Kerr-fully Diving into the Abyss: Analytic Solutions to Plunging Geodesics in Kerr

TL;DR

This paper derives explicit analytical solutions for plunging geodesics in Kerr spacetime, including special cases from the ISSO, and provides practical tools for black hole physics research.

Contribution

It introduces closed-form solutions for generic and ISSO plunging geodesics in Kerr spacetime, extending previous work and implementing these in a computational toolkit.

Findings

Solutions expressed in elementary and elliptic functions

Almost elementary solutions for ISSO plunges depending only on spin and radius

New equation for radial inflow from ISSO to horizon including inclination

Abstract

We present closed-form solutions for plunging geodesics in the extended Kerr spacetime using Boyer-Lindquist coordinates. Our solutions directly solve for the dynamics of generic timelike plunges, we also specialise to the case of test particles plunging from a precessing innermost stable circular orbit (ISSO). We find these solutions in the form of elementary and Jacobi elliptic functions parameterized 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. This extends recent work on the case of equatorial plunges from the innermost stable circular orbit. Furthermore, we introduce a new equation that characterizes the radial inflow from the ISSO to the horizon, taking into account the inclination. For ease of application, our results have been implemented in the KerrGeodesics package in the Black Hole Perturbation Toolkit.