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

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.

Some details corrected, one figure and several references added

References in corpus (8)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole
• First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole
• First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring
• First M87 Event Horizon Telescope Results. II. Array and Instrumentation
• First M87 Event Horizon Telescope Results. III. Data Processing and Calibration
• Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. IV. H$\beta$ Time Lags and Implications for Super-Eddington Accretion
• Understanding photon sphere and black hole shadow in dynamically evolving spacetimes

Cited by in corpus (6)

• Dark matter spike around Bumblebee black holes
• Dynamical photon sphere and time evolving shadow around black holes with temporal accretion
• Parameter estimation of Kerr-Bertotti-Robinson black holes using their shadows
• Image of the time-dependent black hole
• Gravitational Black Hole Shadow Spectroscopy
• Optical appearance of regularized compact objects without an exterior photon sphere