Scalar Polynomial Curvature Invariants in the Context of the Cartan-Karlhede Algorithm

TL;DR

This paper uses the Cartan-Karlhede algorithm to identify minimal scalar polynomial curvature invariants that uniquely characterize G"odel-like spacetimes in three-dimensional gravity, classifying subclasses by Ricci tensor types.

Contribution

It provides a systematic method to derive minimal scalar invariants for G"odel-like spacetimes, enhancing their geometric classification.

Findings

Generated a minimal set of scalar polynomial invariants for G"odel-like spacetimes.

Classified subclasses based on Ricci tensor P-types.

Expressed invariants in terms of Cartan invariants at each order.

Abstract

We employ the Cartan-Karlhede algorithm in order to completely characterize the class of G\"odel-like spacetimes for three-dimensional gravity. By examining the permitted Segre types (or P-types) for the Ricci tensor we present the results of the Cartan-Karlhede algorithm for each subclass in terms of the algebraically independent Cartan invariants at each order. Using this smaller subset of Cartan invariants we express the scalar polynomial curvature invariants for the G\"odel-like spacetimes in terms of this subset of Cartan invariants and generate a minimal set of scalar polynomial curvature invariants that uniquely characterize metrics in the class of G\"odel-like spacetimes and identify the subclasses in terms of the P-types of the Ricci tensor.