Special relativity I: postulates, time dilation and length contraction

In 1905 Albert Einstein, a 26-year-old technical examiner at the Swiss patent office in Bern, published a paper with the modest title On the Electrodynamics of Moving Bodies. It began not with the Michelson–Morley experiment but with a puzzle about magnets and coils. Move a magnet past a stationary coil and a current flows; move the coil past a stationary magnet and the same current flows. Yet the textbook explanations were completely different in the two cases, an electric field in one and a magnetic force in the other. Einstein took the sameness of the result as a clue that only relative motion matters, for electromagnetism as well as for mechanics. From that idea and one more postulate he derived the whole of special relativity.

This lesson states the postulates, uses a simple light clock to derive time dilation, deduces length contraction, and checks both against experiments with muons, atomic clocks and particle accelerators. Lesson 3 then builds the full Lorentz transformation and the geometry of spacetime.

Frames of reference

An inertial frame is a frame of reference in which a free particle, one with no forces on it, moves in a straight line at constant speed. A train moving smoothly along a straight track is, to a very good approximation, an inertial frame; so is the platform. Galileo pointed out in 1632 that no experiment inside a smoothly moving ship's cabin can tell whether the ship is moving: butterflies fly the same way, water drips straight down into a bowl. This is the principle of relativity for mechanics. In Galilean relativity, if the train moves at speed \(v\) along \(x\), coordinates in the two frames are related by \(x' = x - vt\) and \(t' = t\), and velocities simply add: a ball thrown forward at \(u'\) on the train moves at \(u = u' + v\) relative to the platform.

Apply that rule to light and you get a contradiction with Maxwell's equations, which give the same speed \(c\) without any reference to the motion of the source or observer. Either Maxwell's equations hold only in one special frame, the ether frame, or the Galilean addition of velocities is wrong. Michelson and Morley found no special frame. Einstein chose to keep the principle of relativity and give up Galilean addition.

The two postulates

Einstein's postulates are short.

• The principle of relativity: the laws of physics take the same form in every inertial frame. No experiment, mechanical, electrical or optical, can single out a frame at absolute rest.
• The constancy of the speed of light: light in vacuum travels at the same speed \(c\) in every inertial frame, independent of the motion of its source.

The second postulate is the strange one. It says that if you chase a light beam at half the speed of light, you still measure it to move away from you at \(c\). Since speed is distance divided by time, something about our measurements of distance and time must differ between frames. Einstein's insight was that the notion of time itself, and in particular of two events being simultaneous, had never been examined carefully. Clocks in different places must be synchronized by some signal, and if the signal is light, the constancy of \(c\) fixes what synchronization means.

Since 1983 the second postulate has also been built into the units. The meter is defined as the distance light travels in vacuum in \(1/299\,792\,458\) of a second, so the speed of light is exactly \(c = 299\,792\,458\ \mathrm{m/s}\) by definition. Measuring the speed of light is now a way of calibrating a ruler.

The light clock

The simplest clock to analyze is a light clock: two parallel mirrors a distance \(L\) apart, with a pulse of light bouncing between them. Each round trip is one tick. In the clock's own rest frame a tick takes

Now put the clock on a train moving at speed \(v\) parallel to the mirrors, and watch it from the platform. The light still goes up and down between the mirrors, but the mirrors move while it does, so in the platform frame the light follows a zigzag path. Each leg is the hypotenuse of a right triangle with height \(L\) and base \(v\,\Delta t/2\), where \(\Delta t\) is the tick time measured on the platform. The second postulate says the light still moves at \(c\) along this longer path, so

Solving for \(\Delta t\) gives

The moving clock ticks slowly, by the Lorentz factor \(\gamma\), which is always at least 1. The interval \(\Delta t_0\) measured by a clock that is present at both events, here the departure and return of the light pulse, is called the proper time. Any other inertial observer, for whom the two events happen at different places, measures a longer time. This is time dilation.

