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

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.

Added footnote No.3 and comments on (i) possible numerical solutions with the inner horizon for k=1, (ii) no-go theorem for charged dS black holes with charged scalar hairs; Corrected typos, Added reference, Accepted in EPJC