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There are several geometric interpretations of the scalar curvature on Riemannian manifolds. The one that compares the volume of an infinitesimal ball on the manifold to the volume of a ball with the same radius in flat space. The analog on the boundaries of the same balls. There is a specific intepretation for 2D that relates the scalar curvature to the convergence rate of two neighbouring and parallel geodesics. Yet all this is for Riemannian geodesics.

Are there similar geometric interpretations for pseudo-Riemannian manifolds? I mean, for manifolds that have a non-positive definite metric, such as the ones encountered in general relativity?

Thanks!

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  • $\begingroup$ Related. $\endgroup$
    – tcamps
    Jun 20, 2019 at 1:18

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