Algebraic classification of 2+1 geometries: a new approach

TL;DR

This paper introduces a new algebraic classification method for 2+1 spacetimes using Cotton scalars, simplifying the process and relating it to null directions and invariants, demonstrated on various explicit examples.

Contribution

It develops a null-triad based algebraic classification method for 2+1 geometries that is equivalent to traditional Petrov-type classification, but more straightforward and invariant-based.

Findings

Method classifies 2+1 spacetimes without field equations.

Classification based on Cotton scalar vanishing patterns.

Applied to Robinson-Trautman and other metrics.

Abstract

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman-Penrose curvature scalars $\Psi_A$ of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions (CANDs) and to the corresponding Bel-Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton-York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson-Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.