Special Relativity: Everyone Gets the Same Speed of Light

The satellites that tell your phone where it is carry atomic clocks, and those clocks run at the wrong rate. Seen from the ground, they tick about 38 microseconds per day too fast: roughly 7 microseconds slower because they are moving, and roughly 46 faster because they sit higher in Earth's gravity. Left uncorrected, your position would drift by around ten kilometres a day. The engineers fix it by building the correction into the clocks' design and software.

Here is a quieter proof. High in the atmosphere, cosmic rays crash into air molecules and make muons, particles that fall apart in about 2.2 millionths of a second on average. Even at nearly the speed of light, that should let a muon travel only about 660 metres. Yet muons made 15 kilometres up reach sea level in huge numbers. Their short lives are stretched because they are moving.

Special relativity is the theory behind both. It starts from one fact that sounds harmless, and follows it with mathematics that never once breaks. A physics graduate comes out of this course able to feel the consequences in their bones: time, length and simultaneity are not what they seemed, and the thing everybody does agree on is a different quantity altogether.

Two rules and a flashlight on a train

Special relativity rests on two postulates. First: the laws of physics are the same for everyone moving at a steady speed. (Drop a ball on a smooth-flying aeroplane and it falls just as it does on the ground; nothing inside tells you that you are moving.) Galileo understood that in 1632. Second, the new one: light travels at the same speed, c, for every observer, no matter how the source or the observer moves. The speed is 299 792 458 m/s, exactly, because since 1983 the metre has been defined from it.

The second rule collides with common sense. If you throw a ball forward at 20 m/s from a train going 30 m/s, the ground sees 50. But flash a torch forward on a train going nearly at c, and the ground still sees light at c, not 2c. The old rule of adding speeds must be only a good approximation. The exact rule, from the new postulates, is:

At everyday speeds uv/c² is vanishingly small and you get u + v back. At high speeds the denominator holds the total below c. Two spaceships each going 0.9 c towards the ground, one launching from the other, do not reach 1.8 c; the result is 0.9945 c. And if u is c, then w = c, whatever v is.

The postulate was not pulled from nowhere. In 1887 in Cleveland, Albert Michelson and Edward Morley compared light travelling along two perpendicular arms of an interferometer, hoping to detect Earth's motion through the supposed ether that carried light. They found essentially nothing. Whether Einstein had that result in mind in 1905 is disputed by historians; he gave different accounts later. What is not disputed is that Hendrik Lorentz and Henri Poincaré had much of the mathematics before him, and that Einstein's contribution was to see it as a fact about space and time themselves, with no ether needed.

Moving clocks run slow

Take the simplest clock imaginable: a pulse of light bouncing between two mirrors, one tick per round trip. Now watch it fly past you sideways. In its own frame the light goes straight up and down. In your frame, because the clock moves while the light is in flight, the light zig-zags along a longer, diagonal path. But it must still go at c. A longer path at the same speed takes more time, so each tick lasts longer for you than for the people riding with the clock. Moving clocks run slow.

The size of the effect follows from Pythagoras applied to that triangle. It is captured by a single factor, which appears in nearly every formula in the subject:

Some values: at 0.1 c, γ = 1.005; at 0.5 c, 1.155; at 0.9 c, 2.29; at 0.99 c, 7.09; at 0.999 c, 22.4. It is close to 1 for anything in daily life, which is why we never see it. Aeroplanes cross oceans with their clocks off by nanoseconds.

And the muons: take a typical cosmic muon at 0.995 c, so γ = 10.0, created 15 km up (an illustrative height). It needs 50 microseconds in our frame to reach the ground, which is 22.9 of its average lifetimes of 2.197 µs. The chance of surviving that without time dilation is e^−22.9, about one in nine billion. With dilation, only 2.29 lifetimes pass in the muon's own time, and about 10% survive. The 1941 experiment by Bruno Rossi and David Hall on Mount Washington counted the extra muons, and it is the famous one, though the authors presented it in qualitative terms; Frisch and Smith's 1963 measurement is usually credited with the clean confirmation.

It is not a trick of the muon. Atomic clocks flown around the world on airliners (Hafele and Keating, 1971) and the clocks in every GPS satellite show the same thing. Everything about the moving thing runs slow, including its biology, if it had any.

Simultaneity depends on who is looking

The next consequence is stranger. Two events that happen at the same moment for you may happen one after the other for someone moving relative to you. Picture lightning striking both ends of a moving train at once, as judged from the platform. A passenger in the middle, moving towards one flash and away from the other, sees the light from the front flash first: it has less distance to cover and she is running into it. Since light has the same speed for her, she concludes that the front strike happened first. Neither is wrong.

The arithmetic is the Lorentz transformation. For two events separated by a distance Δx and a time Δt, an observer moving at v finds a time separation Δt′ = γ(Δt − vΔx/c²) and a distance Δx′ = γ(Δx − vΔt). Worked: two flashes 3 light-seconds apart, simultaneous for you; for an observer moving at 0.6 c (γ = 1.25), they are 2.25 seconds apart, with the order reversed if she runs the other way.

