A Riemannian geometric approach for timelike and null spacetime geodesics

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.