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 $\alpha=2$ leaves
$$
\boxed{
\nabla^e\nabla_eR_{abcd}
+2R_a{}^e{}_c{}^fR_{bdef}
-2R_a{}^e{}_d{}^fR_{becf}
+R_{abef}R_{cd}{}^{ef}=0
}.
$$
This nonlinear, gauge-independent curvature equation is the <Penrose wave equation>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-309/2/b/ii/solution