The final Kasner regime inside black holes with scalar or vector hair

TL;DR

This paper investigates the near-singularity behavior inside black holes with scalar or vector hair, showing most solutions settle into a Kasner regime, with some rare solutions exhibiting chaotic oscillations.

Contribution

It demonstrates that the interior dynamics of hairy black holes conform to the cosmological billiard model and identifies conditions for chaotic behavior.

Findings

Most solutions settle into a Kasner regime near the singularity.

The hyperbolic billiard region has infinite volume, indicating non-chaotic behavior.

A measure-zero set of solutions exhibits chaotic BKL oscillations.

Abstract

The final (close to the singularity) dynamical behavior of the metric inside black holes with massive charged scalar or vector hair is analyzed for general anisotropic and inhomogeneous initial conditions. These solutions are relevant to a holographic realization of superconductivity. It is shown that the dynamics falls within the scope of the "cosmological billiard" description and that in both cases, the corresponding hyperbolic billiard region has infinite volume so that the system ultimately settles down to a final Kasner regime. For massive vector hair, the conclusion holds because the longitudinal mode plays the same role as a scalar field. There exists, however, a measure-zero subset of solutions characterized by vanishing longitudinal modes that exhibit a chaotic behavior with an infinite number of BKL oscillations as one goes to the singularity.