Nothing in the argument depends on the clock being made of mirrors. If a light clock and a wristwatch sat side by side on the train and ran at different rates, a passenger could detect the train's motion by comparing them, violating the first postulate. So every process, a heartbeat, a quartz crystal, the decay of a particle, slows by the same factor. Time dilation is a property of time, not of clocks.

How big is gamma?

For everyday speeds \(\gamma\) is indistinguishable from 1. Expanding for small \(\beta\) gives \(\gamma \approx 1 + \tfrac12 \beta^2\), so a jet airliner at 250 m/s has \(\gamma - 1 \approx 3.5 \times 10^{-13}\). The program below tabulates \(\gamma\) from walking speed to the speed of the protons in the Large Hadron Collider, and shows why the expansion is useful: computing \(1 - \sqrt{1 - \beta^2}\) directly for tiny \(\beta\) loses precision to rounding.

import math
c = 299792458.0
cases = [("walking person", 1.4), ("highway car", 30.0), ("jet airliner", 250.0),
         ("GPS satellite", 3874.0), ("Earth around the Sun", 29780.0),
         ("electron in a TV-style tube", 0.3 * c), ("muon from cosmic rays", 0.998 * c),
         ("LHC proton at 6.8 TeV", None)]
mp_GeV = 0.938272
for name, v in cases:
    if v is None:
        g = 6800 / mp_GeV                     # gamma = E / (m c^2)
        one_minus_beta = 1 / (2 * g * g)      # 1 - beta for gamma >> 1
        print(f"{name:28s} gamma = {g:12.1f}   1 - beta = {one_minus_beta:.2e}")
        continue
    b = v / c
    g_direct = 1 / math.sqrt(1 - b * b)
    g_minus_1 = b * b / 2 + 3 * b**4 / 8      # series, accurate for small beta
    if b < 0.01:
        print(f"{name:28s} beta = {b:.3e}   gamma - 1 = {g_minus_1:.3e}")
    else:
        print(f"{name:28s} beta = {b:.4f}      gamma = {g_direct:.4f}")

walking person               beta = 4.670e-09   gamma - 1 = 1.090e-17
highway car                  beta = 1.001e-07   gamma - 1 = 5.007e-15
jet airliner                 beta = 8.339e-07   gamma - 1 = 3.477e-13
GPS satellite                beta = 1.292e-05   gamma - 1 = 8.349e-11
Earth around the Sun         beta = 9.934e-05   gamma - 1 = 4.934e-09
electron in a TV-style tube  beta = 0.3000      gamma = 1.0483
muon from cosmic rays        beta = 0.9980      gamma = 15.8193
LHC proton at 6.8 TeV        gamma =       7247.4   1 - beta = 9.52e-09

Relativity matters when \(\beta\) is a sizeable fraction of 1. At 30% of light speed time runs about 5% slow; at 99.8% it runs about sixteen times slow. A proton in the Large Hadron Collider has \(\gamma\) above 7,000: one second of its time corresponds to about two hours in the laboratory.

Muons: time dilation in the sky

Cosmic-ray protons strike nuclei in the upper atmosphere and produce showers of particles, among them muons, which are heavy cousins of the electron. Muons are unstable, decaying with a mean lifetime of \(\tau = 2.197\ \mu\mathrm{s}\) in their own rest frame. Most are produced 10 to 15 km up. Even at nearly the speed of light, a muon travels only \(c\tau \approx 660\) m in one lifetime. Without time dilation, only a tiny fraction would survive to sea level, yet a detector the size of a dinner plate at sea level counts about one muon per second.

