Analyticity of the Black Hole S-Matrix

TL;DR

This paper analyzes the complex-frequency analytic structure of the black hole S-matrix in a Schwarzschild background, revealing how causality, quasinormal modes, and IR regulators influence its properties.

Contribution

It establishes the detailed analytic structure of the black hole S-matrix, including the effects of singularities, quasinormal modes, and IR cutoffs, advancing understanding of wave scattering in curved spacetime.

Findings

Partial-wave amplitudes are analytic except for quasinormal-mode poles and branch cuts.

Retarded Green's function and absorption amplitude are analytic in the upper-half plane.

IR regulators impose a lower bound on the cutoff related to the Schwarzschild radius.

Abstract

We establish the analytic structure of the S-matrix in the complex-frequency plane for classical wave scattering on a Schwarzschild background in four space-time dimensions. Our argument relies on the analytic continuation of the gravitational potential, with the singularity behind the horizon playing a crucial role. We find that in the lower half-plane the partial-wave amplitudes are analytic except for the quasinormal-mode poles and the branch cut associated with late-time tails. As a direct consequence of causality, the retarded Green's function and absorption amplitude are analytic in the upper-half plane. We show, however, that Stokes phenomena can obstruct this analyticity domain from carrying over to the elastic amplitude, which instead develops a branch-cut in the upper-half plane. We also determine the effect of infrared (IR) regulators on the analytic structure, showing that polynomial boundedness requires a sharp lower bound on the IR cutoff in terms of the Schwarzschild radius.