Searching for non-minimally coupled scalar hairs

TL;DR

This paper investigates black hole solutions with scalar fields in a non-minimally coupled theory, establishing conditions under which scalar hairs are absent depending on the coupling parameter , _c, and the scalar field magnitude.

Contribution

It provides a comprehensive analysis of scalar hair existence in higher-dimensional, non-minimally coupled scalar-tensor theories, identifying parameter regimes with no scalar hairs.

Findings

Scalar hairs are absent for <0.

No scalar hairs exist for >_c with |(r)|<1/ or |(r)|>1/.

For 0<<_c, scalar hairs are absent within specific bounds on (r).

Abstract

In this work we study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action $$S = \int d^nx\sqrt{-g} \left\{\left(1-\xi\phi^2 \right)R - g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi\right\},$$ with $n>3$ and arbitrary $\xi$. We demonstrate the absence of scalar hairs for $\xi<0$. For $\xi>\xi_c=\frac{n-2}{4(n-1)}$, we show that there is no scalar hair obeying $|\phi(r)| < 1/\sqrt{\xi}$ or $|\phi(r)| > 1/\sqrt{\xi}$. For $0<\xi<\xi_c$, we prove the absence of scalar hairs such that $|\phi(r)| < 1/\sqrt{\xi}$ or $\frac{1}{\xi} < \phi^2(r) < \frac{\xi_c}{\xi(\xi_c-\xi)}$.