Algebraic classification of 2+1 geometries

TL;DR

Problem: algebraic classification of 2+1 geometries in three dimensions where the Weyl tensor vanishes and curvature is encoded by the Cotton tensor. Approach: project the Cotton tensor onto a real null frame to obtain five scalars $\\Psi_0$ through $\\Psi_4$ and classify by their vanishing patterns; determine Cotton-aligned null directions by solving the quartic $\\Psi_4 K^4 - 2\\sqrt{2}\\Psi_3 K^3 - 6\\Psi_2 K^2 + 2\\sqrt{2}\\Psi_1 K - \\Psi_0 = 0$, with boost-transform behavior $\\Psi'_A = B^{2- A}\\Psi_A$, and use a frame-independent route based on invariants $I$ and $J$ of the Cotton tensor and the Cotton–York tensor $Y_{ab}$ together with covariants $G,H,N$. Contributions: explicit Bel–Debever criteria in 3D, a complete invariant-based diagnostic with $I,J$ and polynomial covariants $G,H,N$, and a flow diagram algorithm; demonstration of equivalence to Cotton–York (Petrov-like) eigenvalue classification. Significance: provides a robust, frame-invariant, computationally practical toolkit for classifying 2+1 geometries in gravity, with implications for exact solutions and quantum gravity in three dimensions.

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.

Figures (1)

• Figure 1: Flow diagram for determining the algebraic type of 2+1 geometry using polynomial invariants.