= Solution
<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, $\nabla_aV_b=\partial_aV_b-\Gamma^c_{ab}V_c$. The <Levi-Civita connection> is torsion-free, so
$$
\partial_aV_b-\partial_bV_a=\nabla_aV_b-\nabla_bV_a
$$
is an antisymmetric <covariant tensor>. Next expand $\nabla_aK_b+\nabla_bK_a$ using $K_b=g_{bc}K^c$ and $2g_{cd}\Gamma^d_{ab}=\partial_ag_{bc}+\partial_bg_{ac}-\partial_cg_{ab}$. Cancelling the first two metric derivatives leaves
$$
\boxed{2\nabla_{(a}K_{b)}=K^c\partial_cg_{ab}+g_{ac}\partial_bK^c+g_{bc}\partial_aK^c=Q_{ab}}.
$$
If $K=\partial_4$ has constant coordinate components, $Q_{ab}=\partial_4g_{ab}$. Thus $Q=0$ implies the metric is independent of $x^4$ and $K$ is a <Killing vector>. Its derivative is antisymmetric, yielding
$$
\boxed{\nabla_aK_b=\tfrac12(\partial_aK_b-\partial_bK_a)}.
$$
For an affinely parametrized <geodesic> with tangent $u$, differentiation gives
$$
\frac d{d\lambda}(-K_au^a)=-u^au^b\nabla_aK_b-K_a\nabla_uu^a=0.
$$
The 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.
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