Relativity, with pictures
The mathematics of general relativity is hard, but the ideas are not. They follow from two or three assumptions, pursued with more patience than most people give them, and what comes out is strange enough to dismiss out of hand, except that the phone in your pocket relies on it every day. Its map needs both of Einstein's theories to put you on the right street.
Special relativity comes first (1905, flat space, no gravity), then general relativity (1915, gravity), with diagrams throughout and a good many things to drag, scrub and switch on and off. The phone turns up again in section 5.
1. Two rules
Special relativity rests on two postulates.
- The laws of physics are the same in every steadily moving frame. If you're in a windowless train gliding at a constant speed on perfect rails, no experiment inside can tell you whether you're moving. "Moving" only means something relative to something else.
- Light travels at the same speed for everyone. In a vacuum, 299,792,458 metres per second (call it 300,000 km/s), no matter how fast you or the light source is moving.
The first one is old. Galileo had it in 1632, with a ship instead of a train. The second is the troublemaker. Throw a ball forward at 50 km/h from a train doing 100, and someone on the platform clocks it at 150, because speeds add. Shine a torch forward from the same train and the platform measures the light at c, with nothing added for the train.
At everyday speeds the difference is far too small to notice, so here is a faster train, and a ball thrown forward at half the speed of light (a very good arm). Drag the train's speed and compare the bars.
Push the train to 0.99c and Newton's ball is doing 1.49c. The real one never reaches c, and the light doesn't budge. Measurement after measurement agrees, from the Michelson–Morley experiment of 1887, which tried to detect Earth's motion through space by timing light and found nothing, to a great many better ones since. (Einstein himself got to rule two mostly through Maxwell's equations of electromagnetism, which have c built in. The experiments bear it out.)
If two people moving relative to each other measure the same speed for the same flash, then something else in "speed = distance ÷ time" has to give, and what gives is distance and time themselves. Most of what follows is the bill for rule two.
2. Moving clocks run slow
Take a thought-experiment clock, two mirrors facing each other a distance L apart with a single photon bouncing between them. One round trip is one tick. It's a real clock in principle, if an inconvenient one.
Put one on the platform and an identical one on a train going past at 0.8c. Below, the train clock is stopped mid-flight with a triangle drawn over its current zig. Press play, and the strip along the bottom lays both clocks' ticks on one time axis so the two rhythms sit side by side.
Seen from the platform, the train clock's light has to go up and sideways between the mirrors, because they moved while it was in flight. Its path is the long side of a right triangle, and by rule two it still travels at c. A longer path at the same speed takes longer, so, measured from the platform, the moving clock ticks slowly.
The triangle gives the amount. Call t the time one zig takes, measured from the platform, and t₀ the time one zig takes on a clock at rest. The upright side is the mirror gap, which light crosses in t₀, so it is c·t₀. The bottom is how far the train moved, v·t, and the diagonal is how far the light went, c·t. By Pythagoras,
(c·t)² = (c·t₀)² + (v·t)²
One assumption is hiding in there. The vertical side has to be L in both frames, which means lengths at right angles to the motion don't change. They can't. Give two identical upright rulers a pencil at the tip each and slide them past each other. If moving rulers shrank sideways, each would expect its own pencil to clear the other ruler and the other's pencil to mark it. A pencil mark is a fact, not a point of view, and rule one says neither ruler is special, so neither can be shorter.
Divide through by c², collect the t terms and take a square root, and out falls the Lorentz factor, gamma, which runs through everything that follows.
t = γ·t₀
γ = 1 / √(1 − v²/c²)
None of this depended on the clock being made of mirrors. Rule one says all physics behaves alike in the moving frame, so atomic clocks, heartbeats and radioactive decay slow by the same factor, and whatever keeps time on the train, measured from the platform, keeps it slow.
"The moving clock runs slow" doesn't mean you would literally see it tick slowly. Light from a clock takes time to reach you, so an approaching clock looks fast and a receding one looks slow, which is the Doppler effect. Time dilation is what's left once you correct for that delay, the figure any careful observer in that frame works out, and on this page that is what "measures" means.
How big is the effect?
At everyday speeds it is tiny. At 0.5c it's 15%. Near c it grows without limit, so the chart's speed axis counts nines, each step right adding another 9 after the decimal point. Drag the slider, or tap a dot on the curve.
At the speed of the International Space Station (7.66 km/s) gamma is 1.0000000003. A year aboard puts an astronaut about 0.01 seconds behind the people on the ground, leaving gravity aside until section 5. On the chart, every vehicle anyone has built sits on the left edge, and the rest of the curve belongs to particles.
