 Solution

ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2019/10/a/solution

For a boost along

+ x

v

E^{'} = γ (E - v p_{x}), p_{x}^{'} = γ (p_{x} - v E / c^{2}), p_{y}^{'} = p_{y}, p_{z}^{'} = p_{z} .

The total four-vector is the sum over the particles. Write

E = \sum_{i} E_{i}

and

P = \sum_{i} p_{i}

. A centre-of-momentum rest frame exists when this total is timelike, with

V_{CM} = c^{2} P / E

. Boost along

P

and use

P = E V_{CM} / c^{2}

E_{CM} = γ_{CM} (E - V_{CM} P) = E / γ_{CM} .

Therefore

γ_{CM} = \frac{\sum _{i} E _{i}}{E _{CM}} = \frac{1}{1 - V _{CM}^{2} / c ^{2}} .

The timelike qualification is needed: exactly parallel photons alone have null total four-momentum and no such rest frame.

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