Invariant classification of vacuum PP-waves

TL;DR

This paper provides a complete invariant classification of vacuum PP-wave spacetimes using Cartan invariants, including the characterization of G2 and G3 subclasses, and confirms the sharpness of the fourth order covariant derivative bound.

Contribution

It introduces a set of Cartan invariants for classifying vacuum PP-waves and demonstrates the necessity of up to fourth order derivatives for full classification.

Findings

Complete invariant classification of vacuum PP-waves achieved.

Invariant characterization of G2 and G3 subclasses derived.

Confirmed the sharpness of the fourth order derivative bound.

Abstract

We solve the equivalence problem for vacuum PP-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [Collins91] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q<=4 bound is sharp and explicitly describe all such maximal order solutions.