# Black hole hair in generalized scalar-tensor gravity

## TL;DR

This paper demonstrates that in shift-symmetric Horndeski theories, black holes generally possess scalar hair unless a specific coupling is finely tuned, challenging previous no-hair theorems.

## Contribution

It proves that generic shift-symmetric Horndeski theories inevitably produce black hole hair, revealing a gap in earlier no-hair theorem assumptions.

## Key findings

- Black holes in these theories have scalar hair unless a specific coupling is tuned.
- The presence of scalar hair contradicts recent no-hair theorems.
- The scalar field configuration is forced in generic Horndeski models.

## Abstract

The most general action for a scalar field coupled to gravity that leads to second order field equations for both the metric and the scalar --- Horndeski's theory --- is considered, with the extra assumption that the scalar satisfies shift symmetry. We show that in such theories the scalar field is forced to have a nontrivial configuration in black hole spacetimes, unless one carefully tunes away a linear coupling with the Gauss--Bonnet invariant. Hence, black holes for generic theories in this class will have hair. This contradicts a recent no-hair theorem, which seems to have overlooked the presence of this coupling.