Chapter 36, Problem 63P

(III) (a) In reference frame S, a particle has momentum $\stackrel{\to }{p}=p_x\hat{i}$ along the positive x axis. Show that in frame S′, which moves with speed v as in Fig. 36–11, the momentum has components

$\begin{array}{c}{p^{\prime }}_x=\frac{p_x-vE/c^2}{\sqrt{1-v^2/c^2}}\\ {p^{\prime }}_y=p_y\\ {p^{\prime }}_z=p_z\\ E^{\prime }=\frac{E-p_{x}v}{\sqrt{1-v^2/c^2}}.\end{array}$

(These transformation equations hold, actually, for any direction of $\stackrel{\to }{p}$, as long as the motion of S′ is along the x axis.) (b) Show that p_x, p_y, p_z, E/c transform according to the Lorentz transformation in the same way as x, y, z, ct.