= Causal reachability on a spatial circle
{title2=$J^\pm(0)=\{(t,x):\pm ct\ge\min_n|x+2\pi Rn|\}$}
For flat <spacetime> with $x$ periodic of circumference $2\pi R$, lift endpoints of <causal curves> to the covering <Minkowski spacetime>. A signal can choose any winding, so the <causal future> and <causal past> use the shortest circular distance rather than a fixed coordinate separation. After a duration $\pi R/c$, every point on the circle is reachable. A constant tangential <electric field> has a nonzero loop <electromotive force>; maintaining it while a charge gains <energy> each lap requires an energy reservoir, not a single-valued electrostatic potential.
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