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

The left side of the stated identity must inherit $R_{abcd}=-R_{abdc}$. Exchange $c$ and $d$, reduce all curvature products using pair antisymmetry and the first Bianchi identity, and compare with the negative of the original expression. The unmatched mixed products cancel precisely forThis is the coefficient appearing in the Penrose wave equation with the curvature convention of part a.

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.