Static electrostatic test-energy first integral

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open and let $\varphi \in C^{\mathrm{\infty }}(\mathrm{\Omega },\mathbb{R})$, assigned volts in SI bookkeeping. On $M={\mathbb{R}}_{x^0}\times \mathrm{\Omega }$ with Minkowski metric, orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, future time-orientation $N={\mathrm{\partial }}_0$, and length coordinate $x^0=ct$, take the static one-form $A=-(\varphi \mathrm{/}c)d{x}^0$ and $F=dA$. Let $I\subseteq \mathbb{R}$ be an open interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $z:I\to M$ be an existing $C^2$ future-timelike curve parametrized by proper time $\tau$ in the sense of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, with $U=dz\mathrm{/}d\tau$, $g(U,U)=-c^2$, and $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$ for constants $m>0$ and $q\in \mathbb{R}$, assigned kilograms and coulombs respectively. Write $\mathbf{w}=d\mathbf{x}\mathrm{/}dt$, $\mathrm{\mid }\mathbf{w}\mathrm{\mid }<c$, and ${\gamma }_L=(1-\mathrm{\mid }\mathbf{w}{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$.

This is a test-particle first integral in a prescribed static background. It does not assert that Minkowski space with nonzero $F$ solves the coupled Einstein equation. The expression uses kinetic momentum and the stated potential gauge; general gauge changes are covered by the compensator in the supplier theorem.