Basis for scalar curvature invariants in three dimensions

TL;DR

This paper aims to construct a minimal basis of scalar polynomial curvature invariants in three-dimensional Lorentzian spacetimes, using computational tools and methods to facilitate spacetime characterization.

Contribution

It introduces a systematic approach to identify algebraically independent scalar curvature invariants up to fifth order in 3D Lorentzian spacetimes, utilizing the Invar software and equivalence methods.

Findings

Computed an overdetermined basis of scalar invariants in 3D

Discussed the use of the Karlhede algorithm for invariants

Provided insights into spacetime characterization using invariants

Abstract

$\mathcal{I}$-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann tensor and its covariant derivatives up to fifth order of differentiation. We use the computer software \emph{Invar} to calculate an overdetermined basis of scalar curvature invariants in three dimensions. We also discuss the equivalence method and the Karlhede algorithm for computing Cartan invariants in three dimensions.