The Cotton tensor in Riemannian spacetimes

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 n. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components as n(n 2 − 4)/3 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. file cott16.tex,Recently, the study of three-dimensional spaces is becoming of great interest; for these spaces the Weyl tensor is always zero and the vanishing of the Cotton tensor depends on the type of relation between the Ricci tensor and the energy-momentum tensor of matter.Any three-dimensional space is conformally flat if the Cotton tensor vanishes. If matter is present, the Ricci tensor is related to the energy-momentum tensor of matter by means of the Einstein equations. Then the vanishing of the Cotton tensor imposes severe restrictions on the energy-momentum tensor. The Cotton tensor also plays a role in the context of the Hamiltonian formulation of general relativity, see [10].The outline of the article is as follows. First we derive the Cotton 2-form in the context of the Bianchi identities. Subsequently we describe its characteristic properties and perform

Generalization of Einstein’s gravitational field equations

“…where we identify here the Cotton tensor C jkl [11,12]. Imposing energy conservation thus gives us the physical value of a:…”

Equivalence of three-dimensional spacetimes

“…The conformal properties of three-dimensional spacetime are described by the Cotton-York tensor [36] …”

Asymptotic expansions of the Cotton–York tensor on slices of stationary spacetimes

“…Due to the specialness of the latter spacetime, the form of the obstructions for more general stationary spacetimes cannot be inferred from merely looking at the Kerrian case. 9 …”