David Díaz-Guerra, Ángel Rincón, Diego Rubiera-Garcia, Diego Saez-Chillon Gomez

Black holes (BHs) can be studied via fundamentally different observational channels that probe complementary aspects of their physics. While BH imaging provides access to the quasi-static space-time geometry via the strong bending of light rays, gravitational wave (GW) observations probe the dynamical response of the space-time to time-dependent processes in the inspiral, merger and ringdown phases. Both messengers -- electromagnetic imaging probes and ringdown GW spectroscopy --, provide access to essentially the same region -- the one between the BH event horizon and the photon region --, but they do it via conceptually different methods, encoding different physical information. However, it has been shown in the literature that physical quantities supposedly exclusive of each such messenger are actually tightly related to each another via a correspondence that occurs in the eikonal limit (i.e. large values of the multipole number $\ell$) of the geometric-optics approximation. In this paper we clarify the actual identification of observables within such a correspondence and test its accuracy for a bunch of modified spherically symmetric BH geometries proposed in the literature. We find that even for low values of $\ell$ the correspondence is surprisingly accurate in relating the real and imaginary parts of quasi-normal modes in the GW ringdown phase with the critical impact parameter and Lyapunov exponent of nearly-bound light trajectories for every such model analyzed. We discuss the applicability of such a result both for each messenger individually, and also for foreseeable tests of BHs combining both messengers.