 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/8/b/i/solution

In the common

c t, x_{1}

(cosh θ sinh θ sinh θ cosh θ) (cosh θ^{'} sinh θ^{'} sinh θ^{'} cosh θ^{'}) = (cosh (θ + θ^{'}) sinh (θ + θ^{'}) sinh (θ + θ^{'}) cosh (θ + θ^{'})) .

This follows from the addition identities for hyperbolic sine and hyperbolic cosine. The transverse coordinates are unchanged by both transformations. Consequently the rapidity of

\overline{O}

relative to

O

θ^{''} = θ + θ^{'} .

The signed rapidities add because both boosts use the same spatial axis and the stated matrix convention.

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