Geodesically equivalent metrics in general relativity

Details Geodesically equivalent metrics in general relativity Geodesically equivalent metrics in general relativity

2011-01-11

arxiv.org/abs/1101.2069v2

J. Geom. Phys. Volume 62, Issue 3, March 2012, Pages 675--691

Mathematics

Differential Geometry

28 pages, one figure. No essential changes w.r.t. (v1): misprints corrected and references updated

Scientific paper

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.

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