# Stable black holes in shift-symmetric Horndeski theories

## TL;DR

This paper investigates stable black hole solutions with linearly time-dependent scalar hair in shift-symmetric Horndeski theories, proposing new models that potentially avoid previous instabilities and analyzing their stability properties.

## Contribution

It introduces new black hole solutions with time-dependent scalar hair in Horndeski theories and examines conditions for their stability, addressing prior instability issues.

## Key findings

- New black hole solutions with linearly time-dependent scalar hair.
- Identification of conditions under which these solutions are stable.
- Discussion of stability in cases with static scalar hair and nonminimal couplings.

## Abstract

In shift-symmetric Horndeski theories, a static and spherically symmetric black hole can support linearly time-dependent scalar hair. However, it was shown that such a solution generically suffers from ghost or gradient instability in the vicinity of the horizon. In the present paper, we explore the possibility to avoid the instability, and present a new example of theory and its black hole solution with a linearly time-dependent scalar configuration. We also discuss the stability of solutions with static scalar hair for a special case where nonminimal derivative coupling to the Einstein tensor appears.