The Photon Sphere and Response Functions in Holography

Abstract

In this letter, we show the Unruh temperature of the photon sphere for an AdS$_4$-Schwarzschild black hole can be determined holographically from the retarded Green's function and is proportional to its circumference according to a boundary observer. We then sharpen the conjecture that the photon sphere, as seen by a boundary observer, is the spatial Fourier transform of the response function. The conjecture is found to be in excellent agreement with our numerics after certain long-lived excitations -- associated with geodesics traveling between boundary points -- are removed from the response, which is then controlled by the short-lived excitations of the AdS black hole.