The same transformation shrinks lengths. A rod measured while it moves along its length is shorter than when at rest, by the same factor γ: a 100 m spaceship at 0.8 c measures 60 m for you, and at 0.99 c only 14 m. It is not that the ship is squashed on its own terms; it is that "its length" requires marking both ends at the same instant, and simultaneity is what moved.

You cannot rescue common sense by clever tricks, but there is a limit on how bad things get. Events that could be linked by a signal at c or slower, such as one causing the other, keep their order for everyone. Events too far apart for light to connect (a "spacelike" separation) can swap order for some observers, but they can never influence each other. The speed of light is not an engineering limit; it is the thing that keeps cause ahead of effect.

Proper time: the clock you carry

If time and distance both change with viewpoint, what is real? There is a combination that all observers agree on. For two events, the squared interval is c²Δt² − Δx² (the time part minus the space part). It is the same number whichever frame you compute it in, which the diagram above shows in its readout. For a clock moving along a path, the sum of intervals along its route is the time it actually records: its proper time.

This is the cure for the famous twin "paradox". One twin stays; the other rockets to a star and returns. Each sees the other's clock run slow, so who is younger? The answer is the traveller, because only the traveller accelerated: she turned around, changing frames, so the situation is not symmetric. Compare paths, and the one that stays in a single frame covers the longest proper time. The numbers can be dramatic: to a star 4.25 light-years away at 0.99 c, one leg takes 4.29 years on Earth's clocks and 0.61 years on the ship's. Go fast enough (γ = 1000) and you reach the centre of the galaxy in 26 years of ship time, while about 26,000 years pass at home.

Do not read this as a way to cheat death for free. The ship's journey needs energy, which is the next section's subject.

Energy, momentum and E = mc²

Push on something and it speeds up, but never past c. Where does the extra effort go? Into energy, and the theory's accounting says energy has weight. Momentum is p = γmv and the total energy of a moving object is E = γmc². Stand still (γ = 1) and an object still has E = mc²: rest energy. The kinetic energy is the rest of it, (γ − 1)mc². The tidy relation tying them is:

For small speeds it reduces to Newton's ½mv² plus the constant mc². For light itself, which has no mass, E = pc. Rest energy is huge: 1 kg corresponds to c² = 9.0 × 10¹⁶ joules, about 21 megatons of TNT. That is why a nuclear reaction that converts even a tenth of a percent of mass releases so much energy. The Sun converts about 4.26 million tonnes of mass into light every second (its luminosity 3.828 × 10²⁶ W, divided by c²).

The slider below pushes an electron or proton with a given amount of kinetic energy. Newton says that a 1 MeV electron should travel at nearly twice the speed of light; in fact it reaches 0.941 c. To get to 0.999 c needs about 10 MeV; 1 GeV gives 0.99999987 c. In the Large Hadron Collider, each proton carries 6.8 TeV (since 2022) and has γ = 7247. It travels only about 3 metres per second slower than light. Almost all the extra energy goes into γ, not into speed.

What a graduate just sees

• Light has the same speed for every observer, so speeds don't add the old way: w = (u+v)/(1+uv/c²).
• A moving clock ticks γ times slower; γ is 1.005 at 0.1 c, 2.3 at 0.9 c, and 22 at 0.999 c.
• Two events that are simultaneous for you needn't be for someone moving; the order of unrelated events is a matter of viewpoint.
• Cause always precedes effect for everyone, because nothing carries a signal faster than light.
• What everybody agrees on is the interval c²Δt² − Δx², and a clock's own (proper) time.
• The twin who turns around is the younger one: acceleration breaks the symmetry.
• E = mc² says rest mass is a store of energy; as speed climbs, the energy goes into γ, not into going faster.
• At everyday speeds all of this vanishes into Newton, which is why Newton was right for 200 years.

Where it shows up next

• General Relativity (34/85): drop the restriction to steady motion and gravity becomes the curvature of the spacetime you have just met; the muon and GPS stories both carry a gravity half.
• Particle & Nuclear Physics (34/86): collisions are computed with four-vectors, mass is the invariant, and every particle-decay lifetime in the tables is a proper time.
• Electromagnetics & Waves (34/38): magnetism is what electricity looks like from a moving frame; Maxwell's equations were already relativistic before relativity.
• Quantum Mechanics (34/83) and the Standard Model (13/07): Dirac combined the two in 1928 and predicted antimatter.
• The real world: GPS and other satellite navigation, PET scanners (matter-antimatter annihilation), synchrotrons, and electron microscopes, whose beams need relativistic corrections at 100 kV and more.

If you want to go deeper

• 📘 Textbook: Edwin Taylor and John Archibald Wheeler, Spacetime Physics (2nd edition, W. H. Freeman, 1992): built around the invariant interval, and free to download from the authors' site. David Morin's Special Relativity for the Enthusiastic Beginner is a friendly, problem-rich alternative.
• 📘 Gentler: The Feynman Lectures on Physics, Vol. I, chapters 15–17 (relativity, space-time, and relativistic dynamics), free online. Or this library's Why Time Is Weirder Than You Think (13/03).
• ▶️ Free course: MIT OpenCourseWare 8.033 Relativity (Fall 2006).
• 🌐 Easier first reads in this library: Why Time Is Weirder Than You Think (13/03) and The Big Bang and Our Expanding Universe (13/08); in this series Electromagnetics & Waves (34/38).