No hair for spherical black holes: charged and nonminimally coupled scalar field with self--interaction

TL;DR

This paper proves three theorems in general relativity that establish the absence of classical scalar hair for static, spherically symmetric black holes, including charged and nonminimally coupled scalar fields with self-interaction.

Contribution

It generalizes Bekenstein's no-hair theorem to charged black holes and nonminimal coupling cases, providing comprehensive no-hair results under various conditions.

Findings

No scalar hair for charged black holes with minimally coupled fields.

No scalar hair for nonminimally coupled fields with certain coupling parameters.

Exclusion of charged scalar hair for all coupling parameters.

Abstract

We prove three theorems in general relativity which rule out classical scalar hair of static, spherically symmetric, possibly electrically charged black holes. We first generalize Bekenstein's no--hair theorem for a multiplet of minimally coupled real scalar fields with not necessarily quadratic action to the case of a charged black hole. We then use a conformal map of the geometry to convert the problem of a charged (or neutral) black hole with hair in the form of a neutral self--interacting scalar field nonminimally coupled to gravity to the preceding problem, thus establishing a no--hair theorem for the cases with nonminimal coupling parameter $\xi<0$ or $\xi\geq {1\over 2}$. The proof also makes use of a causality requirement on the field configuration. Finally, from the required behavior of the fields at the horizon and infinity we exclude hair of a charged black hole in the form of a charged self--interacting scalar field nonminimally coupled to gravity for any $\xi$.