Time-Dependent Black Hole Lensing and the Unified Weak-to-Strong Deflection Framework

TL;DR

This work presents a fully analytic, gauge-safe framework that unifies weak-field and time-dependent strong-deflection lensing around a Schwarzschild black hole perturbed by a single axisymmetric even-parity quasinormal mode with frequency $\omega$. A master time-dependent Born integral maps the perturbation onto the observer's screen, yielding a $\boldsymbol{\alpha}(b,t_o)$ that encodes both the familiar $1/b$ weak-field deflection and a time-varying strong-field logarithmic limit, with all effects modulated at the QNM frequency. A key result is the explicit matching between the weak-field and near-photon-sphere regimes, showing that the SDL coefficients and the photon-ring modulations are encoded in the same kernel that produces the weak-field centroid wobble, enabling phase-locked imaging diagnostics without numerical ray tracing. The framework thus connects ringdown spectroscopy to imaging observables, providing analytic predictions for centroid wobble, relativistic-image spacing, ring radius, and inter-image delays, and it offers clear pathways to extend the method to rotation and higher multipoles.

Abstract

We present a fully analytical framework that unifies 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 $ω$. 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 like centroid motion, multi-image hierarchy, and time delays, and 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. The result is a coherent, purely theoretical account of how ringdown physics imprints on imaging observables without numerical ray tracing.

Figures (10)

• Figure 1: Geometry and observer-screen schematic. The Schwarzschild background with horizon at $r=2M$ (filled disk) and photon sphere at $r_{\rm ph}=3M$ (dashed circle) is shown together with a near-critical null trajectory with closest approach $r_0(b)$. A static observer at $r_o\gg M$ carries the tetrad of Eq. \ref{['8']} and measures screen coordinates $(X,Y)$ via Eq. \ref{['12']}. The critical impact parameter $b_c=3\sqrt{3}\,M$ is indicated. This setup underlies the background bending integral in Eq. \ref{['16']} and the time-dependent Born map (Eqs. \ref{['18']}-\ref{['22']}).
• Figure 2: Far-field, QNM-driven deflection amplitude vs impact parameter. The $\mathcal{O}(\varepsilon)$ correction from the Born map ($\omega_I<0$) behaves as $|\delta\boldsymbol{\alpha}(b,t_o)|\sim |c_A(\omega)\,\mathcal{F}_A(\omega b)|/b$, leading to the $1/b$ falloff summarized in Eq. \ref{['28']}. Mild oscillations from the Fresnel factor $\mathcal{F}_A$ encode the finite-frequency sampling of the QNM along the nearly straight path, Eq. \ref{['24']}.
• Figure 3: Centroid wobble in the weak field from Eq. \ref{['30']}. The components $\theta_{c,X}(t_o)$ and $\theta_{c,Y}(t_o)$ execute a damped harmonic motion at the QNM frequency $\omega=\omega_R+i\omega_I$ with $\omega_I<0$, with complex amplitude set by $\mathcal{B}(\omega)$. This panel visualizes the $\mathcal{O}(\varepsilon)$, gauge-safe centroid modulation predicted by the Born expansion in Eqs. \ref{['25']}-\ref{['28']}.
• Figure 4: Near-critical deflection from the time-dependent SDL law. Shown is $\alpha(b,t_o)$ versus $\ln\Delta$ with $\Delta=b/b_c-1\ll1$, as predicted by Eq. \ref{['36']} and its linearized form Eq. \ref{['39']}. The dominant $1/\Delta$ sensitivity arises from the modulation of $b_c(t_o)$ through $\beta_1(\omega)$, while the slope receives a small time-dependent correction from $a_1(\omega)$. Parallel, slightly offset curves at different $t_o$ illustrate the coherent QNM-driven modulation.
• Figure 5: Relativistic image hierarchy and photon ring. The angular positions $\theta_n(t_o)$ are shown versus winding number $n$ for several snapshots in $t_o$, with the instantaneous ring radius $\theta_{\rm ring}(t_o)=b_c(t_o)/r_o$ indicated. The geometric approach $\theta_n\to\theta_{\rm ring}$ is governed by $s_n^{(0)}=\exp[(\bar{b}-2\pi n)/\bar{a}]$, while the coherent, small-amplitude wobble of all $\theta_n$ is driven by the QNM via the response amplitudes in Eq. \ref{['42']}.