Analytical approach for calculating shadow of dynamical black hole

TL;DR

The paper addresses time-dependent black-hole shadows in slowly evolving, spherically symmetric spacetimes by deriving a force-decomposed radial equation for null geodesics in ingoing Eddington-Finkelstein coordinates and a gauge-invariant energy-flux relation $d(E^2)/dv = -\Lambda/r$ with $\Lambda = \varepsilon\dot M\dot v^2$. In the adiabatic regime, it produces a first-order shift of the photon-sphere radius $r_{ph}(v)=r_0-a_i/(a_g'+a_c')$ and an observable shadow radius via $b_{\rm crit}(v)=\sqrt{r_{ph}^2/f(r_{ph})}$, showing that accretion expands the photon sphere and shadow while mass loss contracts them. The framework unifies force-balance intuition with time-dependent photon surfaces and yields simple, observer-ready expressions for shadow evolution, serving as analytic benchmarks for horizon-scale imaging with dynamical inflow/outflow. It also outlines clear pathways for extending to rapid dynamics, rotating or radiating backgrounds, plasma effects, and numeric-ray-tracing comparisons.

Abstract

We develop a compact and transparent framework for photon dynamics and shadow formation in slowly evolving, spherically symmetric spacetimes. Starting from the Eddington-Finkelstein action, we derive a force-decomposed radial equation in which the radial acceleration splits into an induced term sourced by mass variation, a centrifugal term, and a purely general-relativistic correction. A key result is a gauge-invariant energy-flux relation, $d(E^2)/dv=-Λ/r$, with $Λ\equiv \varepsilon\,\dot M\,\dot v^2$, which controls how time dependence modifies the canonical energy of null geodesics. In the adiabatic regime we obtain an explicit first-order shift of the photon-sphere radius, $r_{ph}(v)=r_0-a_i/(a_g'+a_c')$, and connect it to the observable shadow through the evolving critical impact parameter, $b_{\rm crit}(v)=\sqrt{r_{ph}(v)^2/f(r_{ph}(v))}$. For Vaidya spacetimes this predicts that accretion ($\dot M>0$) expands the photon sphere and increases the shadow angle, whereas mass loss has the opposite effect. Our formulation refines classic force-balance ideas to dynamical settings, provides a constructive link to time-dependent photon surfaces, and yields simple, observer-ready expressions for the evolution of the shadow. The framework offers a baseline for confronting time-variable horizon-scale imaging with dynamical inflow/outflow models.