Kasner interiors from analytic hairy black holes

TL;DR

This paper explores the interior geometries of hairy black holes in AdS space, revealing Kasner singularities, proposing new complexity measures, and analyzing the effects of symmetry breaking on these singularities.

Contribution

It provides an exhaustive analysis of black hole interiors with scalar hair, introduces a new complexity variant linked to Kasner exponents, and studies symmetry breaking effects on singularities.

Findings

Black holes exhibit Kasner or timelike singularities beyond the horizon.

A new complexity measure relates late-time growth to Kasner exponents.

Kasner singularities occur with both explicit and spontaneous symmetry breaking.

Abstract

We conduct an exhaustive study of the interior geometry of a family of asymptotically AdS$_{d+1}$ hairy black holes in an analytically controllable setup. The black holes are exact solutions to an Einstein-Maxwell-Dilaton theory and include the well-known Gubser-Rocha model. After reviewing the setup, we scrutinize the geometry beyond the horizon, finding that these backgrounds can exhibit timelike or Kasner singularities. We generalize the no inner-horizon theorem for hairy black holes to accommodate these findings. We then consider observables sensitive to the geometry behind the horizon, such as Complexity = Anything and the thermal $a$-function. In the Kasner case, we propose a new variant of complexity that characterizes the late-time rate by the Kasner exponents, extending previous work by J{\o}rstad, Myers and Ruan. Additionally, we elucidate the power-law behavior of the thermal $a$-function near the singularity, directly relating it to the Kasner exponents. Finally, we introduce axion-like fields in the Gubser-Rocha model to study the impact of translational symmetry breaking on the black hole interior. We show that Kasner singularities occur for both explicit and spontaneous symmetry breaking, with the Kasner exponents depending on the strength of broken translations only in the latter case.