Dynamics of the General Bianchi IX Model near a Cosmological Singularity
DOI:
https://doi.org/10.15407/ujpe67.2.93Keywords:
general relativity, cosmology, singularity, general solutionAbstract
Half a century ago, Belinsky and Khalatnikov proposed a generic solution of the Einstein equations near their cosmological singularity, basing on a generalization of the homogeneous model of Bianchi type IX. The consideration of the evolution of the most general non-diagonal case of this model is significantly simplified, if it is assumed that, when approaching the singularity t = 0, it reduces to the so-called asymptotic dynamics, at which inequality Γ1 ≫ Γ2 ≫ Γ3 holds. It has been suggested that this inequality continues to be true from the moment of its first fulfilment up to the singularity of space-time. We analyze this assumption and show that it is incorrect in the general case. However, it is shown that in any case there exists a time t0, after which this assumption becomes true. The value of t0 is the smaller, the less is the degree of non-diagonality of the model. Some details of the behavior of the non-diagonal homogeneous model of Bianchi type IX are considered at the stage of asymptotic dynamics of approaching the singularity.
References
V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz. A general solution of the Einstein equations with a time singularity. Adv. Phys. 31, 639 (1982).
https://doi.org/10.1080/00018738200101428
E.M. Lifshitz, I.M. Khalatnikov. Investigations in relativistic cosmology. Adv. Phys. 12, 185 (1963).
https://doi.org/10.1080/00018736300101283
L.D. Landau, E.M. Lifshitz. The Classical Theory of Fields (Butterworth-Heinenann, 1975) [ISBN: 0-7506-2768-9].
J.M.M. Senovilla, D. Garfinkle. The 1965 Penrose singularity theorem. Class. Quant. Grav. 32, 124008 (2015).
https://doi.org/10.1088/0264-9381/32/12/124008
V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz. Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. 19, 525 (1970).
https://doi.org/10.1080/00018737000101171
O.I. Bogoyavlenskii, S.P. Novikov. Singularities of the cosmological model of the Bianchi IX type according to the qualitative theory of differential equations. Sov. Phys. JETP 37, 747 (1973).
O.I. Bogoyavlenskii. Some properties of the type IX cosmological model with moving matter. Sov. Phys. JETP 43, 187 (1976).
V.A. Belinskii, I.M. Khalatnikov, M.P. Ryan. The oscillatory regime near the singularity in Bianchi-type IX universes. Preprint 469 (1971), Landau Institute for Theoretical Physics, Moscow.
M.P. Ryan. The oscillatory regime near the singularity in bianchi-type IX universes. Ann. Phys. 70, 301 (1972).
https://doi.org/10.1016/0003-4916(72)90269-2
V.A. Belinski. On the cosmological singularity. Int. J. Mod. Phys. D 23, 1430016 (2014).
https://doi.org/10.1142/S021827181430016X
E. Czuchry, W. Piechocki. Bianchi IX model: Reducing phase space. Phys. Rev. D 87, 084021 (2013).
https://doi.org/10.1103/PhysRevD.87.084021
S.L. Parnovskii. Electromagnetic and scalar fields around an infinite thread and other Kasner-type naked singularities. Sov. Phys. JETP 49, 589; ЖЭТФ 76, 1162 (1979).
S.L. Parnovskii. Effects of electric and scalar fields on timelike singularities. Sov. Phys. JETP 67, 2400; ЖЭТФ 94, 15 (1988).
S.L. Parnovsky. Gravitational fields near the naked singularities of the general type. Physica A 104, 210 (1980).
https://doi.org/10.1016/0378-4371(80)90082-5
S.L. Parnovsky. A general solution of gravitational equations near their singularities. Class. Quant. Grav. 7, 571 (1990).
https://doi.org/10.1088/0264-9381/7/4/008
S.L. Parnovsky, W. Piechocki. Classical dynamics of the Bianchi IX model: spacelike and timelike singularities. Gen. Rel. Grav. 49, id.87 (2017).
https://doi.org/10.1007/s10714-017-2249-0
D. Kramer, H. Stephani, M.Maccallum, C.Hoenselaers, E. Herlt. Exact solution of the Einsteins field equations (Cambridge University Press, 2003).
https://doi.org/10.1017/CBO9780511535185
E. Kasner. Geometrical theorems on Einstein's cosmological equations. Amer. J. Math. 43, 217 (1921).
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