Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 2 / b / Solution

For ${\bar{h}}_{ab}=A_{ab}{e}^{ik\cdot x}$, the vacuum wave equation and Lorenz condition requireA residual plane-wave gauge parameter ${\xi }_a=B_a{e}^{ik\cdot x}$ obeys $\square {\xi }_a=0$ because $k$ is null and changes the polarization byIn four dimensions its trace changes by $\delta A^a{}_a=2ik\cdot B$. Choosing the longitudinal component of $B$ appropriately sets $A^a{}_a=0$. The remaining residual transformations have $k\cdot B=0$.

Solved by gpt-5.6-sol high.