# Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

## TL;DR

This paper revisits Jacobi metrics in stationary spacetimes, providing explicit formulas for timelike and null geodesics, and connects these metrics to electromagnetic properties like magnetoelectric effects in curved spaces.

## Contribution

It introduces a unified Jacobi-Maupertuis Randers-Finsler metric framework for both timelike and null geodesics in stationary spacetimes, correcting previous misconceptions.

## Key findings

- Explicit formulas for Jacobi-Maupertuis Randers-Finsler metrics in stationary spacetimes
- Application to Taub-NUT and Kerr spacetimes
- Connection between metric and electromagnetic magnetoelectric effects

## Abstract

In this paper we return to the subject of Jacobi metrics for timelike and null geodsics in stationary spactimes, correcting some previous misconceptions. We show that not only null geodesics, but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulae. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behaviour of Maxwell's equations in the curved spacetime. In particular, we see in very concrete terms how the magnetolectric susceptibility enters the Jacobi-Maupertuis-Randers-Finsler function.