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

Degree of mobility for metrics of Lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

“…In our case the whole manifold M does not admit a solution of (19), but, as we show below, each indecomposable Riemannian block does admit a solution of (19).…”

“…Then there exist coordinates (r, x), such thatĝ has the form (13). By direct calculations, we see that the function v = 1 2 r 2 satisfies (19). ⇐ Suppose that v is a positive function in U (P ) satisfying (19).…”

“…From the first equaiton of (19) we see that a solution v of (19) has nonzero differential at every point of a certain everywhere dense open subset of M . Then, v is not zero at every point of a certain everywhere dense open subset of M .…”

“…The metric g is then the restriction of g to the hypersurface {v = const}, where v is the function satisfying (19).…”

“…Cone structure as the existence of a positive solution of (19) By the metric cone over (M, g), we understand the product manifoldM = R >0 (r) × M (x) equipped with the metricĝ such that in the coordinates (r, x) its matrix has the form g(r, x) = 1 0 0 r 2 g(x)…”