Conformal boundaries and the photon sphere of AdS black holes

Abstract

We study linearized gravitational perturbations of four-dimensional Schwarzschild-AdS black holes with finite timelike boundaries subject to conformal boundary conditions. These fix the conformal class of the induced metric and the mean curvature, while allowing the boundary Weyl mode to fluctuate. The photon sphere plays a distinguished role in this problem: a static boundary at this radius is totally umbilic, and the scalar problem separates exactly into independent bulk and boundary sectors. We prove the absence of exponentially growing scalar and vector modes for spherical harmonics of degree two and higher. We determine how large angular momentum instabilities are suppressed as the boundary approaches umbilicity. Near the horizon, we recover the universal Rindler spectrum, including sound and shear excitations. We numerically follow these fluid modes as the boundary is moved outwards and find that they approach standard gravitational quasinormal modes at the asymptotic AdS$_4$ boundary. For large black holes, these asymptotic endpoints correspond to the sound and shear poles of the dual three-dimensional conformal field theory. We also study perturbations of the AdS$_4$ black brane, where we find an analogous continuation of the fluid modes in the absence of an intermediate umbilic boundary.