A new effective approach to the algebraic classification of geometries in $2+1$ gravity is presented. It uses five real Cotton scalars ${\mathrm{\Psi }}_{\mathrm{A}}$ of distinct boost weights, which are $3D$ analogs 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 ${\mathrm{\Psi }}_{\mathrm{A}}$, 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.

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• Alternative gravity theories
• General relativity
• General relativity equations & solutions
• General relativity formalism
• Gravity in dimensions other than four
• Quantum gravity
• Mathematical physics methods