Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 3 / b

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- Solution b

Solution

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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