Geodesically equivalent metrics in general relativity
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.
Cite
@article{arxiv.1101.2069,
title = {Geodesically equivalent metrics in general relativity},
author = {Vladimir S. Matveev},
journal= {arXiv preprint arXiv:1101.2069},
year = {2013}
}
Comments
28 pages, one figure. No essential changes w.r.t. (v1): misprints corrected and references updated