The photon sphere and response functions in AdS/CFT

Abstract

We demonstrate that the photon sphere plays a surprising role in the holography for small AdS-Schwarzschild black holes. A precise formula is derived to show how a black hole amplifies or attenuates an oscillating source dual to boundary operators across the bulk. We estimate the quasi-normal modes of small black holes at large angular momenta $l$. The lowest mode matches the phase transition for the massless scalar fields and the null geodesics, which are identified with the short-lived excitations of the black hole at large $l$. This sharpens the conjecture by Hashimoto that the photon sphere seen by a boundary observer is the spatial Fourier transform of the response function. The conjecture agrees excellently with numerics after certain long-lived excitations -- associated with geodesics connecting boundary points, predicted by Festuccia and Liu -- are removed from the response, which is then controlled by the short-lived excitations of the black hole.