 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2017/3/b/solution

□ (A^{α β} e^{i k_{γ} x^{γ}}) = - (k_{γ} k^{γ}) A^{α β} e^{i k_{γ} x^{γ}} .

The wave equation therefore requires a null wave vector. The Lorenz gauge in linearized gravity supplies transversality:

k_{γ} k^{γ} = 0, k_{α} A^{α β} = 0.

The amplitude is symmetric because it represents a metric perturbation; its real part is the physical wave. The two displayed conditions do not by themselves make it spatially transverse and traceless: the additional reduction to transverse-traceless gauge uses the remaining residual gauge symmetry of linearized gravity.

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