Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 2 / b / ii

The vacuum Einstein field equations imply $R_{ab}=0$ and hence $R=0$. Every term in the supplied general identity involving the Ricci tensor or its covariant derivatives therefore vanishes. Substituting $\alpha =2$ leavesThis nonlinear, gauge-independent curvature equation is the Penrose wave equation.