= Charged-particle radial equation in Reissner-Nordstrom spacetime
{title2=$\dot r^2=(E-eQ/r)^2/m^2-F(1+l^2/r^2)$}
For a massive <test particle> with <electric charge> $e$, choose the <electromagnetic four-potential> $A_t=-Q/r$ and <proper time> $\tau$. The <cyclic coordinates> $t,\phi$ in the equatorial <Lagrangian> give the <conserved quantities> $E=mF\dot t+eQ/r$ and $l=r^2\dot\phi$. Here $l$ is <angular momentum> per unit <mass>. Substituting these quantities into the <four-velocity> normalization $g(U,U)=-1$ gives
$$
\dot r^2=\frac{(E-eQ/r)^2}{m^2}-F\left(1+\frac{l^2}{r^2}\right).
$$
The <energy> includes the gauge-dependent electrostatic term; the displayed convention fixes that gauge by taking the potential to vanish at infinity.
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