Why black hole ringing stops converging: a geometric answer

Quasinormal modes (QNMs) describe how a perturbed black hole rings down, but their expansion of the Green's function converges only within a finite time window — a fact previously unexplained. Arnaudo and Withers show analytically that this convergence boundary is set by a singularity in the complex time plane, arising from a null geodesic that reaches the Schwarzschild singularity and reflects back — a 'bouncing singularity' familiar from AdS/CFT. The seemingly arbitrary coordinate location r*=0 marking the convergence boundary in earlier work is demystified: it simply mirrors the distance of this bouncing singularity in the complex plane. The same geometry also controls an annular convergence region for the Matsubara sum governing early-time behavior near the horizon.