Analytical approach for calculating shadow of dynamical black hole

Abstract: A real astrophysical black hole, surrounded by an accretion disc, experiences time-dependent mass variation due to accretion, impacting observable features like the shadow. In this work, we analyzed the dynamical shadow by extending the concept of the photon sphere to time-dependent spacetimes, incorporating induced, centrifugal, and relativistic forces. We demonstrated that the shadow grows when light loses energy and shrinks when it gains energy, with a surprising result that the shadow remains static when the photon sphere radius coincides with the NEC horizon radius. Our method, assuming a small accretion rate, provides a foundation for studying dynamical black holes, offering insights into processes like gravitational collapse, which could make shadow evolution detectable and reveal details about black hole formation.