= Geodesic potential of the magnetic dilaton black hole
{title2=$\widetilde V=\tfrac12[V(\epsilon+L^2/R^2)-E^2]$}
Rotate any <geodesic> into an equatorial plane of the <magnetically charged dilaton black hole>. Conserved energy and angular momentum are $E=V\dot t$ and $L=R^2\dot\phi$. Normalizing its tangent to $g(\dot x,\dot x)=-\epsilon$, with $\epsilon=1,0,-1$ for timelike, null and spacelike geodesics, gives $\dot r^2=E^2-V(\epsilon+L^2/R^2)$. Thus its radial equation is $\dot r^2/2+\widetilde V=0$. Radial null rays have $\dot r=\pm E$ and use the same <Schwarzschild tortoise coordinate> with $2M$ replaced by $r_+$.
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