Black hole tomography: Unveiling black hole horizon dynamics via ringdown observations

During the postmerger regime of a binary black hole merger, the gravitational wave signal consists of a superposition of quasinormal modes (QNMs) of the remnant black hole. It has been observed empirically, primarily through numerical simulations and heuristic arguments, that the infalling radiation at the horizon is also composed of a superposition of QNMs. In this paper we provide an analytic explanation for this observation in the perturbative regime. Our analysis is based on a characteristic initial value formulation where data is prescribed on the horizon (modeled as a perturbed isolated horizon), and on a transversal null hypersurface which registers the outgoing radiation. This allows us to reformulate the traditional QNM problem in a fully four-dimensional setting. Using a mode decomposition, we demonstrate that the radiation modes crossing $\mathcal{H}$ are highly correlated with the outgoing modes crossing $\mathcal{I}$, and provide explicit expressions linking ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Psi}}}}_{0}$ at the horizon with ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Psi}}}}_{4}$ at null infinity. Hence, we analytically demonstrate that measuring the ringdown signal at null infinity allows us to probe the dynamics of the horizon.