Analytical approach for calculating shadow of dynamical black hole

Abstract

In this paper, 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, 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, bcrit(v). For Vaidya spacetimes, this predicts that accretion (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.