Black Hole Tomography: Unveiling Black Hole Ringdown via Gravitational Wave Observations

TL;DR

This paper provides an analytic explanation for the superposition of quasi-normal modes in black hole ringdown signals, linking horizon and outgoing radiation modes through a 4D perturbative analysis.

Contribution

It introduces a 4D perturbative framework that analytically relates horizon and outgoing radiation modes in black hole mergers, advancing understanding of gravitational wave signals.

Findings

Modes crossing the horizon are highly correlated with outgoing modes at infinity.

Explicit expressions link horizon data with null infinity radiation.

Provides a fully 4D reformulation of the QNM problem.

Abstract

During the post-merger regime of a binary black hole merger, the gravitational wave signal consists of a superposition of quasi-normal 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 4-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 $\widetilde{\Psi}_0$ at the horizon with $\widetilde{\Psi}_4$ at null infinity.