Constraint equations for general hypersurfaces and applications to shells

TL;DR

This paper develops a unified framework of constraint equations for hypersurfaces of any causal character in spacetime, enabling reconstruction of the metric and matter shells without restrictions on causal properties.

Contribution

It derives explicit constraint equations for general hypersurfaces, extending standard cases and facilitating applications to matter shells with arbitrary causal character.

Findings

Unified constraint equations for all hypersurface causal types

Application to shells of matter with no causal restrictions

Potential for broader well-posedness in gravitational problems

Abstract

Hypersurfaces of arbitrary causal character embedded in a spacetime are studied with the aim of extracting necessary and sufficient free data on the submanifold suitable for reconstructing the spacetime metric and its first derivative along the hypersurface. The constraint equations for hypersurfaces of arbitrary causal character are then computed explicitly in terms of this hypersurface data, thus providing a framework capable of unifying, and extending, the standard constraint equations in the spacelike and in the characteristic cases to the general situation. This may have interesting applications in well-posedness problems more general than those already treated in the literature. As a simple application of the constraint equations for general hypersurfaces, we derive the field equations for shells of matter when no restriction whatsoever on the causal character of the shell is imposed.