Kerr-fully diving into the abyss: analytic solutions to plunging geodesics in Kerr

Abstract

We present closed-form solutions for the generic class of plunging geodesics in the extended Kerr spacetime using Boyer–Lindquist coordinates. We also specialise to the case of test particles plunging from the innermost precessing stable circular orbit (ISSO) and unstable spherical orbits. We find these solutions in the form of elementary and Jacobi elliptic functions parameterised by Mino time. In particular, we demonstrate that solutions for the ISSO case can be determined almost entirely in terms of elementary functions, depending only on the spin parameter of the black hole and the radius of the ISSO. Furthermore, we introduce a new equation that characterises the radial inflow from the ISSO to the horizon, taking into account the inclination. For ease of application, our solutions have been implemented in a Mathematica package that is available as part of the KerrGeodesics package in the Black Hole Perturbation Toolkit.