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

Table of contents
- i d
  - Solution i
- ii d
  - Solution ii
- iii d
  - Solution iii

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.

ii

Solution

The same Lorentz transformation maps the past light cone to itself. In the primed axes the causal past is

J^{-} (0) = {(t^{'}, x^{'}) : c t^{'} \leq - ∣ x^{'} ∣},

the orange region in the upper-right spacetime diagram of part c. Future and past are not interchanged by the standard boost.

iii

Solution

The invariant spacelike condition is

x^{'2} - c^{2} t^{'2} > 0

. Hence the primed-coordinate events in neither the causal future nor the causal past obey

∣ x^{'} ∣ > c ∣ t^{'} ∣.

The green regions in part c retain their shape in primed coordinates. A particular spacelike event's coordinate time may change sign, but it never enters either light cone.

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