What is the theory of relativity? Space, time and E=mc²

The theory of relativity is Albert Einstein's description of how space, time and gravity work, and it comes in two parts. Special relativity, from 1905, says that the speed of light is the same for everyone, so time runs slower and lengths shrink for things moving fast, and that mass is a form of energy, E = mc². General relativity, from 1915, says gravity is not a pull but the curving of space and time by mass.

In short

• Special relativity: the laws of physics and the speed of light are the same for every steady observer, so time and distance have to give instead.
• General relativity: mass bends spacetime, and objects, light included, follow the bends. That is what gravity is.
• Both are tested daily: GPS satellites would drift by about 10 km a day if their clocks weren't corrected for them.

What problem was Einstein solving?

By the late 1800s physics had two great theories that didn't fit together. Newton's laws said that speeds simply add: throw a ball forward at 20 km/h from a train moving at 100 km/h, and someone on the platform sees it move at 120 km/h. But James Clerk Maxwell's equations of electricity and magnetism gave light one fixed speed, about 300,000 km per second, and experiments could find no sign of it changing with Earth's motion.

In 1905, in a paper called "On the Electrodynamics of Moving Bodies", Einstein took both at face value. He built special relativity on two postulates: the laws of physics are the same for every observer moving at a steady speed, and the speed of light in a vacuum is the same for all of them, however fast they move. If the speed of light can't change, then something else must: time and distance.

What does special relativity say?

Einstein's train thought experiment shows the first surprise. Lightning strikes both ends of a moving train at once, as seen by a woman on the platform standing halfway between the strikes. A man at the middle of the train is moving toward the front flash and away from the back one, so the front flash reaches him first. Since light moves at the same speed for him as for her, he concludes the front strike happened first. Two events that are simultaneous for one observer are not simultaneous for another. Neither is wrong.

From that follow two effects. Time dilation: a moving clock ticks slower than one at rest, as measured by the observer it moves past. Length contraction: a moving object is shorter along its direction of motion. Both are tiny at everyday speeds and huge near light speed. The size of the effect is set by a factor called gamma, which equals 1 divided by the square root of (1 − v²/c²). At about 87% of light speed, gamma is 2: a clock on the spaceship ticks once for every two ticks of a clock at home.

Special relativity also gave the most famous equation in physics, in a short follow-up paper later in 1905: E = mc², energy equals mass times the speed of light squared. Mass and energy are two forms of the same thing, and because c² is enormous, a little mass is a lot of energy. Converting 1 gram of mass entirely into energy releases 0.001 kg × (300,000,000 m/s)², which is 9 × 10¹³ joules, about the energy of 21 kilotons of TNT. That is more than the Hiroshima bomb released, and nuclear weapons and the Sun turn far less than all of their fuel's mass into energy to do what they do.

Does "relativity" mean everything is relative?

No, and that's the most common misreading. The theory is as much about what does not change as what does. The speed of light is absolute. The laws of physics are the same for all observers. And a combination of distance and time called the spacetime interval comes out the same for everyone, even when they disagree about the distance and the time separately. Relativity replaces Newton's absolute space and absolute time with a single absolute spacetime. It says nothing about morals, opinions or culture being relative.

What does general relativity add?

Special relativity left out gravity and acceleration. In 1907, Einstein had what he later called the happiest thought of his life: a person falling freely doesn't feel their own weight. Picture someone in a closed elevator. If the elevator sits on Earth, a dropped apple falls to the floor. If the elevator is in deep space accelerating upward at 9.8 m/s², the apple "falls" to the floor in exactly the same way. No experiment inside the box can tell the two apart. This is the equivalence principle: gravity and acceleration are, locally, the same thing.

It took Einstein eight more years to turn that idea into equations. He submitted the final form to the Prussian Academy of Sciences in Berlin on 25 November 1915, the last of four papers on consecutive Thursdays that month, as Physics Today's account describes. The field equations say how mass and energy curve spacetime, and how curved spacetime tells mass and light how to move. A planet orbits the Sun not because a force reaches across space but because it follows the straightest possible path through spacetime that the Sun has curved.

General relativity explained what Newton's laws couldn't. Mercury's orbit slowly swings around the Sun about 43 arcseconds per century more than Newton predicted; Einstein's equations gave exactly that. They also predicted that starlight passing the Sun would bend, which the British eclipse expeditions of 1919 measured, making Einstein world famous. They predicted black holes, regions where spacetime is curved so steeply that nothing escapes, and gravitational waves, ripples in spacetime itself. LIGO first detected those on 14 September 2015, from two black holes of about 29 and 36 times the Sun's mass merging 1.3 billion years ago.

How does GPS prove relativity?

Your phone's location is the everyday proof. GPS satellites orbit about 20,000 km up, carrying atomic clocks, and a receiver works out its position from how long each satellite's timed signal took to arrive. Light travels about 30 cm in a nanosecond, so the clocks have to agree to within tens of nanoseconds. Two relativity effects pull on them in opposite directions, using the figures from Ohio State astronomer Richard Pogge:

Effect	Cause	Satellite clock vs ground, per day
Special relativity	The satellite moves at about 14,000 km/h, so its clock runs slow	About 7 microseconds behind
General relativity	It sits higher in Earth's gravity, where time runs faster	About 45 microseconds ahead
Net		About 38 microseconds ahead

Now multiply. An error of 38 microseconds in timing, at the speed of light, is 0.000038 s × 300,000 km/s ≈ 11 km. Left uncorrected, GPS fixes would drift by roughly 10 km every day. The satellites' clocks are set to tick slightly slower before launch so that, in orbit, they keep time with the ground. Each effect has also been measured directly, with atomic clocks flown around the world on airliners in 1971 and, in 2010, with laboratory clocks raised less than a meter above one another.

What is relativity still missing?

Relativity has passed every test it has faced for more than a century, and it is one of the best-tested ideas in science. Yet Einstein won his 1921 Nobel Prize not for it but for explaining the photoelectric effect, a founding result of quantum physics.

That split survives. General relativity describes gravity and the universe on large scales; quantum mechanics describes particles and forces on tiny ones. Both work superbly in their own domain, but they disagree at the center of a black hole and at the first instant of the Big Bang, where both matter at once. A theory of quantum gravity that joins them is one of the largest open problems in physics.

Check yourself

• Without looking back: why does a constant speed of light force time and distance to change for moving observers?
• Explain the elevator thought experiment, and what it says about gravity.
• GPS satellite clocks run fast overall. Which effect makes them fast, which makes them slow, and which is bigger?

Sources

• Real-World Relativity: The GPS Navigation System — Richard W. Pogge, The Ohio State University, 2017
• Arch and scaffold: How Einstein found his field equations — Michel Janssen and Jürgen Renn, Physics Today, November 2015
• Gravitational Waves Detected 100 Years After Einstein's Prediction — LIGO Caltech, 2016
• The Nobel Prize in Physics 1921, Albert Einstein facts — Nobel Prize Outreach
• Theory of relativity — Wikipedia