No Scalar-Haired Cauchy Horizon Theorem in Charged Gauss-Bonnet Black Holes

TL;DR

This paper extends the no inner horizon theorem to charged Gauss-Bonnet black holes, showing that planar black holes cannot have scalar-hair Cauchy horizons, while non-planar cases are more complex and often allow them, supported by numerical examples.

Contribution

It generalizes the no scalar-hair Cauchy horizon theorem to Einstein-Maxwell-Gauss-Bonnet theories, including numerical validation and a no-go theorem for de Sitter black holes.

Findings

No inner horizon with scalar hairs for planar black holes.

Non-planar black holes may have scalar-hair Cauchy horizons due to Gauss-Bonnet effects.

Numerical solutions support the theorem's predictions.

Abstract

Recently, a ``no inner (Cauchy) horizon theorem" for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories and also in Einstein-Maxwell-Horndeski theories with the non-minimal coupling of a charged (complex) scalar field to Einstein tensor. In this paper, we study an extension of the theorem to the static black holes in Einstein-Maxwell-Gauss-Bonnet-scalar theories, or simply, charged Gauss-Bonnet (GB) black holes. We find that no inner horizon with charged scalar hairs is allowed for the planar (k=0) black holes, as in the case without GB term. On the other hand, for the non-planar (k=+1,-1) black holes, we find that the haired inner horizon can not be excluded due to GB effect generally, though we can not find a simple condition for its existence. As some explicit examples of the theorem, we study numerical GB black hole solutions with charged scalar hairs and Cauchy horizons in asymptotically anti-de Sitter space, and find good agreements with the theorem. Additionally, in an Appendix, we prove a ``no-go theorem" for charged de Sitter black holes (with or without GB terms) with charged scalar hairs in arbitrary dimensions.