Hidden conformal symmetry on the black hole photon sphere

Abstract

We consider a class of static and spherically symmetric black hole geometries endowed with a photon sphere. On the one hand, we show that close to the photon sphere, a massless scalar field theory exhibits a simple dynamical $SL(2,\mathbb{R})$ algebraic structure which allows to recover the discrete spectrum of the weakly damped quasinormal frequencies in the eikonal approximation, and the associated quasinormal modes from the algebra representations. On the other hand, we consider the non-radial motion of a free-falling test particle, in the equatorial plane, from spatial infinity to the black hole. In the ultrarelativistic limit, we show that the photon sphere acts as an effective Rindler horizon for the geodesic motion of the test particle in the $(t, r)$-plane, with an associated Unruh temperature $Tc = \hbarΛ_c/2πk_B$, where $Λ_c$ is the Lyapunov exponent that characterizes the unstable circular motions of massless particles on the photon sphere. The photon sphere then appears as a location where the thermal bound on chaos for quantum systems with a large number of degrees of freedom, in the form conjectured a few years ago by Maldacena et al., is saturated. The study developed in this paper could hopefully shed a new light on the gravity/CFT correspondence, particularly in asymptotically flat spacetimes, in which the photon sphere may also be considered as a holographic screen.