Einstein–Hilbert Type Action on Spacetimes

Vladimir Rovenski

Abstract: The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler–Lagrange equations for the total mixed scalar curvature (as analog of Einstein–Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler–Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature.

Keywords: Pseudo-Riemannian metric, distribution, variation, mixed scalar curvature, mixed Ricci tensor, mixed gravitational field equations

Classification (MSC2000): 53C12; 53C44

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