Shadow Ringing of Black Holes from Photon Sphere Quasinormal Modes

TL;DR

This work introduces a linear-perturbation framework to predict time-dependent black hole shadows during ringdown, modeling the shadow boundary as an instantaneous separatrix that responds to a single QNM perturbation. By reconstructing metric perturbations via Regge-Wheeler and Zerilli formalisms and applying an adiabatic, geodesic-based mapping, the authors derive a gauge-invariant transfer law that ties $h_{\mu\nu}$ to the shadow boundary displacement $\delta R(\varphi,t)$. The key result is that the shadow boundary rings at the QNM real frequency with damping set by the imaginary part, and its azimuthal structure directly reveals the mode content $(\ell,m)$ through a mode-by-mode transfer coefficient, enabling a potential spectroscopic probe of strong gravity from horizon-scale imaging. The framework is anchored at the photon sphere and naturally extends to Kerr and beyond-GR scenarios, offering a novel channel to combine gravitational-wave and shadow observations for black-hole diagnostics.

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 $h_{μν}$ and the displacement of the shadow boundary, $δR(\varphi,t)$. 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 $ω_{\rm Re}$ with an exponential damping rate set by $|ω_{\rm Im}|$. 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.

Figures (3)

• Figure 1: Schwarzschild unit circle (thin curve) with instantaneous shadow contours $\hat{R}(\varphi,t_k)$ at evenly spaced phases $t_k$. The azimuthal periodicity exposes $m$, while the shrinking amplitude reflects the QNM damping.
• Figure 2: Time series $\delta\hat{R}(t,\varphi_0)$ at fixed angle (here $\varphi_0=0$), with analytic envelopes $\pm 2\varepsilon|\mathcal{T}|e^{-\omega_{\rm Im}t}$. The period measures $\omega_{\rm Re}$; the decay constant measures $\omega_{\rm Im}$; the phase identifies $m$ in tandem with $\varphi_0$.
• Figure 3: Azimuthal spectrum $|\hat{R}_m(t)|$ at several times. Nonzero content identifies the allowed $m$'s (selection rules); a common decay trend across time snapshots signals the QNM damping.