The consequences of special relativity

Time dilation

Einstein's two postulates cannot both hold while time stays universal. The first thing to go is the idea that all observers share one clock. The proper time t₀ between two events is the time measured in the inertial frame where both events happen at the same place, on a clock riding along with them. Any observer moving at speed v relative to that clock measures a longer time:

The root is smaller than one, so t is bigger than t₀. A moving clock runs slow, as judged from the frame it moves through. At everyday speeds the correction is invisibly tiny. Near c it is enormous.

The muons that should not arrive

Nature runs this experiment constantly. Cosmic rays striking the upper atmosphere create muons about 10 km up, travelling near c. One undilated lifetime at 0.996c covers 650 m, so the 10 km descent lasts fifteen lifetimes, and since decay is exponential, fewer than three muons in every ten million should survive it. Ground-level detectors measure far more than that. Around a quarter of them arrive.

Time dilation closes the gap exactly. At 0.996c the factor is 11.2, the laboratory-frame lifetime stretches to 25 μs, the descent lasts only 1.4 dilated lifetimes, and the predicted survival is 26 per cent, matching what detectors count. Muon showers are time dilation measured daily, with the sky itself for apparatus.

Length contraction

The same root reaches the metre rule. The proper length l₀ of an object is its length measured in its own rest frame; an observer it passes at speed v measures it shorter along the direction of motion:

Nothing is being crushed. Lengths, like durations, are relations between frames. The muon's own frame shows this. In the muon's rest frame its lifetime really is 2.2 μs, and the atmosphere is length-contracted from 10 km to under 900 m. At 0.996c that takes 3.0 μs, the same 1.4 lifetimes the ground observer counted. The two frames describe the descent differently and predict the same survival rate.

Mass, energy, and Bertozzi's electrons

Speed changes the mass as well. Push on an object near light speed and the work done produces ever less increase in speed, as though the inertia itself were growing. AQA writes that growth as a relativistic mass above the rest mass m₀, m = m₀/√(1 − v²/c²), the convention this option examines; modern treatments keep the mass itself invariant and let energy and momentum carry the growth, and both languages describe the same physics. Tying the AQA form to the equivalence of mass and energy gives the total energy of a moving object, and the booklet prints the whole chain as one line:

Subtract off the rest energy m₀c² and what remains is the kinetic energy, which the booklet does not print:

As v approaches c the denominator collapses towards zero, so the mass, and with it the kinetic energy, grows without limit. That is what stops any accelerator, however strong, from pushing an electron past c. It is also why the check question in Cathode rays and the electron broke. eV = ½mv² assumes all the work becomes kinetic energy of a fixed mass, when past a few tenths of c it increasingly becomes mass-energy instead. An electron's rest energy m₀c² is 8.2 × 10⁻¹⁴ J, about 0.51 MeV, and beyond that most of the energy an accelerator supplies goes into mass-energy rather than speed.

In 1964 William Bertozzi tested this about as directly as anyone could. Accelerate electrons through a known pd, so the kinetic energy is known from eV, then time their flight over 8.4 m to measure the speed outright. Classically v² should climb in proportion to the energy forever. His electrons instead saturated just below c, their measured speed flattening while the energy, delivered as heat to the target, kept rising exactly as supplied. Kinetic energy varies with speed the relativistic way, by direct measurement.