Scalar Field Perturbation of Hairy Black Holes in EsGB theory

TL;DR

This paper studies the stability and phase transition of hairy black holes in Einstein-scalar-Gauss-Bonnet theory, revealing how spontaneous symmetry breaking leads to stable hairy black holes and analyzing their perturbations and quasinormal modes.

Contribution

It provides a detailed analysis of scalar perturbations and stability of hairy black holes in EsGB theory, highlighting the role of spontaneous symmetry breaking in black hole phase transitions.

Findings

Hairy black holes in the symmetric phase become unstable beyond a critical coupling.

Hairy black holes in the symmetry-broken phase are always stable at the critical point.

Late-time behavior of perturbations differs from Schwarzschild only by a mass term.

Abstract

We investigate scalar field perturbations of the hairy black holes involved with spontaneous symmetry breaking of the global U(1) symmetry in Einstein-scalar-Gauss-Bonnet theory for asymptotically flat spacetimes. We consider the mechanism that black holes without hairs become unstable at the critical point of the coupling constant and undergo a phase transition to hairy black holes in the symmetry-broken phase driven by spontaneous symmetry breaking. This transition occurs near the black hole horizon due to the diminishing influence of the Gauss-Bonnet term at infinity. To examine such process, we introduce a scalar field perturbation on the newly formed background spacetime. We solve the linearized perturbation equation using Green's function method. We begin by solving the Green's function, incorporating the branch cut contribution. This allows us to analytically investigate the late-time behavior of the perturbation at both spatial and null infinity. We found that the late-time behavior only differs from the Schwarzschild black hole by a mass term. We then proceed to calculate the quasinormal modes (QNMs) numerically, which arise from the presence of poles in the Green's function. Our primary interest lies in utilizing QNMs to investigate the stability of the black hole solutions both the symmetric and symmetry-broken phases. Consistent with the prior study, our analysis shows that hairy black holes in the symmetric phase become unstable when the quadratic coupling constant exceeds a critical value for a fixed value of the quartic coupling constant. In contrast, hairy black holes in the symmetry-broken phase are always stable at the critical value. These numerical results provide strong evidence for a dynamical process that unstable black holes without hairs transition into stable hairy black holes in the symmetry-broken phase through the spontaneous symmetry breaking.