Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1 / e / i / 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 distancethe causal future isIn 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 c.