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

Abstract

This paper systematically revisits the critical orbits of test particles in various black hole backgrounds, including Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman spacetimes. We identify the critical orbits directly from the root structure of the radial equation, and we provide explicit expressions that relate the relevant parameters - energy, angular momentum, and charge-to-mass ratio - to the critical radius, as well as explicit formulas for the critical orbits in each case. Special attention is given to the relationships among the photon spheres, black hole shadows, and critical null geodesics. We also present extensive numerical results.