Time-dependent black hole lensing from ringdown quasinormal mode

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.