 Solution

ID: past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2015/4/b/i/solution

Let

A = (c t_{A}, f (t_{A}))

on the rear rocket and

B = (c t_{B}, h (t_{B}))

on the front rocket. The instantaneous inertial rest frame of the rear rocket at

A

moves at

v = f^{'} (t_{A})

relative to S. Transform the time difference of these two events:

Δ t^{'} = γ [(t_{B} - t_{A}) - \frac{v}{c ^{2}} (h (t_{B}) - f (t_{A}))] .

Setting

Δ t^{'} = 0

gives

c^{2} (t_{B} - t_{A}) = v [h (t_{B}) - f (t_{A})] .

The velocity belongs to the rear event

A

. It is not to be evaluated at laboratory time

t_{B}

, nor does the simultaneity condition mean

t_{A} = t_{B}

in S. At nonzero velocity the front event on the same instantaneous-rest slice is at a different laboratory time.

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