Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 3 / c / i / Solution

Let $s=k_{\rho }{x}^{\rho }$. In vacuum,A nonzero localized profile cannot have $H^{\prime \prime }=0$, because an affine function does not decay at both ends. HenceDecay removes the integration constant, so the second condition is equivalently $k^{\mu }{H}_{\mu \nu }=0$. Thus a nontrivial plane gravitational wave in linearized gravity has a null wavevector and transverse amplitude.