Past exam of the physics 1 paper of the University of Oxford / 2022 / 7

Align the spatial axes and make the origins coincide at $t=t^{\prime }=0$. For an inertial frame moving at velocity $+v$ along $x$, the Lorentz transformation isThe inverse Lorentz transformation follows by interchanging primed and unprimed coordinates and replacing $v$ by $-v$.

Let the object be at rest in $S^{\prime }$, with proper length $L_0=x_2^{\prime }-x_1^{\prime }$. Measuring its length in $S$ requires two endpoint events simultaneous in $S$, so $\Delta t=0$. The Lorentz transformation then givesThis is length contraction. The simultaneous endpoint events in $S$ are not simultaneous in $S^{\prime }$; that relativity of simultaneity is essential to the measurement.

For the two rockets, take the first rocket's Earth-frame velocity as $+c/2$ and the second's as $-c/2$. The velocity-addition formula gives the second rocket's velocity in the first rocket's rest frame:Its Lorentz factor is ${\gamma }_{\mathrm{rel}}=5/3$, and therefore its measured length isHere “appear” is interpreted as a length measured using simultaneous endpoint positions in the observer's inertial frame. A photograph, involving unequal light travel times, is a different measurement.

Write the incident and scattered photon energies as $\epsilon =hc/\lambda$ and ${\epsilon }^{\prime }=hc/{\lambda }^{\prime }$. Conservation of relativistic energy gives the recoiling electron energy $E_e=m{c}^2+\epsilon -{\epsilon }^{\prime }$. By momentum conservation, its momentum is the difference of the two photon momenta; henceInsert these into the relativistic energy-momentum relation for the electron:Expanding and cancelling terms givesSince $1/\epsilon =\lambda /(hc)$, the Compton scattering formula follows:The scale $h/(mc)$ is the electron Compton wavelength. At $\theta =0$, ${\lambda }^{\prime }=\lambda$: there is no energy or momentum transfer in this ideal forward limit, so the electron remains at rest. At $\theta =\pi$, the wavelength shift is maximal:The electron gains its largest kinetic energy, $\epsilon -{\epsilon }_{\min }^{\prime }$, and recoils forward while the photon reverses direction.