Time-Dependent Black Hole Lensing from Ringdown Quasinormal Mode

TL;DR

This paper develops an analytic model for gravitational lensing effects caused by black hole ringdowns, revealing how quasinormal mode perturbations influence observable lensing phenomena in both weak and strong gravitational fields.

Contribution

It introduces a first-order analytic framework linking black hole ringdown modes to lensing observables, bridging weak and strong deflection regimes without numerical geodesic integration.

Findings

Derived a time-dependent deflection formula incorporating ringdown frequencies.

Identified harmonic centroid wobble and impact parameter oscillations in lensing signals.

Connected near-critical and weak-field coefficients through the same integral kernel.

Abstract

Is it possible to find imprints of a black hole ringdown through gravitational lensing? To address this question, we formulate an analytic description of weak-field and strong-deflection lensing of light in a time-dependent, perturbed Schwarzschild spacetime. The spacetime dynamics are modeled by a single, axisymmetric, even-parity quasinormal mode with \(\ell=2\), \(m=0\) and complex frequency \(\omega\). Working to first order in a small perturbation amplitude while keeping background null geodesics exact, we derive a time-dependent line-of-sight (Born) expression for the screen-plane deflection measured by a static observer at large radius. From the same integral, an asymptotic expansion yields the familiar weak-field \(1/b\) law with a ringdown-frequency correction that drives a harmonic centroid wobble, whereas a near-photon-sphere expansion produces a time-dependent generalization of the logarithmic strong-deflection limit with modulated coefficients, including a small oscillation of the critical impact parameter. An observer tetrad built from the background static frame ensures that all screen-plane quantities, such as centroid motion, multi-image hierarchy, and time delays, as well as photon-ring morphology, are gauge-safe at first order. We provide explicit matching across regimes, showing that the near-critical coefficients governing spacing and ring-radius modulations are encoded in the same Born kernel that controls the weak-field correction. This provides an analytic account of how ringdown-scale perturbations enter imaging observables, without resorting to numerical integration of null geodesics.