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

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.

Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory

“…Our result is different from the result in [22] where the author discussed the interior dynamical behavior of the metric using the "cosmological billiard" description [46,47]. It was found that the corresponding hyperbolic billiard region has infinite volume so that the system will settle down to a final Kasner regime.…”

“…For a certain class of black holes with charge scalar hair, it was recently argued that [22] the interior dynamics falls within the scope of the "cosmological billiard" description and the corresponding hyperbolic billiard region has infinite volume so that the system ultimately settles down to a final Kasner regime. Interestingly, as we will show below, in the top-down model we consider, the final dynamic behavior can not settle down to a Kasner regime to the singularity at z → ∞.…”

“…As our Lagrangian is different, the resulting billiard table is different. The study of [22] only considered a massive charged scalar with a non-minimal term ψ 2 F µν F µν . As shown in Appendix B, the billiard table structure of [22] may not apply to our case due to the presence of the exponential form of couplings sinh ψ and cosh ψ in the top-down theory (2.8).…”

“…The study of [22] only considered a massive charged scalar with a non-minimal term ψ 2 F µν F µν . As shown in Appendix B, the billiard table structure of [22] may not apply to our case due to the presence of the exponential form of couplings sinh ψ and cosh ψ in the top-down theory (2.8). It is an interesting question to see if our result can be obtained from the billiard approach.…”

“…In a recent paper [22], the author analyzed the metric behavior near the singularity using the Hamiltonian phase space formalism which offers a clear way to present the transition law of Kanser exponents in different Kasner epochs. It was found that the singularity of a black hole with a massive charged scalar hair or vector hair is of the 4D BKL type transformation law p t → |pt| 1−2|pt| .…”