Shadow of the generalized Vaidya black hole

Abstract

In this paper, we investigate the shadow of the generalized Vaidya black hole described by the Husain solution. Assuming a barotropic equation of state for the additional matter sector, we first identify the self-similar class admitting a homothetic Killing vector and transform the geometry to a conformally static form. This allows us to derive the effective potential for null geodesics in the conformally static representative, determine the corresponding homothetic photon surface, and then transform the result back to the original generalized Vaidya coordinates. We show that, within the weak-energy-condition branches, the parameter $α$ controlling the equation of state determines whether the shadow is enlarged or reduced relative to the Vaidya case: the branch $0\leqα<1/2$ increases the photon-sphere and shadow radii, whereas the branch $1/2<α\leq 1$ decreases them. We then turn to the genuinely time-dependent Husain metric and derive quasistatic slow-evolution equations for the instantaneous photon surface and critical impact parameter. In particular, we show that the growth or contraction of the shadow is governed by the local effective influx in the photon region rather than by the signs of $\dot M$ and $\dot N$ separately. For the charge-like branch, this yields a transparent criterion for when accretion enlarges the shadow and when rapid charge growth instead causes it to shrink.