Reconstructed black hole solutions in the scalar-tensor theory with nonminimal coupling

TL;DR

This paper develops a reconstruction method for scalar-tensor theories with nonminimal coupling to find black hole solutions based on a given static spherically symmetric metric, illustrated with Reissner-Nordström and Bekenstein solutions.

Contribution

It introduces a systematic reconstruction procedure for scalar-tensor black hole solutions from a specified metric form, including explicit relations for potential and coupling functions.

Findings

Derived exact solutions for specific metrics

Provided relations for potential and coupling functions

Illustrated the method with known black hole metrics

Abstract

We consider the scalar-tensor theory witn non-minimal coupling in the Jordan frame. The action of the model contains a potential term $U(\varphi)$, a coupling function $f(\varphi)$. We explore a reconstruction procedure for a generic static spherically symmetric metric written in the Buchdahl parametrization: $ds^2 = \left(A(u)\right)^{-1}du^2 - A(u)dt^2 + C(u)d\Omega^2$, with given $A(u) > 0$ and $C(u) > 0$. The procedure gives the relations for $U(\varphi(u))$, $f(\varphi(u))$ and $d\varphi/du$, which lead to exact solutions to equations of motion with a given metric. A key role in this approach is played by the solutions to a first order linear differential equation for the function $f(\varphi(u))$. The formalism is illustrated by two examples when: a) the Reissner-Nordstr\"om-(Anti-)de Sitter metric and b) the Bocharova-Bronnikov-Melnikov-Bekenstein-(Anti)de-Sitter metric are chosen as a starting point.