Scalar Hair from a Derivative Coupling of a Scalar Field to the Einstein Tensor

TL;DR

This paper demonstrates that a scalar field coupled to the Einstein tensor can form hairy black holes, evading the no-hair theorem, with a second-order phase transition from Reissner-Nordström black holes at a calculable critical temperature.

Contribution

It introduces a novel scalar-Einstein tensor coupling that allows black holes to develop scalar hair, providing explicit calculations of the phase transition and hairy black hole properties.

Findings

Reissner-Nordström black holes undergo a second-order phase transition to hairy black holes.

The critical temperature for the phase transition is explicitly computed.

Hairy black holes are energetically favored near the critical temperature.

Abstract

We consider a gravitating system of vanishing cosmological constant consisting of an electromagnetic field and a scalar field coupled to the Einstein tensor. A Reissner-Nordstr\"om black hole undergoes a second-order phase transition to a hairy black hole of generally anisotropic hair at a certain critical temperature which we compute. The no-hair theorem is evaded due to the coupling between the scalar field and the Einstein tensor. Within a first order perturbative approach we calculate explicitly the properties of a hairy black hole configuration near the critical temperature and show that it is energetically favorable over the corresponding Reissner-Nordstr\"om black hole.