Weak Cosmic Censorship Conjecture of Hairy Black Holes in Einstein Gravity

TL;DR

This paper proves the weak cosmic censorship conjecture for static, spherically symmetric hairy black holes in Einstein gravity using a generalized Sorce-Wald method, showing they cannot be destroyed under certain perturbations.

Contribution

It extends the Sorce-Wald method to hairy black holes without explicit metric expressions, demonstrating the universality of WCCC in Einstein gravity for such black holes.

Findings

Nearly extremal static hairy black holes cannot be destroyed under second-order perturbations.

The weak cosmic censorship conjecture holds universally for hairy black holes satisfying the null energy condition.

The method does not require explicit matter field Lagrangians or spherical symmetry of perturbations.

Abstract

Apart from the Kerr-Newman black holes, the hairy black holes in general relativity have been widely investigated in the gravity and cosmology. In this paper, we extend the Sorce-Wald method to prove the weak cosmic censorship conjecture (WCCC) of the static and spherically symmetric hairy black holes in the Einstein gravity without using the explicit expressions of the metric and lagrangian for matter fields. We examine the WCCC in the collision process without requiring the spherically symmetry of the perturbation matter fields. After assuming the stability condition of spacetime and applying the Gaussian null coordinates into the variational identities, we derive the first two order perturbation inequalities which reflects the null energy condition of the matter fields. As a result, we find that nearly extremal static hairy black hole cannot be destroyed in the above perturbation process under the second-order approximation. Our result implies that the valid of the WCCC is universal for the hairy black hole in the Einstein gravity as long as the matter fields satisfy the null energy condition under the perturbation level.