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

Under $h_{\mu \nu }^{\prime }=h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, trace reversal givesTaking a divergence yieldsApplying $\square$ to the transformed field likewise produces only derivatives of $\square \xi$. Therefore $\square {\xi }^{\mu }=0$ preserves both the gauge condition and the sourced wave equation. These are the residual gauge transformations.