Bouncing singularities in Schwarzschild: a geometric origin of the QNM convergence region

TL;DR

This paper analytically links the convergence of Schwarzschild QNM expansions to a bouncing singularity caused by null geodesics, revealing a geometric origin for the convergence region in complex time.

Contribution

It identifies a bouncing singularity in the complex plane as the geometric origin of QNM convergence limits in Schwarzschild spacetime.

Findings

QNM convergence is limited by a bouncing singularity in the complex time plane.

The bouncing singularity is caused by a null geodesic bouncing from the black hole singularity.

The same singularities determine the convergence region for the Matsubara mode sum.

Abstract

We show analytically that the convergence of the QNM expansion of the retarded Green's function of the Schwarzschild spacetime is set by a singularity in the complex time plane. The singularity has a simple geometric origin: it is an example of a `bouncing singularity' in the language of AdS/CFT literature, caused by a null geodesic which bounces from the black hole singularity. Our work explains why the QNM convergence region at real times is bounded by null ray which scatters from the gravitational potential at a seemingly unremarkable point ($r_* = 0$ in the conventions of previous work) -- this ray is the same distance from the origin as the bouncing singularity in the relevant complex plane. The same set of singularities are responsible for an annular region of convergence for the Matsubara mode sum which describes the early time behaviour of the Schwarzschild Green's function for perturbations close to the horizon.