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Null hypersurfaces in general relativity: Intrinsic symmetries and differential invariants

G. Dautcourt

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.

Null hypersurfaces in general relativity: Intrinsic symmetries and differential invariants

Abstract

This paper investigates intrinsic Killing symmetries of null hypersurfaces within the framework of general relativity. To this end we consider 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.
Paper Structure (37 sections, 304 equations, 33 tables)

This paper contains 37 sections, 304 equations, 33 tables.

Table of Contents

  1. Introduction
  2. A triad formalism for $\mathcal{N}_3$
  3. Differential invariants
  4. The intrinsic Killing equation for $\mathcal{N}_3$.
  5. Groups of motion with a null Killing vector
  6. Basic equations
  7. $\delta K \neq 0$.
  8. $\delta K = 0$
  9. Example: horizons of the Kerr metric
  10. Groups $G_1,G_2$ with spacelike generators
  11. Abelian groups $G_2I$ with space-like generators
  12. Abelian groups $G_2I$ with spacelike generators coplanar with the null direction
  13. Non-Abelian groups $G_2II$ with spacelike generators
  14. Non-Abelian groups $G_2II$ with spacelike generators coplanar with the null direction
  15. Groups $G_3$ with spacelike generators
  16. ...and 22 more sections