 Solution

ID: past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2019/1/d/i/solution

For the standard passive Lorentz boost, write

β = v / c

and

c t^{'} = γ (c t - β x), x^{'} = γ (x - β c t) .

A proper orthochronous Lorentz transformation preserves the sign of the time component of a nonzero future causal vector and preserves its Minkowski norm. Therefore

J^{+} (0) = {(t^{'}, x^{'}) : c t^{'} \geq ∣ x^{'} ∣} .

The upper-right panel of part c uses

x^{'}

horizontally and

c t^{'}

vertically. Its null rays still have slopes

\pm 1

. The grey dashed old axes satisfy

x^{'} = - β c t^{'}

for the old time axis and

c t^{'} = - β x^{'}

for the old spatial axis; changing the coordinate grid has not changed which events are reachable.

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