 Solution

ID: 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 + n P

on the covering space. A lift is reachable from the origin if

c t \geq ∣ x + n P ∣

for at least one integer

n

. With shortest circular distance

d (x, 0) = min_{n \in Z} ∣ x + n P ∣ \leq \frac{P}{2},

J^{+} (0) = {(t, \overset{x}{ˉ}) : c t \geq d (\overset{x}{ˉ}, 0)} .

In the centred fundamental strip,

d = ∣ \overset{x}{ˉ} ∣

. For

t \geq P / (2 c) = π 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.

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