Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 8B (Special Relativity)

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- Solution 8B

Solution

Take

S^{'}

to move along

+ x

V

, with the origins coincident at zero time. The Lorentz transformation is

x^{'} = γ_{V} (x - V t), t^{'} = γ_{V} (t - V x / c^{2}), γ_{V} = (1 - V^{2} / c^{2})^{- 1/2} .

Differentiating along the particle trajectory gives the relativistic velocity-addition formula

u^{'} = \frac{u - V}{1 - u V / c ^{2}} .

For a timelike particle,

∣ u ∣ < c

and

∣ V ∣ < c

. Direct algebra gives

1 - \frac{u ^{'2}}{c ^{2}} = \frac{( 1 - u ^{2} / c ^{2} ) ( 1 - V ^{2} / c ^{2} )}{( 1 - u V / c ^{2} ) ^{2}} > 0,

so $∣ u^{'} ∣ < c$ , in particular

u^{'} < c

. Strictness requires a massive, subluminal particle; for a photon with

u = \pm c

, the transformation preserves

u^{'} = \pm c

instead.

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