McGehee regularization of general SO(3)-invariant potentials and applications to stationary and spherically symmetric spacetimes

TL;DR

This paper extends the McGehee regularization method to analyze the singularities of causal geodesics in stationary, spherically symmetric spacetimes, providing a unified phase space description for Schwarzschild and Reissner-Nordström geometries.

Contribution

It generalizes the McGehee transformation to regularize geodesic singularities in Kerr-Schild spacetimes, enabling a global phase space analysis of particle trajectories.

Findings

Geodesics can be modeled as Newtonian particles in a radial potential.

The method applies to both massive and massless particles.

The phase space for geodesics has a non-trivial topology due to causality constraints.

Abstract

The McGehee regularization is a method to study the singularity at the origin of the dynamical system describing a point particle in a plane moving under the action of a power-law potential. It was used by Belbruno and Pretorius to perform a dynamical system regularization of the singularity at the center of the motion of massless test particles in the Schwarzschild spacetime. In this paper, we generalize the McGehee transformation so that we can regularize the singularity at the origin of the dynamical system describing the motion of causal geodesics (timelike or null) in any stationary and spherically symmetric spacetime of Kerr-Schild form. We first show that the geodesics for both massive and massless particles can be described globally in the Kerr-Schild spacetime as the motion of a Newtonian point particle in a suitable radial potential and study the conditions under which the central singularity can be regularized using an extension of the McGehee method. As an example, we apply these results to causal geodesics in the Schwarzschild and Reissner-Nordstr\"om spacetimes. Interestingly, the geodesic trajectories in the whole maximal extension of both spacetimes can be described by a single two-dimensional phase space with non-trivial topology. This topology arises from the presence of excluded regions in the phase space determined by the condition that the tangent vector of the geodesic be causal and future directed.