Equivalence of three-dimensional spacetimes

TL;DR

This paper develops a coordinate-invariant method for classifying three-dimensional spacetimes using curvature tensors and their derivatives, adapting techniques from four-dimensional general relativity to analyze local geometry and equivalence.

Contribution

It introduces a practical adaptation of Karlhede's invariant classification for 3D gravity, providing a complete local geometry characterization via spinor formalism and classifying Goedel-type spacetimes.

Findings

Complete classification of local geometries using curvature invariants

Conditions for homogeneity in 3D Goedel-type spacetimes

Comparison with four-dimensional spacetime classifications

Abstract

A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton-York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Goedel-type spacetimes in three-dimensional General Relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Goedel-type spacetimes is studied and the results are compared with previous works on four-dimensional Goedel-type spacetimes.