Weak Cosmic Censorship Conjecture of Hairy Black Holes in Einstein Gravity

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.