# Asymptotically flat black holes in Horndeski theory and beyond

## TL;DR

This paper discovers asymptotically flat black hole solutions in shift-symmetric Horndeski and beyond theories, including hairy and singular solutions, and shows that some GR solutions like Kerr also solve these theories.

## Contribution

It provides new exact black hole solutions in Horndeski and beyond theories, including conditions under which GR solutions are also solutions in these modified theories.

## Key findings

- Found shift-symmetric black hole solutions with non-trivial scalar fields.
- Identified conditions where GR solutions like Schwarzschild and Kerr are also solutions.
- Discovered hairy black holes and singular solutions in the theories.

## Abstract

We find spherically symmetric and static black holes in shift-symmetric Horndeski and beyond Horndeski theories. They are asymptotically flat and sourced by a non trivial static scalar field. The first class of solutions is constructed in such a way that the Noether current associated with shift symmetry vanishes, while the scalar field cannot be trivial. This in certain cases leads to hairy black hole solutions (for the quartic Horndeski Lagrangian), and in others to singular solutions (for a Gauss-Bonnet term). Additionally, we find the general spherically symmetric and static solutions for a pure quartic Lagrangian, the metric of which is Schwarzschild. We show that under two requirements on the theory in question, any vacuum GR solution is also solution to the quartic theory. As an example, we show that a Kerr black hole with a non-trivial scalar field is an exact solution to these theories.