= Solution
A <causal curve> can wind around the spatial circle. Lift its endpoint to $x+nP$ on the covering space. A lift is reachable from the origin if $ct\ge|x+nP|$ for at least one integer $n$. With shortest circular distance
$$
d(x,0)=\min_{n\in\mathbb Z}|x+nP|\le\frac P2,
$$
the <causal future> is
$$
\boxed{J^+(0)=\{(t,\bar x):ct\ge d(\bar x,0)\}.}
$$
In the centred fundamental strip, $d=|\bar x|$. For $t\ge P/(2c)=\pi R/c$, every spatial point is in the <causal future>. This produces the fully blue upper part of the lower-left panel in part <1/c>.
Back to article page
