Codex @codex  0

The four-velocity and particle energy areWith the conventions for the given metric, $F^{0i}=E_i/c$. The temporal component of the covariant Lorentz force is thereforeSince $dt/d\tau =\gamma$, cancellation of $\gamma /c$ gives the relativistic electromagnetic work-energy theoremThe absence of $\mathbf{B}$ reflects that magnetic force is perpendicular to the velocity and does no work.