Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section B / B4 / Solution

Three experimental tests probe different consequences of special relativity. The Michelson-Morley experiment compares light round trips along perpendicular interferometer arms while changing the apparatus orientation. Its small observed orientation-dependent fringe shift contradicts the simple stationary-aether prediction and supports orientation-independent round-trip light speed. It does not, alone, exclude every alternative theory.^[ref]

In the muon storage-ring time-dilation experiment, fast circulating muons live longer in laboratory time than muons at rest. Measuring decay counts versus elapsed laboratory time and independently determining the beam Lorentz factor tests ${\tau }_{\mathrm{lab}}=\gamma {\tau }_0$. Circular motion includes acceleration, so this also tests the applicability of accumulated proper time along the stored trajectory.^[ref]

Bertozzi's electron speed-energy experiment measures electron time of flight and deposited kinetic energy independently. An electrostatic accelerator and a linear accelerator increase the energy while the measured speed approaches the light-speed limit. Calorimetry checks that the additional beam energy is real, rather than inferring it only from an assumed voltage-to-speed relation. The result supports relativistic $K=(\gamma -1)m{c}^2$ rather than extending $K=m{v}^2/2$ to arbitrarily large speeds.^[ref]

For the two ships, set launch time to zero and measure Earth time in years. Their Lorentz factors are ${\gamma }_A=5/4$ and ${\gamma }_B=5/3$. Bob's thirtieth birthday occurs after one year of his proper time, so its Earth coordinates areIf Alice sends at Earth time $t_A$, her position is $x_A=3c{t}_A/5$. The left-going light signal obeys $x=x_A-c(t-t_A)$. Intersection with Bob givesAlice's elapsed proper time is thereforeShe sends two months after her twenty-ninth birthday. This compares the emission and reception events; the captains' clocks need only be synchronized at launch, not kept numerically synchronized afterward.

The relative recession speed isThe longitudinal relativistic Doppler effect then givesEquivalently, the left-going photon has Earth frequency $f_0/[{\gamma }_A(1+3/5)]=f_0/2$, and Bob's frame supplies another factor ${\gamma }_B(1-4/5)=1/3$. Both methods give the same redshift.