Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 4 / 4B / Solution

Write the standard Lorentz transformation as $x^{\prime }=\gamma (x-vt)$, $c{t}^{\prime }=\gamma (ct-vx/c)$, with Lorentz factor $\gamma =(1-v^2/c^2{)}^{-1/2}$ and $|v|<c$. In the null coordinates in two-dimensional Minkowski spacetime, addition and subtraction giveSince both factors are positive,Their product is one, so Lorentz invariance of the interval follows immediately:Two successive collinear Lorentz boosts multiply the positive-coordinate factor by $\lambda (v_2)\lambda (v_1)$ and the negative-coordinate factor by its reciprocal. Thus they have the same form with a velocity $v_3$ determined byCross multiplication yields the relativistic velocity-addition formula formulaFor $|v_1|,|v_2|<c$, the denominator is positive andso $v_3$ is admissible. The positive square roots then verify the factor identity itself, not merely its square. In terms of rapidity $v/c=\tanh \eta$, this same calculation is addition of rapidities because $\lambda (v)=e^{-\eta }$.