Dynamics of the General Bianchi IX Model near a Cosmological Singularity
Keywords:general relativity, cosmology, singularity, general solution
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.
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).
E.M. Lifshitz, I.M. Khalatnikov. Investigations in relativistic cosmology. Adv. Phys. 12, 185 (1963).
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).
V.A. Belinskii, I.M. Khalatnikov, E.M. Lifshitz. Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. 19, 525 (1970).
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).
V.A. Belinski. On the cosmological singularity. Int. J. Mod. Phys. D 23, 1430016 (2014).
E. Czuchry, W. Piechocki. Bianchi IX model: Reducing phase space. Phys. Rev. D 87, 084021 (2013).
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).
S.L. Parnovsky. A general solution of gravitational equations near their singularities. Class. Quant. Grav. 7, 571 (1990).
S.L. Parnovsky, W. Piechocki. Classical dynamics of the Bianchi IX model: spacelike and timelike singularities. Gen. Rel. Grav. 49, id.87 (2017).
D. Kramer, H. Stephani, M.Maccallum, C.Hoenselaers, E. Herlt. Exact solution of the Einsteins field equations (Cambridge University Press, 2003).
E. Kasner. Geometrical theorems on Einstein's cosmological equations. Amer. J. Math. 43, 217 (1921).
How to Cite
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.