Past exam of the short options course of the University of Oxford / 2021 / 8 / b

In the common $ct,x_1$ plane, multiplication of the two Lorentz boost matrices givesThis 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 $O$ isThe signed rapidities add because both boosts use the same spatial axis and the stated matrix convention.

By definition $v/c=\tanh \theta$ and $v^{\prime }/c=\tanh {\theta }^{\prime }$. Using the addition identity for the hyperbolic tangent and the rapidity sum from (i),Thus the velocity-addition formula isThe denominator is positive for the permitted subluminal signed velocities. For the nonnegative speeds in (iii), the composite is also nonnegative and remains below $c$.

When both speeds are much smaller than $c$, expand the denominator of the velocity-addition formula:This is the Galilean transformation limit. For $0\le \beta =v/c<1$ and $0\le {\beta }^{\prime }=v^{\prime }/c<1$,If both $v,v^{\prime }\to c$ from below, the right side tends to zero, soNo sum of these subluminal speeds exceeds the speed of light.