Kerr Geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass functions

Abstract

We revisit the theory of timelike and null geodesics in the (extended) Kerr spacetime. This work is a sequel to a recent paper by Cie\'{s}lik, Hackmann, and Mach, who applied the so-called Biermann-Weierstrass formula to integrate Kerr geodesic equations expressed in Boyer-Lindquist coordinates. We show that a formulation based on the Biermann-Weierstrass theorem can also be applied in horizon-penetrating Kerr coordinates, resulting in solutions that are smooth across Kerr horizons. Horizon-penetrating Kerr coordinates allow for an explicit continuation of timelike and null geodesics between appropriate regions of the maximal analytic extension of the Kerr spacetime. A part of this work is devoted to a graphic visualisation of such geodesics.