Analytical approach for calculating shadow of dynamical black hole

TL;DR

This paper presents a new analytical framework for understanding how black hole shadows evolve over time in dynamical, spherically symmetric spacetimes, linking photon dynamics to observable shadow changes.

Contribution

It introduces a gauge-invariant energy-flux relation and explicit formulas for the evolving photon sphere and shadow in dynamical black hole spacetimes.

Findings

Mass accretion enlarges the photon sphere and shadow.

Mass loss reduces the photon sphere and shadow.

Provides explicit first-order shift formulas for photon-sphere radius.

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, $d(E^2)/dv=-\Lambda/r$, with $\Lambda\equiv \varepsilon\,\dot M\,\dot v^2$, 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.