On scalar curvature invariants in three dimensional spacetimes

TL;DR

This paper develops a method to identify a minimal set of algebraically independent scalar curvature invariants in 3D Lorentzian spacetimes using the Cartan-Karlhede algorithm, demonstrated on Szekeres cosmological models.

Contribution

It introduces a systematic approach to determine minimal scalar curvature invariants in 3D spacetimes, linking Cartan invariants to scalar polynomial invariants.

Findings

At most twelve algebraically independent Cartan invariants in 3D Szekeres spacetimes.

Explicit relations between Cartan invariants and scalar polynomial curvature invariants.

Identification of minimal sets of invariants up to second order.

Abstract

We wish to construct a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some order of differentiation in three dimensional (3D) Lorentzian spacetimes. In order to do this we utilize the Cartan-Karlhede equivalence algorithm since, in general, all Cartan invariants are related to scalar polynomial curvature invariants. As an example we apply the algorithm to the class of 3D Szekeres cosmological spacetimes with comoving dust and cosmological constant $\Lambda$. In this case, we find that there are at most twelve algebraically independent Cartan invariants, including $\Lambda$. We present these Cartan invariants, and we relate them to twelve independent scalar polynomial curvature invariants (two, four and six, respectively, zeroth, first, and second order scalar polynomial curvature invariants).