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Newman-Penrose formalism Created 2026-09-28 Updated 2026-11-05

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.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 3 / b / Solution 2026-09-28

The phase is

p_{ρ} x^{ρ} = - ω t + ω z = - ω u .

Because the transverse-traceless gauge perturbation has no component along

k = \frac{1}{2} (\partial_{t} - \partial_{z}),

only the third term of the linearized Riemann curvature operator contributes to the required contraction:

Ψ_{4} = - \frac{1}{2} (k^{ρ} \partial_{ρ})^{2} (h_{μν} \overset{m}{ˉ}^{μ} \overset{m}{ˉ}^{ν}) .

The derivatives and transverse polarization contraction are

k^{ρ} \partial_{ρ} e^{- iω u} = - i 2 ω e^{- iω u},

H_{μν} \overset{m}{ˉ}^{μ} \overset{m}{ˉ}^{ν} = H_{+} - i H_{\times}, \overset{m}{ˉ} = \frac{1}{2} (\partial_{x} - i \partial_{y}) .

Therefore the Newman--Penrose scalar Psi4 is

Ψ_{4} = ω^{2} (H_{+} - i H_{\times}) e^{- iω u} .

Its real and imaginary parts encode the plus and cross gravitational wave polarizations, up to the stated Riemann curvature tensor and complex null tetrad conventions.

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