Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 36B (General Relativity)

Use metric compatibility and the symmetric Levi-Civita connection. Expanding $k_a=g_{ac}{k}^c$ givesThe Christoffel formula makes the bracket ${\partial }_c{g}_{ab}$. Thus the Killing equation is equivalent to the displayed coordinate formula in the question, also the vanishing Lie derivative of the metric.

Along an affinely parametrized geodesic with tangent $V$,The first term vanishes by the geodesic equation and the second by the Killing equation, because $V^a{V}^b$ is symmetric. This proves the geodesic conserved quantity from a Killing vector.

For $d{s}^2=-d{u}^2+u^{2}d{v}^2$, the three independent coordinate Killing conditions areThe field $(0,1)$ satisfies all three immediately. For $e^{-v}(1,u^{-1})$, the second equation reads $u{e}^{-v}-u{e}^{-v}=0$, and the third reads $e^{-v}-e^{-v}=0$. For $e^v(-1,u^{-1})$, they read $-u{e}^v+u{e}^v=0$ and $e^v-e^v=0$. The first equation holds for both, verifying all three Killing vector fields.

Their conserved contractions with $(\dot{u},\dot{v})$ may be denotedAdding the last two expressions after multiplying by $e^v,e^{-v}$ givesTo identify these loci, set $T=u\cosh v$, $X=u\sinh v$. The metric becomes $-d{T}^2+d{X}^2$, and the equation becomes $(\alpha +\beta )T+(\alpha -\beta )X=2\gamma$, a straight line in Minkowski spacetime. Nontrivial such lines, parametrized affinely, are exactly the geodesics in this coordinate patch. The all-zero choice of constants is a vacuous equation rather than an individual geodesic.