A Riemannian geometric approach for timelike and null spacetime geodesics

TL;DR

This paper introduces a generalized Riemannian approach to analyze geodesics in Lorentzian spacetimes by projecting metrics along Killing vectors, enabling new insights into spacetime geometry and cosmic censorship.

Contribution

It presents a generalized Jacobi metric derived from Lorentzian metrics, applicable to various spacetime geometries and dimensions, extending the classical Jacobi metric approach.

Findings

The method reproduces spacetime geodesics through projected Riemannian metrics.

Calculated Gaussian and geodesic curvatures for the 2D metrics.

Applied the technique to study near horizon, asymptotic limits, and cosmic censorship violations.

Abstract

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a $3-$dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The $3-$dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting $2-$dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.