Regularization of geodesics in static spherically symmetric Kerr-Schild spacetimes

TL;DR

This paper introduces a generalized regularization method for analyzing causal geodesics near singularities in static spherically symmetric Kerr-Schild spacetimes, extending previous techniques to a broader class of spacetimes.

Contribution

It develops a generalized McGehee regularization approach for causal geodesics in Kerr-Schild spacetimes, enabling detailed analysis near singularities.

Findings

Regularization method applicable to Schwarzschild and Reissner-Nordström spacetimes.

Enhanced understanding of geodesic behavior near singularities.

Potential for broader application to other Kerr-Schild spacetimes.

Abstract

We describe a method to analyze causal geodesics in static and spherically symmetric spacetimes of Kerr-Schild form which, in particular, allows for a detailed study of the geodesics in the vicinity of the central singularity by means of a regularization procedure based on a generalization of the McGehee regularization for the motion of Newtonian point particles moving in a power-law potential. The McGehee regularization was used by Belbruno and Pretorius to perform a dynamical system regularization of the central singularity of the motion of massless test particles in the Schwarzschild spacetime. Our generalization allows us to consider causal (timelike or null) geodesics in any static and spherically symmetric spacetime of Kerr-Schild form. As an example, we apply these results to causal geodesics in the Schwarzschild and Reissner-Nordstr\"om spacetimes.