Special Relativity

In June 1905 Albert Einstein, a 26-year-old examiner at the Swiss patent office, took that step. He assumed only two things: the principle of relativity holds for all of physics, and light always moves at $c$. Instead of asking how the ether distorts clocks, he asked what "at the same time" means for distant events. It can only mean what clocks synchronised by light signals say, and observers in relative motion synchronise differently. Simultaneity is relative.

From there everything followed as kinematics, with no mechanism needed. Moving clocks run slow by the factor $\gamma = 1/\sqrt{1 - v^2/c^2}$, moving lengths contract by the same factor, and velocities combine so that nothing overtakes light. The ether became, in his word, "superfluous". A three-page sequel that September showed that a body's mass is a measure of its energy content, the result now written $E = mc^2$.

Hermann Minkowski saw the geometry inside the theory. Observers disagree about time intervals and distances separately, but all agree on one combination,

$$s^2 = (c\,\Delta t)^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 ,$$

just as rotated observers in ordinary space disagree about $x$ and $y$ but agree on distance. Special relativity is the geometry of a four-dimensional spacetime, and a change of velocity is a kind of rotation in it. The minus sign makes the geometry strange: the space of possible velocities turns out to be a hyperbolic space, the non-Euclidean geometry of Lobachevsky and Bolyai.

Einstein at first is said to have called this "superfluous learnedness". Within a few years he found he could not build a theory of gravity without it.

Cosmic rays striking the upper atmosphere produce muons, unstable particles that decay with a half-life, in their own rest frame, of about 1.5 microseconds, and an average lifetime of 2.2 microseconds. Suppose a muon is made 10 km up and travels down at 99.5% of the speed of light.

Without relativity. The trip takes

$$t = \frac{10{,}000 \text{ m}}{0.995 \times 3.00 \times 10^8 \text{ m/s}} \approx 33.5 \ \mu\text{s} ,$$

about fifteen average lifetimes. The fraction surviving would be $e^{-33.5/2.2} \approx 2 \times 10^{-7}$, roughly one in four million. Almost none should reach the ground.

With relativity. A clock moving at $v = 0.995c$ runs slow by the factor

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - 0.995^2}} \approx 10 .$$

The muon's own clock records only $33.5 / 10 \approx 3.35$ microseconds for the trip, about one and a half lifetimes, so the fraction surviving is $e^{-3.35/2.2} \approx 0.22$. About one muon in five arrives.

Muons do reach the ground in large numbers, about one per square centimetre per minute. Rossi and Hall measured it in 1941, and in 1963 David Frisch and James Smith compared the muon counts on the summit of Mount Washington and at sea level, 1,907 metres lower. Far more survived the descent than the muons' lifetime would allow without time dilation, in close agreement with Einstein's factor.

From the muon's point of view, its clock is normal. Instead the atmosphere, rushing past at 0.995c, is contracted by the same factor of 10, to about 1 km thick. Both descriptions give the same count of surviving muons. They are the same prediction, seen from two frames.

The predictions are strange but testable. In 1941 Bruno Rossi and David Hall showed that muons from cosmic rays survive the trip through the atmosphere only because their internal clocks run slow. Particle accelerators, atomic clocks flown on aircraft and the GPS constellation have confirmed special relativity many times over, to extraordinary precision.

What it could not include was gravity. Newton's gravity acts instantly across any distance, and in a theory where nothing outruns light, "instantly" has no meaning. Fixing that took Einstein another ten years and produced general relativity.