Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 36A (General Relativity)

Normal coordinates at an event have $g_{ab}={\eta }_{ab}$ and first metric derivatives, equivalently connection coefficients, zero there. They describe freely falling frames in which nongravitational laws take their special-relativistic form to first order, expressing the Equivalence principle. Curvature and tidal effects cannot generally be removed throughout a neighbourhood.

For a covector, ${\nabla }_a{V}_b={\partial }_a{V}_b-{\Gamma }_{ab}^c{V}_c$. The Levi-Civita connection is torsion-free, sois an antisymmetric covariant tensor. Next expand ${\nabla }_a{K}_b+{\nabla }_b{K}_a$ using $K_b=g_{bc}{K}^c$ and $2g_{cd}{\Gamma }_{ab}^d={\partial }_a{g}_{bc}+{\partial }_b{g}_{ac}-{\partial }_c{g}_{ab}$. Cancelling the first two metric derivatives leavesIf $K={\partial }_4$ has constant coordinate components, $Q_{ab}={\partial }_4{g}_{ab}$. Thus $Q=0$ implies the metric is independent of $x^4$ and $K$ is a Killing vector. Its derivative is antisymmetric, yieldingFor an affinely parametrized geodesic with tangent $u$, differentiation givesThe first term vanishes by antisymmetry and the second by the geodesic equation, proving conservation in every coordinate system. A stationary metric requires the Killing vector to be timelike in the region concerned. For particle energy, the tangent should be future timelike for a massive particle, with affine parameter proportional to proper time, or future null with appropriate momentum normalization for a photon; normalize the future timelike Killing vector relative to the chosen stationary observers or at infinity. Then the conserved quantity is proportional to their Killing energy. A spacelike Killing vector instead represents momentum, not stationary time translation.