The constraint tensor for null hypersurfaces

TL;DR

This paper introduces a fully covariant, explicit definition of the constraint tensor for null hypersurfaces, applicable to any topology, and explores its geometric implications and applications in spacetime geometry.

Contribution

It provides a new, explicit, covariant definition of the constraint tensor for null hypersurfaces, extending its applicability and simplifying related geometric analyses.

Findings

Constraint tensor coincides with the pull-back of ambient Ricci tensor.

Derived simple gauge-invariant quantities on transverse submanifolds.

Generalized near horizon equation for totally geodesic null hypersurfaces.

Abstract

In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold $S$ of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to $S$ takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.