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

TL;DR

This paper develops a method to describe Kerr geodesics smoothly across horizons using Weierstrass functions in horizon-penetrating coordinates, enhancing understanding of black hole spacetime structure.

Contribution

It extends the application of the Biermann-Weierstrass formula to horizon-penetrating Kerr coordinates, enabling smooth geodesic solutions across horizons.

Findings

Geodesic solutions are smooth across Kerr horizons.

Explicit continuation of geodesics between spacetime regions.

Visualizations of geodesic trajectories in extended Kerr spacetime.

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.