Newman–Penrose formalism

Summary

The Newman–Penrose formalism is a way to write the geometry of a four-dimensional Lorentzian manifold in a frame of four null vectors, a null tetrad. Every tensor is replaced by its complex components in this frame. The connection becomes twelve complex spin coefficients, and the Weyl tensor becomes five complex scalars $\Psi_0, \dots, \Psi_4$. It is the natural language for the Petrov classification, for gravitational radiation and for algebraically special solutions such as the Kerr metric.