Geodesically equivalent metrics in general relativity

TL;DR

This paper explores the problem of reconstructing a metric in general relativity from its unparameterized geodesics, providing algorithms and characterizing when metrics are geodesically equivalent, especially in Ricci-flat cases.

Contribution

It introduces an effective algorithm for reconstructing 4D metrics from unparameterized geodesics and characterizes geodesic equivalence in Lorentzian metrics.

Findings

Algorithm effectively reconstructs Ricci-flat metrics from geodesics

Most metrics do not admit nontrivial geodesic equivalence

Constructed all pairs of 4D Lorentzian geodesically equivalent metrics

Abstract

We discuss whether it is possible to reconstruct a metric by its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for general relativity. We show how to understand whether all curves from a sufficiently big family are umparameterized geodesics of a certain affine connection, and how to reconstruct algorithmically a generic 4-dimensional metric by its unparameterized geodesics. The algorithm works most effectively if the metric is Ricci-flat. We also prove that almost every metric does not allow nontrivial geodesic equivalence, and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorenz signature.