Three-dimensional spacetimes of maximal order

TL;DR

This paper establishes that the equivalence problem for three-dimensional Lorentzian manifolds can be resolved using up to the fifth covariant derivative of the curvature tensor, with a sharp bound demonstrated through specific examples.

Contribution

It introduces a three-dimensional analogue of the Newman-Penrose formalism and proves the maximal order needed for the equivalence problem is five, which is shown to be sharp.

Findings

The equivalence problem requires at most the fifth covariant derivative of the curvature tensor.

A class of 3D Lorentzian manifolds is constructed to demonstrate the sharpness of this bound.

The analysis employs a spinorial classification of the Ricci tensor.

Abstract

We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of the Newman-Pen-rose formalism, and spinorial classification of the three-dimensional Ricci tensor.