Hypersurface Data: General Properties and Birkhoff Theorem in Spherical Symmetry

Abstract

The notions of (metric) hypersurface data were introduced in Mars (Gen Relativ Gravit 45:2175–2221, 2013) as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-Riemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived.