A no-hair theorem for spherically symmetric black holes in $R^2$ gravity

TL;DR

This paper proves a no-hair theorem for spherically symmetric black holes in pure $R^2$ gravity, showing such black holes cannot have additional scalar hair, unlike in more general $f(R)$ theories.

Contribution

It extends no-hair theorems to pure $R^2$ gravity without assuming asymptotic conditions, using scalar tensor representation.

Findings

No hairy black holes in pure $R^2$ gravity for spherical symmetry.

Includes an example of a black hole with a conformally coupled scalar field.

Limits previous theorems to more general $f(R)$ theories.

Abstract

In a recent paper Ca\~nate (CQG, {\bf 35}, 025018 (2018)) proved a no hair theorem to static and spherically symmetric or stationary axisymmetric black holes in general $f(R)$ gravity. The theorem applies for isolated asymptotically flat or asymptotically de Sitter black holes and also in the case when vacuum is replaced by a minimally coupled source having a traceless energy momentum tensor. This theorem excludes the case of pure quadratic gravity, $f(R) = R^2$. In this paper we use the scalar tensor representation of general $f(R)$ theory to show that there are no hairy black hole in pure $R^2$ gravity. The result is limited to spherically symmetric black holes but does not assume asymptotic flatness or de-Sitter asymptotics as in most of the no-hair theorems encountered in the literature. We include an example of a static and spherically symmetric black hole in $R^2$ gravity with a conformally coupled scalar field having a Higgs-type quartic potential.