasner epochs. We have calculated the angles of this rotation. In our paper we give the fmal results of the work and discuss physical derivation these resultstheir will be given meaning. in a more The detailed paper.of The Einstein equations have the form R~ç 1~R6~ç = —FkIF’1 + t~6~FimFlm
a
~ b tP2, c ~ <~
1’~,
t
Passing to the new basis vectors without changing the structure constants [4,5] we can get the components of the vectors L, M, N in the new basis in the following form: L =(1, 0,0), M=(cos 0m’ ~ 0m’ 0), (2) N (cos O~,sin °n cos ~n’ sin O~sin ~Pn). All three-dimensional vectors and tensors are projected onto three basis vectors. The components of such projections are functions of time only. Let us introduce the vectors of the “reduced” electric and magnetic fields B’1 = _ 71/2çOa D’~= X FbcI2, where a, b, c run through the values 1, 2,123 3, detII7abII, e’~~ is the antisymmetric symbol (eare == +1). Here and elsewhere all tensorial operations performed with the help of metric (1).
where ElkKasner is the EMF In the epochtensor. [5], when the energy—momentum tensor of matter and the three-dimensional Ricci tensor can be neglected in the Einstein equations, the three-dimensional metric has in the general case the form [4]
Since a(t) b(t), c(t); b(t) 0, c(t) -÷0fort —*0, it may be shown that for Bianchi types V1 1)2 + (D1)2 const. [the 0, V110, VIII, IX, I, II we have metric coefficient a2(t)(Bincreases in the direction 1]. All the remaining quantities: B2, B3 and D2, D3 are almost constant (up to terms -~-t~, where F> 0). The existence of the Belinsky—Khalatnikov—Lifshitz (BKL) oscillation regime depends only on whether
c2(t)N~N 1, where L, M, N are constant vectors,
1)2 + (D1)2 are zero[5]simultaneously. the “main curvatures” and the quantity U0 = (B
—
Yab
,
a2 (t)LaLb + b2 (t)MaMb
~‘
+
,
(1)
0 031—9163/81/0000—0000/$ 02.50 © North-Holland Publishing Company
—~
493
Volume 81A, number 9
PHYSICS LETTERS
When analysing the Einstein equations, we have found that in the metrics of Bianchi types VI0, VII0, VIII and IX there occurs an oscillatory regime, whereas the other types of metric have the Kasner asymptotics. caseV1the constant quantity U0 in the modelsInofthis types 0 and Vu0 is assumed to be nonzero (in other words, there is an electric or a magnetic field in the direction 1); otherwise, these models have a Kasner asymptotics too. For types VIII and IX such a requirement (U0 > 0) is absent. The above-mentioned results generalize the results of refs. [1—3]. The spaces of type III, IV, VIa, Vila do not permit the existence of a nonzero electric or magnetic field component in the direction of the “vector” a [5]. This direction is physically separated for these models. In the direction of the “vector” a the “main curvature” equals zero. And, hence, the spaces of types III, IV, VIa, VIIa have the Kasner asymptotics with a negative index in the above-mentioned direction. A nonzero quantity U0 which “plays the role” of the “main curvature” in the direction of the “vector” a is absent in this case. It should be noted that in the spaces of type v, in the presence an EMF therein appears a Kasner asymptotics with a of negative index the direction of the “vector” a. For the sake of definiteness, suppose that the “vector” a lies in the direction 1. The negative index P 2D1 B1D2. 1 equals (1 +exists q)/3, for where q =q B< —1. This asymptotics —2 ~ The law of rotation of the Kasner axes of successive Kasner epochs for the “oscillating” models of types V1 0, V110, VIII and IX has the form 0m + a cotg 0’m + a1 2u 1 cotg 1 2u + (3) —
—
u+2 cotg o’ = cotg o ~ u—2 —~---~
(a1 cos
+ a2
sin
)
where u is the BKL parameter [4]. If the main curvature in the direction 1 of the
494
increasing Kasner axis equals zero, which may be the case for the models of types V10 and V110, then we have in eq. (3) 2/2B1 = D212D1, a 3/2B1 = D3/2D1. (4) a1 = B 2 =B If the main curvature in the direction 1 of the increasing Kasner axis is not zero, we have in eq. (3) 1B2 + D1D2)/u 1B3 +D1D3)/u a1 = (B 0, a2 = (B 0. (5) The quantities a1 and a2 in eq. (4) are constants near the singularity up to terms ~ where F> 0. In eq. (5) the quantitiesa1 and a2 are constant for a finite number of epochs near the singularity, and they are twice as large as in eq. (4). The quantitiesa1 and a2 have the meaning of the relative “non-parallelity” of the EMF to the increasing axis. When the electric and magnetic fields lie in the direction 1 we have a1 = a2 = 0 and expression (3) co~ incides with the known BKL law [4] for models of type IX without an EMF. The law (3) has a simple physical meaning. The theas“non-parallel” 2, B3, D2greater D3 are, compared to EMF paralcomponents B lel B1 and D1 the “greater” are the angles of rotation of the axes. ,
,
I am very grateful to Professor I.M. Khalatnikov for constant interest in this work and for useful discussions of the results of the paper. References [1] V.A. Belinsky and I.M. Khaiatnikov, preprint, Landau Institute of Theoretical Physics, Chernogolovka, USSR (1976). [2] 0.!. Bogoyavlensky, Teor. Mat. Fjz. 27(1976)184 (in [3] Russian). V.A. Ruban, preprint, Leningrad Institute of Nuclear Physics, USSR, N411 (1978).
—
+
23 February 1981
[4] V.A. Lifshitz and I.M. Zh. Eksp. Belinsky, Teor. Fiz.E.M. 60 (1970)1969 [SoY.Khalatnikov, Phys. JETP 33 (1971) 1061].
[5] L.D. Landau and E.M. Lifshitz, Field theory (Nauka, Moscow, 1973) section 116.