Self-Gravitating Static Non-Critical Black Holes in 4$D$ Einstein-Klein-Gordon System with Nonminimal Derivative Coupling

TL;DR

This paper investigates static black hole solutions with scalar hair in a four-dimensional Einstein-Klein-Gordon system featuring nonminimal derivative coupling, revealing non-critical solutions with constant scalar curvature geometries.

Contribution

It introduces a new ansatz simplifying the field equations, leading to the discovery of non-critical hairy black holes with secondary-like scalar hair.

Findings

Solutions exhibit non-critical scalar potential behavior.

Effective geometries approach constant scalar curvature near boundaries.

Scalar charge depends on a non-constant mass-like quantity.

Abstract

We study static non-critical hairy black holes of four dimensional gravitational model with nonminimal derivative coupling and a scalar potential turned on. By taking an ansatz, namely, the first derivative of the scalar field is proportional to square root of a metric function, we reduce the Einstein field equation and the scalar field equation of motions into a single highly nonlinear differential equation. This setup implies that the hair is secondary-like since the scalar charge-like depends on the non-constant mass-like quantity in the asymptotic limit. Then, we show that near boundaries the solution is not the critical point of the scalar potential and the effective geometries become spaces of constant scalar curvature.