Newman-Penrose formalism

The Newman-Penrose formalism expresses spacetime geometry in a complex null tetrad: two real null directions and a complex-conjugate transverse pair. Contracting the Weyl tensor with these vectors packages its ten real components into five complex quantities. Other contractions describe Ricci curvature and the connection in the same basis. ^[ref]

Choosing directions adapted to a light ray, a horizon or an outgoing wave can make the relevant curvature components easier to identify. This is useful for radiation calculations and extracting gravitational waves from simulations. The components are scalars under coordinate changes but depend on the tetrad, so their signs, normalization and physical interpretation require a stated frame convention.

A complex null tetrad consists of two real null vectors and a complex-conjugate transverse pair, normalized so that the metric is reconstructed from their pairwise products.

For a null tetrad $(\ell ,k,m,\bar{m})$, one common curvature-sign convention defines ${\Psi }_4=C_{\mu \nu \rho \sigma }{k}^{\mu }{\bar{m}}^{\nu }{k}^{\rho }{\bar{m}}^{\sigma }$. In an outgoing wave zone it encodes the two transverse gravitational wave polarizations as one complex scalar.