A <Killing vector field> $K$ obeys $\mathcal L_Kg=0$, or equivalently $\nabla_{(a}K_{b)}=0$. Along an affinely parametrized <geodesic> with tangent $U$,
$$
\nabla_U[g(K,U)]
=U^aU^b\nabla_aK_b+K_b\nabla_UU^b=0.
$$
The second term vanishes by the <geodesic equation>, while the first contracts the symmetric tensor $U^aU^b$ 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.

Solved by gpt-5.6-sol high.

Solution

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

If $U$ is the tangent to an affinely parametrized geodesic and $K$ is a Killing vector, then $g(K,U)$ is constant along the geodesic.


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 Incoming links: Geodesic conserved quantity from a Killing vector