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2.9 On the validity of the BKL conjecture – A status report

Providing a complete rigorous justification of the above description of the behavior of the gravitational field in the vicinity of a spacelike singularity is a formidable task that has not been pushed to completion yet. The task is formidable because the Einstein equations form a complicated nonlinear system of partial differential equations. We shall assume throughout our review that the BKL description is correct, based on the original convincing arguments put forward by BKL themselves [16Jump To The Next Citation Point] and the subsequent fruitful investigations that have shed further important light on the validity of the conjecture. The billiard description will thus be taken for granted.

For completeness, we provide in this section a short guide to the work that has been accumulated since the late 1960’s to consolidate the BKL phenomenon.

As we have indicated, there are two aspects to the BKL conjecture:

  1. The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)
  2. The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasner-like spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs. (“Oscillatory, or mixmaster, spacelike singularities.”)

A third element of the original conjecture was that the matter could be neglected asymptotically. While generically true in four spacetime dimensions (the exception being a massless scalar field, equivalent to a fluid with the stiff equation of state p = ρ), this aspect of the conjecture does not remain valid in higher dimensions where the p-form fields might add relevant walls that could change the qualitative asymptotic behavior. We shall thus focus here only on Aspects 1 and 2.

The connection between the billiard variables and the scale invariant variables has been investigated recently in the interesting works [92162].

Finally, taking for granted the BKL conjecture, one might analyze the chaotic properties of the billiard map (when the volume is finite). Papers exploring this issue are [3032121132] (four dimensions) and [68] (five dimensions).

Let us finally mention the interesting recent paper [40Jump To The Next Citation Point], in which a more precise formulation of the BKL conjecture, aimed towards the chaotic case, is presented. In particular, the main result of this work is an extension of the Fuchsian techniques, employed, e.g., in [49], which are applicable also for systems exhibiting chaotic dynamics. Furthermore, [40] examines the geometric structure which is preserved close to the singularity, and it is shown that this structure has a mathematical description in terms of a so called “partially framed flag”.

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