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

The upper-left panel shows the three causal structure regions, with null boundaries included in the appropriate causal future and causal past. For the charged trajectory, use part b with $a=q{E}_x/m$, choosing the $x$ direction so that $a>0$. The electric field gives the constant three-force $q{E}_x$ and hence the shifted hyperbolaIt starts with a vertical tangent at the origin, stays strictly inside the future light cone, and asymptotes to $x=ct-c^2/a$. The other panels implement the coordinate and topology changes used below; they are original spacetime diagrams, not copies of source artwork.

A future-directed causal curve from the origin has $|dx/dt|\le c$. Integrating this inequality gives the causal futureThe straight timelike or null segment proves sufficiency. This is the blue region above the two null rays in the upper-left spacetime diagram of part c.

Reverse the endpoints of the future-directed causal curve. The origin can be reached precisely from its causal past,This is the orange region below the two null rays in part c. The origin itself belongs to both the causal future and causal past by the usual zero-length-curve convention.

Outside both light cones, the separation from the origin is spacelike. Thus the events in neither the causal past nor the causal future areNo causal curve can connect either endpoint to the other. These are the two green side regions in part c; the strict inequality excludes the causally reachable null boundaries.