Black hole hair in generalized scalar-tensor gravity: An explicit example

TL;DR

This paper demonstrates that in shift-symmetric Horndeski theory, black holes generally possess scalar hair, and provides explicit numerical and analytic solutions showing how the scalar-Gauss-Bonnet coupling influences black hole properties, including size and geometry.

Contribution

The paper presents explicit numerical and perturbative solutions for black holes with scalar hair in scalar-Gauss-Bonnet gravity, highlighting the effects of non-linearities and the minimum black hole size.

Findings

Black holes have scalar hair in the considered theory.

Solutions show a minimum black hole size due to the scalar-Gauss-Bonnet coupling.

Deviations from Schwarzschild are very small across parameter ranges.

Abstract

In a recent Letter we have shown that in shift-symmetric Horndeski theory the scalar field is forced to obtain a nontrivial configuration in black hole spacetimes, unless a linear coupling with the Gauss-Bonnet invariant is tuned away. As a result, black holes generically have hair in this theory. In this companion paper, we first review our argument and discuss it in more detail. We then present actual black hole solutions in the simplest case of a theory with the linear scalar-Gauss-Bonnet coupling. We generate exact solutions numerically for a wide range of values of the coupling and also construct analytic solutions perturbatively in the small coupling limit. Comparison of the two types of solutions indicates that non-linear effects that are not captured by the perturbative solution lead to a finite area, as opposed to a central, singularity. Remarkably, black holes have a minimum size, controlled by the length scale associated with the scalar-Gauss-Bonnet coupling. We also compute some phenomenological observables for the numerical solution for a wide range of values of the scalar-Gauss-Bonnet coupling. Deviations from the Schwarzschild geometry are generically very small.