Probing the Singularity of Scalar-Haired Black Holes with Holographic Complexity

TL;DR

This paper investigates how holographic complexity observables probe the near-singularity geometry of scalar-haired black holes, revealing how scalar hair influences the depth and nature of this probing in AdS spacetimes.

Contribution

It introduces a study of two holographic complexity observables in scalar-haired black holes, analyzing their ability to probe near-singularity Kasner geometries with varying exponents.

Findings

Observables can probe the Kasner regime near the singularity.

Scalar hair allows continuous variation of Kasner exponents.

Different scalar potentials affect the depth of probing.

Abstract

It has been shown that the "complexity=anything" observables allow more possibilities to probe the geometry behind the horizon of AdS black holes compared to the volume complexity. For uncharged black holes, these observables access the geometry all the way to the vicinity of the singularity, while for charged black holes, they only probe up to the inner horizon. Under appropriate conditions, the near-singularity geometry takes the universal form of a Kasner spacetime, characterized by the Kasner exponents. By introducing scalar hair, it is possible to continuously vary the Kasner exponents away from their vacuum values. In this work, we study the behavior of two different observables to determine whether they remain viable holographic duals of complexity in the presence of scalar hair. We also investigate how deeply these observables can probe the Kasner regime near the singularity. To this end, we consider two scalar potentials: an exponential potential, which admits analytic solutions, and a pure mass term, which requires numerical analysis.