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 [1] 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-Nordstrom spacetimes.
Highlights
The motion of freely falling particles in static spherically symmetric spacetimes constitutes a basic topic in General Relativity, which has been studied from several perspectives
The Hamiltonian His a standard Hamiltonian in Newtonian mechanics for a point particle in a central potential. This is a substantial simplification over the original problem of solving the geodesic equations in a stationary and spherically symmetric spacetime of Kerr-Schild form, because we can exploit all the information known for trajectories of point particles in Newtonian mechanics under the influence of a radial potential
The method that we develop in the following theorem, see [2], is called “McGehee regularization” because it provides a generalization of the original approach by McGehee in [4]
Summary
The motion of freely falling particles in static spherically symmetric spacetimes constitutes a basic topic in General Relativity, which has been studied from several perspectives. We show that the dynamics described by the Hamiltonian of causal geodesics in a static, spherically symmetric spacetimes of Kerr-Schild form is equivalent to the dynamics of a Newtonian point particle under the action of a suitable central potential. The Hamiltonian His a standard Hamiltonian in Newtonian mechanics for a point particle in a central potential This is a substantial simplification over the original problem of solving the geodesic equations in a stationary and spherically symmetric spacetime of Kerr-Schild form, because we can exploit all the information known for trajectories of point particles in Newtonian mechanics under the influence of a radial potential. The method that we develop in the following theorem, see [2], is called “McGehee regularization” because it provides a generalization of the original approach by McGehee in [4] This procedure will allow us to obtain information of the geodesics at the vicinity of the singularity. We will see below an example of this behavior when considering the Schwarzschild limit of the dynamical system describing causal geodesics in the Reissner-Nordstrom spacetime
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