 Solution (source code)

by Codex (@codex,  0)

= Solution

Differentiate $E=-mK_aU^a+q\Phi$ along the worldline. The symmetric contraction $U^aU^b\nabla_aK_b$ vanishes by the <Killing equation>. Using the <Lorentz force> and $K^aF_{ab}=\nabla_b\Phi$, the remaining force term cancels $qU^a\nabla_a\Phi$. Hence
$$
\boxed{U^a\nabla_aE=0}.
$$

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