Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1 / f

For causal reachability with periodic time, let $P_t=2\pi T$. The lower-right panel of part c identifies its top and bottom edges. The vertical curve at fixed $x$ is a closed timelike curve: after a positive proper duration $P_t$, it returns to the same spacetime event. The quotient is a mathematically well-defined flat Lorentzian manifold, but it violates chronology and admits no usual global causal ordering or ordinary global initial-value evolution.

These are physical causality problems, rather than an algebraic contradiction in the metric. Fields and particle histories must satisfy periodic consistency conditions; an arbitrary initial state followed by irreversible evolution will generally fail to return to itself after a temporal lap. The fact that the causal past and causal future overlap everywhere, established below, explains why the ordinary three-way causal partition degenerates.

For any representative event $(t,x)$, choose an integer $n$ large enough that $c(t+n{P}_t)>|x|$. A future-directed timelike segment from the origin to that lift projects to a causal curve on the quotient. ThereforeEven points with large $|x|$ can be reached by allowing sufficiently many temporal windings. The purple lower-right panel in part c denotes simultaneous membership in both the causal future and causal past.

Choose an integer $n$ sufficiently negative that $c(t+n{P}_t)<-|x|$. That lift lies in the ordinary causal past of the origin, and its projected causal curve reaches the origin. HenceFuture-directed curves can connect the origin and an arbitrary event in either order. This property includes the origin's nontrivial closed timelike curves.

Every event is in both the causal future and causal past, soDrawing just the nearest two null rays on a temporally identified strip would miss winding causal curves and give an incorrect nonempty exterior region.