A study on the Belinski-Khalatnikov-Lifshitz scenario through quadrics of kinetic energy

TL;DR

This paper explores the asymptotic behavior near cosmic singularities in the BKL scenario using geometric and Lagrangian methods, confirming chaos and instability in the collapse process.

Contribution

It provides a geometric interpretation of BKL dynamics through quadrics of kinetic energy and analyzes the stability of solutions near singularities.

Findings

Collapse is inevitable for decreasing volume initial conditions.

Oscillatory behavior occurs along paths reflecting from a hyperboloid.

The exact solution is uniquely differentiable and unstable, indicating chaos.

Abstract

A detailed description of the asymptotic behaviour in the Belinski-Khalatnikov-Lifshitz (BKL) scenario is presented through a simple geometric picture illustrating the geometry of their ordinary differential equations (ODE), which describe a neighbourhood of the cosmic singularity. The Lagrangian version of the dynamics governed by these equations is described in terms of trajectories inside a conical subset of the corresponding space of the generalised velocities. The calculations confirm that the initial conditions of decreasing volume inevitably result in eventual total collapse, while oscillations along paths reflecting from a hyperboloid, similar to those predicted by Kasner's solutions, occur on the way. The exact solution, found in our previous work, proves to be the only one that shrinks to a point along a differentiable path. Therefore, its instability means that the collapse is always chaotic. It is also shown that the BKL equations are not satisfied by the Kasner solutions exactly, even in the asymptotic regime, although the precision of their approximation may be high.