Revisiting critical orbits of test particles traveling in a black hole background

TL;DR

This paper systematically analyzes the critical orbits of test particles in various black hole spacetimes, providing explicit formulas and numerical results to deepen understanding of photon spheres, shadows, and null geodesics.

Contribution

It offers a comprehensive derivation of critical orbit conditions and explicit parameter relations for multiple black hole backgrounds, enhancing previous analyses.

Findings

Explicit expressions for critical radii and parameters in different black hole spacetimes

Identification of root structures of radial equations for critical orbits

Numerical results illustrating photon spheres and black hole shadows

Abstract

This paper systematically revisits the critical orbits of test particles moving in various black hole backgrounds, including the Schwarzschild, Reissner-Nordstr\"{o}m, Kerr, and Kerr-Newman spacetimes. We identify the critical orbit cases directly from the root structure of the radial equation, and provide explicit expressions relating the relevant parameters -- energy, angular momentum, and charge-to-mass ratio -- to the critical radius, as well as explicit expressions for the critical orbits in each scenario. Special attention is given to the relationship between the photon spheres, black hole shadows and the critical null geodesics. Extensive numerical results are also provided.