Special Theory of Relativity

Einstein built the whole of special relativity on two statements, both of which look harmless.

First, the laws of physics are the same in all inertial frames — in any laboratory moving at constant velocity. There is no experiment you can do inside a smoothly moving train that tells you it is moving. This idea was already old; Galileo knew it.

Second, the speed of light in a vacuum is the same for every observer, whatever their own motion or the motion of the source. This is the one that breaks everything.

Ordinarily speeds add. Throw a ball forward at 20 m/s from a train doing 30 m/s and the ground sees 50 m/s. The second postulate says light does not behave that way: shine a torch forward from a spaceship travelling at half the speed of light and both you and the observer outside measure the beam at exactly c.

Since speed is distance over time, and everybody must agree on the speed, they cannot all agree about distance and time. Something has to give — and what gives is the assumption that time and length are absolute.

A clock moving relative to you runs slow. Not because of anything mechanical — a moving clock of any kind, including a biological one, ticks more slowly as measured by you.

The factor is called gamma, and it depends only on speed. At everyday speeds gamma is so close to 1 that no effect is measurable. At 10% of light speed it is 1.005 — half a percent. At 99% of light speed it is 7.1.

The time measured in the frame where the two events happen at the same place is called the proper time. Every other observer measures a longer interval.

This is not a philosophical curiosity; it is measured routinely. Muons created high in the atmosphere have a half-life so short they should decay long before reaching the ground, yet they arrive in large numbers — because at 99.5% of light speed their internal clocks run about ten times slow. GPS satellites must correct for it too, and would drift by kilometres a day if they did not.

γ

Lorentz factor

v

relative speedm s⁻¹

c

3.0 × 10⁸m s⁻¹

t₀

proper times

Nothing happens until you are travelling very fast indeed. The curve is almost flat up to about half light speed, then climbs without limit as v approaches c — which is why relativity was never noticed in ordinary experience, and why nothing with mass can reach c.

The same logic applied to distance gives length contraction: an object moving relative to you is measured shorter along its direction of motion, by the same factor gamma. Its dimensions across the motion are unaffected.

The two effects are two views of one situation. From the muon's own frame it does not live longer — the atmosphere is contracted, so it has less distance to travel. Both descriptions give the same answer, which is the point.

Mass also changes. As an object is accelerated its relativistic mass increases, and it takes ever more energy to gain each additional metre per second. As v approaches c the mass tends to infinity, so an infinite amount of energy would be needed to reach light speed. That is why nothing with mass can travel at c, and why particle accelerators need so much energy for so little extra speed.

Note carefully what each observer sees. The traveller notices nothing odd — their own clock, ruler and mass are perfectly normal. It is only measurements made of a system moving relative to you that change.

L₀

proper lengthm

m₀

rest masskg

γ

Lorentz factor

The most famous consequence is that mass and energy are the same thing in different forms, related by E = mc². Because c² is about 9 × 10¹⁶, a very small mass corresponds to an enormous energy — one gram is worth roughly 9 × 10¹³ joules, comparable to a large power station running for a day.

This is not an exotic effect confined to bombs. Every energy change is a mass change. A stretched spring is very slightly more massive than a relaxed one; a hot cup of tea is more massive than a cold one. The differences are far too small to measure in those cases.

Where it does show up is in nuclear physics. The mass of a nucleus is always less than the total mass of its separate nucleons, and the difference — the mass defect — is the binding energy that holds it together, released when the nucleus formed.

Fusion in the Sun converts about four million tonnes of mass into energy every second, and that is what makes it shine. Fission in a reactor does the same on a smaller scale. In both cases nothing is destroyed: mass has simply been converted into an equivalent quantity of energy.

E

energyJ

m

masskg

c

3.0 × 10⁸m s⁻¹

Δm

mass defectkg