Synchronous spacetimes with pure trace extrinsic curvature slices

Abstract

We consider a synchronous spacetime with pure trace extrinsic curvature slices, and first prove that spatial metric is Einstein if and only if the electric part of the spacetime Weyl tensor vanishes, and then that spatial metric has a constant curvature if and only if the spacetime Weyl tensor vanishes. We also consider the cases when the spacetime has a harmonic Weyl tensor and is vacuum with and without a cosmological constant. In the last case, the spacetime reduces to Minkowski, de Sitter and anti-de Sitter spacetimes.