Photon sphere and shadow of a time-dependent black hole described by a Vaidya metric

TL;DR

This paper derives exact formulas for the evolution of the photon sphere and shadow of a black hole in a Vaidya spacetime with a linearly changing mass, modeling accreting or radiating black holes.

Contribution

It provides analytical solutions for photon spheres and shadows in time-dependent Vaidya black hole models with linear mass functions, extending previous static analyses.

Findings

Analytical formulas for photon sphere evolution

Explicit calculations of black hole shadow radius

Analysis of red-shift effects in dynamic spacetimes

Abstract

In this paper we derive exact analytical formulas for the evolution of the photon sphere and for the angular radius of the shadow in a special Vaidya spacetime. The Vaidya metric describes a spherically symmetric object that gains or loses mass, depending on a mass function $m(v)$ that can be freely chosen. Here we consider the case that $m(v)$ is a linearly increasing or decreasing function. The first case can serve as a simple model for an accreting black hole, the second case for a (Hawking) radiating black hole. With a linear mass function the Vaidya metric admits a conformal Killing vector field which, together with the spherical symmetry, gives us enough constants of motion for analytically calculating the light-like geodesics. Both in the accreting and in the radiating case, we first calculate the light-like geodesics, the photon sphere, the angular radius of the shadow, and the red-shift of light in coordinates in which the metric is manifestly conformally static, then we analyze the photon sphere and the shadow in the original Eddington-Finkelstein-like Vaidya coordinates.