Dressing a black hole with a time-dependent Galileon

TL;DR

This paper addresses finding black hole solutions in shift-symmetric Horndeski scalar-tensor theories with a John-type interaction that allows non-trivial, regular scalar fields in static, spherically symmetric spacetimes. By allowing a linear time dependence in the scalar field, $\phi(t,r)= q t + \psi(r)$, together with a geometric regularity condition $\beta G^{rr}-\eta g^{rr}=0$, it derives a class of exact solutions in which the scalar is regular and the spacetime metric can be GR-like. The authors present a stealth Schwarzschild solution with a nontrivial scalar, a Schwarzschild solution embedded in an Einstein static universe, and a self-tuned Schwarzschild–de Sitter solution with an effective cosmological constant $\Lambda_{\text{eff}} = -\zeta \eta/\beta$, illustrating how vacuum energy can be screened by integration constants. These results show that time-dependent Galileon fields can evade no-hair arguments and yield hair-like scalar structures, motivating further study of stability and phenomenology.

Abstract

We present a class of exact scalar-tensor black holes for a shift-symmetric part of the Horndeski action. The action includes a higher order scalar tensor interaction term. We find that for a static and spherically symmetric space-time, the scalar field, if time dependent, can be non-trivial and regular thus circumventing in an interesting way no-hair arguments for gallileons. Furthermore, within this class we find a stealth Schwarzschild and a partially self-tuned de-Sitter Schwarzschild black hole, both exhibiting a non trivial and regular space and time dependent scalar. In the latter solution the bulk vacuum energy is screened from a necessarily smaller geometric effective de Sitter vacuum via an integration constant associated to the time dependent scalar field.