Special Relativity: Time, Space and the Speed of Light

In the 1800s most physicists thought light was a wave in an invisible "aether" that filled space. If so, Earth's motion through the aether should change the speed of light we measure, just as a boat's speed changes when it heads upstream or downstream.

Between April and July 1887, Albert Michelson and Edward Morley built an instrument sensitive enough to detect that change. It split a beam of light in two, sent the halves along arms at right angles, then brought them back together to make a pattern of bright and dark fringes. Turning the instrument should have shifted the fringes. They found no real shift.

In 1905 Einstein took the result at face value and built a theory on two rules, called postulates:

1. The principle of relativity. The laws of physics are the same in every inertial frame. No experiment inside a smoothly moving, windowless train can tell you how fast it's going.
2. The speed of light is constant. Light in a vacuum travels at c for every inertial observer, however fast the source or the observer is moving.

The second rule sounds harmless, but it breaks everyday sense. If you run at a light beam, you still measure it passing you at exactly c. Something else has to give, and it turns out to be time and space themselves.

Picture a light clock: a pulse of light bouncing between two mirrors a distance L apart. Each round trip is one tick. On a spaceship, the crew see the light go straight up and down.

Now watch the ship fly past at speed v. The mirrors move sideways while the light travels, so you see the light take a longer, slanted path. Light can't speed up to make up for it (rule 2), so each tick takes longer by your clock. Using Pythagoras on the triangle in Figure 2 gives:

t = γt₀ where γ = 1 ÷ √(1 − v² ÷ c²)

Here t₀ is the proper time on the moving clock and t is the longer time you measure. It isn't just light clocks: by rule 1, every clock on the ship, including the crew's heartbeats, must slow by the same factor, or the crew could use the mismatch to tell they were moving.

The Lorentz factor γ (gamma) is extremely close to 1 at everyday speeds, so we never notice. At 60% of the speed of light it's 1.25, at 80% about 1.67, and at 99.5% about 10.0. As v gets close to c, γ grows without limit.

The effect is symmetrical. To the crew, it's your clock that runs slow. Both views are correct, because each observer is in an inertial frame and neither is "really" at rest.

Einstein explained this one with a train. A lamp flashes in the exact middle of a moving carriage. For a passenger, the light travels the same distance to the front and back walls at the same speed, so it reaches both walls at the same moment.

Someone on the platform also measures the light moving at c in both directions. But while it travels, the carriage moves forward: the back wall comes toward the light and the front wall runs away from it. For the platform observer, the back wall is lit first.

Neither observer is wrong. Two events in different places that are simultaneous in one frame are not simultaneous in another frame moving relative to it. This is called the relativity of simultaneity, and it's the key to understanding length contraction and the twin paradox.

To measure a moving spaceship, you must mark where its front and back are at the same time. Since observers disagree about "the same time," they also disagree about lengths. The result is:

L = L₀ ÷ γ

L₀ is the proper length, measured by someone at rest beside the ship. Anyone watching it fly past measures a shorter length L. A 100 m ship moving at 0.8c (γ ≈ 1.67) is measured as 60 m long.

Only the length along the direction of motion shrinks. The ship's height and width stay the same. And again it's symmetrical: to the crew, it's the platforms and planets streaming past that are squashed.

Muons. When cosmic rays hit the upper atmosphere they make muons, heavy cousins of the electron. A muon at rest lasts on average only about 2.2 μs before it decays. Even at nearly the speed of light, that's only enough time to travel about 660 m, yet huge numbers of muons reach sea level.

In a famous experiment published in 1963, David Frisch and James Smith counted about 563 muons per hour near the top of Mount Washington in New Hampshire, 1,917 m up. The muons took about 6.4 μs to fall to sea level at Cambridge, Massachusetts. With no time dilation, only about 27 per hour should have survived the trip. About 412 per hour arrived. The muons' clocks were running slow, just as relativity predicts.

Clocks on planes. In October 1971 Joseph Hafele and Richard Keating flew atomic clocks around the world on airliners, once eastward and once westward, and compared them with clocks at the U.S. Naval Observatory. The results matched relativity's predictions. Their predictions also include an effect of gravity from Einstein's later general relativity: clocks higher up run slightly fast.

Time gained by the flying clocks compared with the clocks on the ground, in nanoseconds (billionths of a second), from Hafele and Keating's 1972 papers. Flying east adds the plane's speed to Earth's spin, so those clocks moved fastest and lost time.

GPS. GPS satellites orbit at about 3.87 km/s. Their speed alone makes their clocks lose about 7.2 μs a day compared with clocks on the ground. Being higher up in Earth's gravity makes them gain about 45.8 μs a day. Together, they run about 38 μs a day fast, so the clocks are set to tick slightly slower before launch. Light travels about 11 km in 38 μs, which is why GPS would quickly become useless without relativity.

Twin A stays on Earth. Twin B flies at 0.8c to a star 4 light-years away and comes straight back. By Earth's clocks the trip takes 8 light-years ÷ 0.8c = 10 years. B's clock runs slow by γ ≈ 1.67, so B ages only 10 ÷ 1.67 ≈ 6 years. B comes home 4 years younger than A.

Here's the "paradox": from B's point of view, isn't it Earth that moves away and back, so A should be younger? No, because the twins aren't equal. A stays in one inertial frame the whole time. B has to turn around, switching from an outbound frame to an inbound one. During that switch, B's idea of "what's happening on Earth right now" jumps forward, thanks to the relativity of simultaneity. Work it through carefully from either twin's view and you get the same answer: the traveler is younger. Paul Langevin worked out this kind of round-trip ageing in 1911.

A ship moving at 0.8c fires a probe forward at 0.6c, measured by the crew. In everyday physics you'd add them and say the probe moves at 1.4c as seen from Earth. Relativity says instead:

w = (u + v) ÷ (1 + uv ÷ c²)

That gives (0.8 + 0.6) ÷ (1 + 0.48) ≈ 0.946c. However fast you combine speeds below c, the answer stays below c. Put in u = c and you get exactly c, as rule 2 demands. At everyday speeds uv ÷ c² is tiny, so ordinary addition works.

Energy tells the same story. The kinetic energy of a moving object is really

KE = (γ − 1)mc²

At low speed this matches ½mv², but as v approaches c, γ and the energy needed grow without limit. That's why no object with mass can reach the speed of light, however long you push it. Even when the Large Hadron Collider sped protons up to 6.5 TeV each, they reached only about 0.999 999 990c. The leftover term mc² is the famous rest energy, E = mc².

Relativity didn't arrive all at once. It grew out of a puzzle about light, and every step since has been checked by experiment.