The Jacobi-metric for timelike geodesics in static spacetimes

TL;DR

This paper demonstrates that massive particle trajectories in static spacetimes can be described by an energy-dependent Riemannian metric, extending classical Jacobi metric concepts to relativistic contexts, especially near black holes.

Contribution

It introduces an energy-dependent Jacobi metric for massive particles in static spacetimes, generalizing classical mechanics concepts to relativistic gravitational settings.

Findings

Massive particle motion is governed by an energy-dependent Riemannian metric.

In the massless limit, the metric reduces to Fermat's optical metric.

The Gaussian curvature of the metric's equatorial sections varies and is not always negative.

Abstract

It is shown that the free motion of massive particles moving in static spacetimes are given by the geodesics of an energy-dependent Riemannian metric on the spatial sections analogous to Jacobi's metric in classical dynamics. In the massless limit Jacobi's metric coincides with the energy independent Fermat or optical metric. For stationary metrics, it is known that the motion of massless particles is given by the geodesics of an energy independent Finslerian metric of Randers type. The motion of massive particles is governed by neither a Riemannian nor a Finslerian metric. The properies of the Jacobi metric for massive particles moving outside the horizon of a Schwarschild black hole are described. By constrast with the massless case, the Gaussian curvature of the equatorial sections is not always negative.