Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 12C (Dynamics and Relativity)

A four-vector is an object whose components transform between inertial frames by a Lorentz transformation,Lorentz transformations are defined bywhere $\eta$ is the Minkowski metric. ConsequentlyThus the Minkowski norm of a four-vector is the same in every inertial frame.

Along the particle's worldline,The boost from $S$ to $S^{\prime }$ givesThereforeand division of the transformed coordinate velocities by this factor gives the relativistic velocity-addition formula

To first order in $V/c$, $\gamma =1+O(V^2/c^2)$ andThusSince $\mathbf{V}=(V,0,0)$ and $\mathbf{V}\cdot \mathbf{u}=V{u}_x$, these components combine into

For a photon,Part (ii) gives the first-order changesFor a vector of fixed leading-order magnitude $c$, its small change of angle isSubstitution yields the relativistic aberration formulato leading order.

Let the incident and final photon momentum magnitudes beIn a nontrivial collinear collision, the photon is backscattered and the electron moves in the original photon direction, so momentum conservation gives electron momentum $P=p+p^{\prime }$. Conservation of relativistic energy givesSubstitute $P=p+p^{\prime }$, isolate the square root, and square. Cancellation leavesDividing by $p{p}^{\prime }$ and using the wavelength definitions gives the Compton scattering shiftThe other collinear possibility is the trivial forward solution ${\lambda }^{\prime }=\lambda$, in which the electron remains at rest.