Lorentzian Metrics Null-Projectively Related to Semi-Riemannian Metrics

Abstract

We say that a Lorentzian metric and a semi-Riemannian metric on the same manifold M are null-projectively related if every null geodesic of the Lorentzian metric is an unparametrized geodesic of the semi-Riemannian one. This definition includes the case of conformally related Lorentzian metrics and the case of projectively equivalent metrics. We characterize the null-projectively relation by means of certain tensor and provide some examples. Then, we focus on the special case in which both metrics share parametrized null geodesics. In this case, it is said that they are null related. We show how to construct projectively equivalent metrics via a conformal transformation from null-related ones and conversely. The classical Levi-Civita theorem on projectively equivalent metrics is adapted to the case of null-related metrics and some results ensuring that two null-related metric are affinely equivalent are proven under curvature conditions.