The Cotton tensor in Riemannian spacetimes

TL;DR

This paper systematically analyzes the Cotton tensor in three-dimensional Riemannian spacetimes, deriving its properties, classifying it, and exploring its relation to Einstein's equations and solutions in topologically massive gravity.

Contribution

It provides the first systematic derivation, irreducible decomposition, and classification of the Cotton tensor in three dimensions, along with applications to Einstein's equations.

Findings

Derived the irreducible components of the Cotton tensor.

Classified the Cotton tensor in three dimensions.

Presented explicit solutions related to Einstein's equations.

Abstract

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.