No Inner-Horizon Theorem for Black Holes with Charged Scalar Hairs

TL;DR

This paper proves that black holes with charged scalar hairs cannot have an inner horizon, showing they approach a spacelike singularity and revealing universal and novel behaviors near the singularity.

Contribution

The paper establishes a no inner-horizon theorem for charged scalar hairy black holes, independent of scalar potential forms and spacetime asymptotics.

Findings

No inner Cauchy horizon exists for charged scalar hairy black holes.

Near the singularity, the geometry adopts a universal Kasner form when the scalar kinetic term dominates.

For hyperbolic horizons, at most one inner horizon can exist, with an example provided.

Abstract

We establish a no inner-horizon theorem for black holes with charged scalar hairs. Considering a general gravitational theory with a charged scalar field, we prove that there exists no inner Cauchy horizon for both spherical and planar black holes with non-trivial scalar hair. The hairy black holes approach to a spacelike singularity at late interior time. This result is independent of the form of scalar potentials as well as the asymptotic boundary of spacetimes. We prove that the geometry near the singularity takes a universal Kasner form when the kinetic term of the scalar hair dominates, while novel behaviors different from the Kasner form are uncovered when the scalar potential become important to the background. For the hyperbolic horizon case, we show that hairy black hole can only has at most one inner horizon, and a concrete example with an inner horizon is presented. All these features are also valid for the Einstein gravity coupled with neutral scalars.