Null hypersurfaces in general relativity: Intrinsic symmetries and differential invariants

TL;DR

This paper classifies null hypersurfaces in general relativity based on their intrinsic symmetries and invariants, providing a detailed taxonomy and normal forms, with implications for understanding horizons.

Contribution

It extends previous classifications by analyzing null hypersurfaces up to fourth order symmetries using differential invariants and triad calculus.

Findings

Classification of null hypersurfaces by symmetry groups

Normal forms and invariants for each class

Discussion of horizons as special null hypersurfaces

Abstract

This paper investigates intrinsic Killing symmetries of null hypersurfaces $\mathcal{N}_3$ within the framework of general relativity. To this end we consider $\mathcal{N}_3$ as detached from the embedding spacetime and equipped with a degenerate metric of signature (0,+,+). As geometrical tools we use a triad calculus and differential invariants. Extending prior work, we present a classification of null hypersurfaces according to groups of motion up to the fourth order. For each type certain normal forms of the metric are given, and their invariants are listed. A discussion of horizons - defined as null hypersurfaces with vanishing shear and divergence - is included.