Analytical approach for calculating shadow of dynamical black hole

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 $\Lambda \equiv \epsilon \dot{M}{\dot{v}}^2$, 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, $r_{ph}(v)=r_0-a_i/(a_g^{\prime }+a_c^{\prime })$, and connect it to the observable shadow through the evolving critical impact parameter, $b_{\mathrm{crit}}(v)=\sqrt{r_{ph}(v{)}^2/f(r_{ph}(v))}$. 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.

8 pages

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