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

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.