BKL singularity information

A spherical body undergoing a chaotic BKL (Mixmaster) dynamics close to singularity according to rules **eq. 35**. Simulation was made in Mathematica with initial $u={\sqrt {13}}$ .^{[note 1]}

A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitation.^[2] According to this model, the universe is chaotically oscillating around a gravitational singularity in which time and space become equal to zero or, equivalently, the spacetime curvature becomes infinitely big. This singularity is physically real in the sense that it is a necessary property of the solution, and will appear also in the exact solution of those equations. The singularity is not artificially created by the assumptions and simplifications made by the other special solutions such as the Friedmann–Lemaître–Robertson–Walker, quasi-isotropic, and Kasner solutions.

The model is named after its authors Vladimir Belinski, Isaak Khalatnikov, and Evgeny Lifshitz, then working at the Landau Institute for Theoretical Physics.

The picture developed by BKL has several important elements. These are:

Near the singularity the evolution of the geometry at different spatial points decouples so that the solutions of the partial differential equations can be approximated by solutions of ordinary differential equations with respect to time for appropriately defined spatial scale factors. This is called the BKL conjecture.
For most types of matter the effect of the matter fields on the dynamics of the geometry becomes negligible near the singularity. Or, in the words of John Wheeler, "matter doesn't matter" near a singularity. The original BKL work posed a negligible effect for all matter but later they theorized that "stiff matter" (equation of state p = ε) equivalent to a massless scalar field can have a modifying effect on the dynamics near the singularity.
The ordinary differential equations describing the asymptotics come from a class of spatially homogeneous solutions which constitute the Mixmaster dynamics: a complicated oscillatory and chaotic model that exhibits properties similar to those discussed by BKL.

The study of the dynamics of the universe in the vicinity of the cosmological singularity has become a rapidly developing field of modern theoretical and mathematical physics. The generalization of the BKL model to the cosmological singularity in multidimensional (Kaluza–Klein type) cosmological models has a chaotic character in the spacetimes whose dimensionality is not higher than ten, while in the spacetimes of higher dimensionalities a universe after undergoing a finite number of oscillations enters into monotonic Kasner-type contracting regime.^[3]^[4]^[5]

The development of cosmological studies based on superstring models has revealed some new aspects of the dynamics in the vicinity of the singularity.^[6]^[7]^[8] In these models, mechanisms of changing of Kasner epochs are provoked not by the gravitational interactions but by the influence of other fields present. It was proved that the cosmological models based on six main superstring models plus eleven-dimensional supergravity model exhibit the chaotic BKL dynamics towards the singularity. A connection was discovered between oscillatory BKL-like cosmological models and a special subclass of infinite-dimensional Lie algebras – the so-called hyperbolic Kac–Moody algebras.^[9]^[10]^[11]

^ Garfinkle, David (2007). "Of singularities and breadmaking". Einstein Online. Band 03. Max Planck Institute for Gravitational Physics. 03–1014. Retrieved 2020-10-15.
^ Belinsky, Khalatnikov & Lifshitz 1970
^ Demaret, Henneaux & Spindel 1985.
^ Demaret et al. 1986.
^ Demaret, de Rop & Henneaux 1989.
^ Damour & Henneaux 2000.
^ Damour et al. 2001.
^ Damour, Henneaux & Nicolai 2003.
^ Kac 1983.
^ Damour 2015.
^ Henneaux, Persson & Spindel 2008.

Cite error: There are <ref group=note> tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page).

[1] Garfinkle, David (2007). "Of singularities and breadmaking". Einstein Online. Band 03. Max Planck Institute for Gravitational Physics. 03–1014. Retrieved 2020-10-15.

[BKL-3] Belinsky, Khalatnikov & Lifshitz 1970

[FOOTNOTEDemaretHenneauxSpindel1985-4] Demaret, Henneaux & Spindel 1985.

[FOOTNOTEDemaretHanquinHenneauxSpindel1986-5] Demaret et al. 1986.

[FOOTNOTEDemaretde_RopHenneaux1989-6] Demaret, de Rop & Henneaux 1989.

[FOOTNOTEDamourHenneaux2000-7] Damour & Henneaux 2000.

[FOOTNOTEDamourHenneauxJuliaNicolai2001-8] Damour et al. 2001.

[FOOTNOTEDamourHenneauxNicolai2003-9] Damour, Henneaux & Nicolai 2003.

[FOOTNOTEKac1983-10] Kac 1983.

[FOOTNOTEDamour2015-11] Damour 2015.

[FOOTNOTEHenneauxPerssonSpindel2008-12] Henneaux, Persson & Spindel 2008.

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