Algebraic classification of 2+1 geometries

TL;DR

This paper introduces a new algebraic classification method for 2+1 geometries using five real scalars derived from the Cotton tensor, providing a frame-independent algorithm and establishing equivalence with previous methods.

Contribution

The authors develop a novel scalar-based classification approach for 2+1 geometries, including criteria and algorithms that are frame-independent and connect to existing classification techniques.

Findings

Derived Bel-Debever criteria for classification

Established multiplicity of Cotton aligned null directions

Provided a frame-independent polynomial invariant algorithm

Abstract

We present a new effective method of algebraic classification of 2+1 geometries. Our approach simply classifies spacetimes using five real scalars, defined as specific projections of the Cotton tensor onto a suitable null basis. The algebraic type of a spacetime is determined by gradual vanishing of these scalars. We derive the Bel-Debever criteria, together with the multiplicity of the Cotton aligned null directions (CANDs). Additionally, we provide a frame-independent algorithm for classification based on the polynomial curvature invariants and show the equivalence to previous methods of classification.