Bruno Rossi and David Hall turned this into a quantitative test in 1941, comparing muon counts on Mount Evans in Colorado and in Denver. A cleaner version by David Frisch and James Smith in 1963 compared the count on Mount Washington in New Hampshire, 1,910 m above sea level, with the count at sea level in Cambridge, Massachusetts, selecting muons with a narrow band of speeds near \(0.995c\). The program below reproduces the logic.

import math
c = 299792458.0
tau = 2.197e-6         # muon mean lifetime at rest, s
h = 1907.0             # height difference, m (Frisch and Smith, 1963)
beta = 0.9952          # speed of the selected muons
gamma = 1 / math.sqrt(1 - beta**2)
t_lab = h / (beta * c)                    # flight time in the Earth frame
survive_newton = math.exp(-t_lab / tau)   # if clocks were not affected
survive_rel = math.exp(-t_lab / (gamma * tau))
N_top = 563            # muons per hour counted at the summit
print(f"gamma = {gamma:.2f}, flight time in Earth frame = {t_lab * 1e6:.2f} microseconds")
print(f"muon's own elapsed time = {t_lab / gamma * 1e6:.3f} microseconds")
print(f"expected at sea level without dilation: {N_top * survive_newton:.0f} per hour")
print(f"expected at sea level with dilation:    {N_top * survive_rel:.0f} per hour")
print("measured by Frisch and Smith:           about 408 per hour")

gamma = 10.22, flight time in Earth frame = 6.39 microseconds
muon's own elapsed time = 0.626 microseconds
expected at sea level without dilation: 31 per hour
expected at sea level with dilation:    424 per hour
measured by Frisch and Smith:           about 408 per hour

Without time dilation only about 30 of the 563 muons per hour should reach sea level. Frisch and Smith counted about 408, in good agreement with the relativistic prediction of a little over 400. The experiment was later filmed as a teaching film, and the same logic is checked every day in particle detectors. A sharper test came at the muon storage ring at CERN in 1977, where muons circulating at \(\gamma = 29.3\) lived 29.3 times longer than at rest, confirming the formula to about one part in a thousand.

Length contraction

Look at the same muon from its own frame. In that frame the muon is at rest and lives 2.2 microseconds on average, so how does it get to sea level? The mountain moves toward it. If the trip takes only \(\Delta t_0 = \Delta t/\gamma\) of the muon's time, the distance between summit and sea level, as measured in the muon's frame, must be

where \(L_0 = v\,\Delta t\) is the height measured on the Earth. Lengths along the direction of motion are contracted by the factor \(\gamma\). The length \(L_0\) measured in the frame where the object is at rest is the proper length, and it is the longest length any observer measures. Lengths perpendicular to the motion are unchanged. To see why, imagine two trains passing on parallel tracks, each carrying a horizontal paint brush at the same height above its own rails. If perpendicular lengths shrank, each brush would pass below the other's, and both crews would see their own mark on top: a contradiction, since the marks are physical facts. The light clock argument relied on this, since we took the mirror separation \(L\) to be the same in both frames.

Time dilation and length contraction are two sides of the same coin. In the Earth frame the muon lives longer and travels the full 1.9 km. In the muon frame it lives its usual life and the atmosphere is squashed to about a tenth of its height. Both frames agree on the one thing that can be checked directly, whether the muon reaches the detector before it decays.

Clocks on jets and satellites

In 1971 Joseph Hafele and Richard Keating flew cesium-beam atomic clocks around the world on commercial jets, once eastward and once westward, and compared them with clocks at the U.S. Naval Observatory. The flying clocks differed from the ground clocks by tens of nanoseconds, in agreement with predictions that combined the time dilation of this lesson with the gravitational effect of general relativity, under which clocks higher in a gravitational field run faster. The eastward clocks lost about 59 ns and the westward clocks gained about 273 ns. In 2010 researchers at NIST compared two aluminum-ion optical clocks and detected time dilation at a relative speed of only 10 m/s, a speed you could reach on a bicycle.

The Global Positioning System must correct for both effects every day. Its satellites orbit at about 3.9 km/s, so their clocks lose about 7 microseconds per day from motion, and gain about 45 microseconds per day from being higher in the Earth's gravitational field. Lesson 12 computes the net correction and shows why ignoring it would put your position off by kilometers within a day.