Scalar polynomial curvature invariants in the context of the Cartan–Karlhede algorithm

Abstract

We employ the Cartan–Karlhede algorithm in order to completely characterize the class of Gödel-like spacetimes for three-dimensional gravity. By examining the permitted Segre types (or P-types) of 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 Gödel-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 Gödel-like spacetimes and identify the subclasses in terms of the P-types of the Ricci tensor.