Shadow Ringing of Black Holes from Photon Sphere Quasinormal Modes

Abstract

The recent convergence of gravitational-wave (GW) observations and black hole imaging provides complementary probes of strong-gravity dynamics. While the black hole shadow is typically modeled as a static feature, a dynamically perturbed spacetime in its ringdown phase must induce temporal modulations in the shadow's apparent size and shape. We develop a theoretical framework within linear perturbation theory to investigate this shadow ringing effect for a Schwarzschild black hole. By modeling the geometry as a small, mode-selected quasinormal mode (QNM) perturbation, we treat the shadow boundary as an instantaneous separatrix of null geodesics. We derive a first-order, gauge-invariant mapping between the metric perturbation $h_{\mu \nu }$ and the displacement of the shadow boundary, $\delta R(\phi ,t)$. By perturbing the effective potential for null geodesics near the unstable photon sphere ($r=3M$), we derive mode-resolved transfer coefficients that quantify how the QNM imprints itself onto the shadow. We predict that the shadow boundary oscillates coherently at the QNM's real frequency ${\omega }_{\mathrm{Re}}$ with an exponential damping rate set by $|{\omega }_{\mathrm{Im}}|$. Furthermore, the azimuthal structure of the modulation encodes the spherical harmonic content $(\ell ,m)$ of the driving QNM, providing a novel, geometric signature for QNM spectroscopy.

29 pages, 3 figures. Published version

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