Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 37A (General Relativity)

Expansion of the Ricci identity gives $R^{\mu }{}_{\nu \alpha \beta }={\partial }_{\alpha }{\Gamma }_{\beta \nu }^{\mu }-{\partial }_{\beta }{\Gamma }_{\alpha \nu }^{\mu }+{\Gamma }_{\alpha \lambda }^{\mu }{\Gamma }_{\beta \nu }^{\lambda }-{\Gamma }_{\beta \lambda }^{\mu }{\Gamma }_{\alpha \nu }^{\lambda }$.

Solved by gpt-5.6-sol high.

$R_{\mu \nu \alpha \beta }=-R_{\nu \mu \alpha \beta }=-R_{\mu \nu \beta \alpha }=R_{\alpha \beta \mu \nu }$ and $R_{\mu [\nu \alpha \beta ]}=0$. In normal coordinates at a point the connection vanishes, so the expression reduces to derivatives of the metric and these symmetries are transparent; tensoriality extends them to all coordinates.

Solved by gpt-5.6-sol high.

Under a coordinate change, the inhomogeneous second-derivative terms arising from each partial derivative cancel between $V^{\alpha }{\partial }_{\alpha }{W}^{\mu }$ and $W^{\alpha }{\partial }_{\alpha }{V}^{\mu }$. The remainder transforms with one contravariant Jacobian, so the commutator is a vector field.

Solved by gpt-5.6-sol high.

Differentiate the Killing equation, permute indices, and combine using the Ricci identity to obtain ${\nabla }_{\alpha }{\nabla }_{\beta }{V}^{\mu }=R^{\mu }{}_{\beta \alpha \nu }{V}^{\nu }$ (up to the equivalent sign convention for $R$).

Solved by gpt-5.6-sol high.

The Lie derivative formulation is ${\mathcal{L}}_{V}g=0$. Since $[{\mathcal{L}}_V,{\mathcal{L}}_W]={\mathcal{L}}_{[V,W]}$, two Killing fields give ${\mathcal{L}}_{[V,W]}g=0$, so their commutator is Killing.

Solved by gpt-5.6-sol high.