Orbits around a Kerr black hole and its shadow

TL;DR

This paper provides a detailed derivation of the shape of the black hole shadow for a Kerr black hole, using photon orbit analysis and impact parameters, including specific case illustrations.

Contribution

It offers an exhaustive derivation of the Kerr black hole shadow shape based on photon orbit conditions and impact parameters, enhancing understanding of black hole imaging.

Findings

Derived conditions for photon capture by Kerr black holes

Formulated the black hole shadow equation in Boyer-Lindquist coordinates

Illustrated shadow shapes for equatorial observers in two cases

Abstract

Since the full General Theory of Relativity has been unveiled to the scientific community in 1915, many solutions to the vacuum Einstein field equations have been found and studied. This paper aims at documenting exhaustively the derivation of the shape of the patch of the sky that is left completely black by a spinning black hole described by the Kerr solutions in Boyer-Lindquist coordinates. This dark zone in the observer's sky is called the black hole shadow. Conserved quantities which allow for the analysis of particle orbits are first introduced, with the help of which the trajectories of photons are uniquely described by two impact parameters (Specific angular momentum in the azimuthal direction L and dimensionless Carter's constant Q). We then derive the conditions on those parameters required for a photon to be captured by the black hole. These conditions are then translated into an equation for the black hole shadow. We conclude the paper by drawing out the black hole shadows for an equatorial observer for two separate cases.