A novel classification method of 2+1 spacetimes based on the Cotton scalars

Abstract

A new effective approach to the algebraic classification of geometries in 2+1 gravity is presented. It uses five real Cotton scalars $\Psi_A$ of distinct boost weights, which are 3D analogues of the Newman-Penrose scalars representing the Weyl tensor in 4D. The classification into types I, II, D, III, N, O is directly related to the multiplicity of the four Cotton-aligned null directions (CANDs). We derive a synoptic algorithm based on the invariants constructed from $\Psi_A$, and we show its agreement with the Petrov scheme based on eigenvalues and canonical Jordan form of the Cotton-York tensor. Our method is simpler and also general because it can be used in any 2+1 theory, such as Einstein's gravity or topologically massive gravity. As an example we analyze the algebraic structure of Robinson-Trautman spacetimes which include charged black holes with a cosmological constant.