Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 1

A Killing vector field $K$ obeys ${\mathcal{L}}_{K}g=0$, or equivalently ${\nabla }_{(a}{K}_{b)}=0$. Along an affinely parametrized geodesic with tangent $U$,The second term vanishes by the geodesic equation, while the first contracts the symmetric tensor $U^a{U}^b$ with the antisymmetric part selected by the Killing equation. Thus $g(K,U)$ is a geodesic conserved quantity from a Killing vector. In Minkowski spacetime, translational Killing fields give conserved energy-momentum and rotational or boost Killing fields give the corresponding angular-momentum and boost charges; the same construction works for any spacetime isometry.

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Write ${\mathcal{L}}_{X}Y=[X,Y]$. Expanding the definition on $fZ$ gives terms proportional to derivatives of $f$ with coefficientAll remaining terms carry an overall factor $f$, soThe same argument gives linearity in $Y$, confirming that the Lie derivative of an affine connection is tensorial.

Using the torsion-free identities $[X,Y]={\nabla }_{X}Y-{\nabla }_{Y}X$ and $[X,Z]={\nabla }_{X}Z-{\nabla }_{Z}X$, expandThis is the required formula.

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The flow of a Killing vector field consists of local isometries. Isometries preserve both the metric and its unique torsion-free metric-compatible Levi-Civita connection, hence ${\mathcal{L}}_K\nabla =0$. Set $X=K$ and $Y=Z=U$ in part (b). Since ${\nabla }_{U}U=0$,and antisymmetry of the first two curvature arguments givesThis is the geodesic deviation equation with connecting field $K$: applying the isometry to the original geodesic produces a neighboring geodesic, and curvature determines their relative acceleration.

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