The Newman–Penrose (NP) formalism is a mathematical framework used in the field of General Relativity and theoretical physics to study the properties of spacetime and gravitational fields. Developed by physicists Ezra Newman and Roger Penrose in the 1960s, this formalism is particularly useful for analyzing asymptotically flat spacetimes, such as those found in models of gravitational radiation and black hole physics.
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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]
https://doi.org/10.1103/PhysRevD.108.064002. Section II: null-tetrad curvature components and tetrad freedom
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.