Past exam of the physics 1 paper of the University of Oxford / 2017 / 9 / b

Let $S^{\prime }$ move at $+v$ along $x$ relative to $S$. The Lorentz transformation of energy and momentum, from primed to unprimed components, isThe inverse equations have $v$ replaced by $-v$ and the primes interchanged. A photon making angle ${\theta }^{\prime }$ with $+x$ in $S^{\prime }$ has $p_x^{\prime }=(E^{\prime }/c)\cos {\theta }^{\prime }$. Thus its general Doppler factor isThe requested blueshifted expression assumes propagation parallel to the boost, ${\theta }^{\prime }=0$:The propagation-direction assumption matters: for an oppositely travelling photon, $D=\gamma (1-\beta )$, the reciprocal of the displayed factor. Relative frame speed alone does not select the longitudinal relativistic Doppler effect.

For the forward ray of i, the ultrarelativistic longitudinal Doppler factor follows from the exact expression in terms of the Lorentz factor:Indeed, $\gamma \beta =\sqrt{{\gamma }^2-1}$ for a positive boost. If $\beta =1-ϵ$, the exact formulas and their consistent asymptotic expansions areEquivalently,The printed equality is not exact. For a concrete ultrarelativistic example, $\gamma =10$ gives the correct $D=10+\sqrt{99}\simeq 19.949874$, whereas $\sqrt{4{\gamma }^2-1}=\sqrt{399}\simeq 19.974984$. The latter expands as $2\gamma -1/(4\gamma )+O({\gamma }^{-3})$, with the wrong first correction.

One can obtain the printed expression by using $D^2=2/ϵ-1$ but replacing ${\gamma }^2$ by only its leading approximation $1/(2ϵ)$. This retains a constant term in the first equation while discarding a same-order term in the second. Thus $D\approx \sqrt{4{\gamma }^2-1}$ is admissible only as a leading relative-accuracy estimate, of the same order as $D\sim 2\gamma$; it is not the stated identity or a correct next-order approximation.