Stealth Schwarzschild solution in shift symmetry breaking theories

TL;DR

This paper discovers stealth Schwarzschild black hole solutions with nontrivial scalar fields in shift symmetry breaking theories, analyzing their properties, stability, and the conditions under which they exist, including the case of time-dependent scalar fields.

Contribution

It provides explicit stealth Schwarzschild solutions with nontrivial scalar profiles in shift symmetry breaking Horndeski theories and analyzes their stability and the conditions for their existence.

Findings

Two types of stealth Schwarzschild solutions with scalar fields are found.

The scalar mode's kinetic term vanishes, indicating strong coupling.

No stealth Schwarzschild solutions exist with time-dependent scalar fields.

Abstract

We find stealth Schwarzschild solutions with a nontrivial profile of the scalar field regular on the horizon in the Einstein gravity coupled to the scalar field with the k-essence and/or generalized cubic galileon terms, which is a subclass of the Horndeski theory breaking the shift symmetry, where the propagation speed of gravitational waves coincides with the speed of light. After deriving sufficient conditions for the shift symmetry breaking theory to allow a general Ricci-flat metric solution with a nontrivial scalar field profile, we focus on the stealth Schwarzschild solution with the scalar field with or without time dependence. For the profile $\phi=\phi_0(r)$, we explicitly obtain two types of stealth Schwarzschild solutions, one of which is regular on the event horizon. The linear perturbation analysis clarifies that the kinetic term of the scalar mode identically vanishes, indicating that the scalar mode is strongly coupled. The absence of the kinetic term of the scalar mode in the quadratic action would inevitably arise for the stealth Schwarzschild solutions in the theory with a general scalar field profile depending only on the spatial coordinates. On the other hand, for the time-dependent scalar field profile, we clarify that there does not exist a stealth Schwarzschild solution in the shift symmetry breaking theories.