Real evidence 1: muons
Cosmic rays hit the top of the atmosphere and make muons, unstable particles with an average lifetime of 2.2 microseconds. Even at nearly c, 2.2 µs is only long enough to cover about 660 metres. The muons are made around 15 km up, so by rights almost none should reach the ground. Here is one batch of muons drawn twice, with identical dice for when each one decays.
They do reach the ground, about one per square centimetre per minute at sea level, which is a couple through your hand every second. They make it because their clocks, measured from the ground, run slow. The 1963 Frisch–Smith experiment counted muons on top of Mount Washington and at sea level and found the survival rate matched time dilation, not Newton. In 1977, muons circling a storage ring at CERN at γ ≈ 29 lasted 29 times longer than muons at rest.
Real evidence 2: GPS
GPS satellites orbit at about 3.9 km/s. That's slow enough that gamma differs from 1 only in the eleventh decimal place, and fast enough to matter: their clocks lose about 7 microseconds a day to special relativity. GPS works by timing radio signals, and light goes 300 metres in a microsecond, so 7 µs is two kilometres of light travel. A second and bigger effect pulls the other way, and it comes from gravity.
A loose end
Rule one says the train's point of view is as valid as the platform's, so flip the light clock to the . Now the train's clock sits still, and the platform's is the one sliding past, zig-zagging and ticking slow. Each measures the other's clock as the slow one, and it is not obvious how both can be right.
3. Space and "now" stretch and shift
Relativity of simultaneity
The answer is hidden in the word now, which this page has so far used carelessly. Lightning strikes both ends of a moving train. M stands on the platform exactly halfway between the two strike points; T sits in the middle of the train. Both flashes reach M at the same moment, from equal distances, so M concludes, correctly, that the strikes were simultaneous.
The rest is easier on a spacetime diagram, where time runs up, space runs across, and each event is a point. Units are chosen so that light goes one square across for each square up, and all the spacetime diagrams on this page use the marks below.
The diagram below holds the two strikes, their light, and the moment that light meets at M. The sweeping line is the observer's "now", and the strip on top draws what lies along it, the train as that observer would sketch it at that instant. Start with M's version, then switch to T's.
In M's version, though, T rides toward the front flash and away from the rear one, so the front flash reaches T first. T knows they're sitting in the middle of their train, and knows from rule two that light from each end travels at c in T's frame too. The front flash arrived first, so by T's perfectly sound reasoning the front was struck first, which is what T's strip shows. M says the strikes were simultaneous, and neither is making a mistake. There is no fact of the matter about whether two distant events happen "at the same time"; it depends on your frame.
Some things hold still, though. The strikes swap order as the speed crosses zero, yet the meeting at M comes after both at every speed. Light connects each strike to it, and any two events that light (or anything slower) can connect come in the same order in every frame. Only events too far apart for light to link get reshuffled, so causes always come before their effects.
That also settles the loose end from section 2, and the numbers are small enough to follow. Let the train pass the platform at 0.6c (γ = 1.25), with both clocks reading 0 as they pass. In the platform's account, at 10 seconds the train is 6 light-seconds down the line and its clock reads 10 ÷ 1.25 = 8, which is 0.8 × 10. In the train's account, its "now" through that same event is tilted, and the tilt is the relativity of simultaneity in a single number. Six light-seconds back along the line, the train's "now" is earlier by v·x/c² = 0.6 × 6 = 3.6 seconds, so it meets the platform clock when that reads 10 − 3.6 = 6.4, which is 0.8 × 8. Each finds the other running at 0.8, because each lines the clocks up along its own "now", and those are different lines.
So is everything relative?
No. Observers disagree about distances and durations separately, but they agree on one combination of the two, the spacetime interval, (c·Δt)² − (Δx)². For a clock that moves steadily from one event to another, the interval is (c × its own elapsed time)². For a clock that changes course, add up the time it shows on each straight piece (each piece's √interval ÷ c). That total, its proper time, is a fact every frame agrees on.
The diagram above shows it. The dotted curve marks everywhere a clock leaving the event where T passes M can be when it reads 2, whatever steady speed it goes at. Drag the speed, and the "now" tilts and every t′ changes, while the diamond where the observer's own clock reads 2 slides along that one curve.
Length contraction
Back to the muon, from its side. In its own frame its clock is fine, and it lives an ordinary 2.2 µs on average. What it measures instead is the atmosphere rushing up at 0.998c, squashed along the motion by that γ ≈ 16, so that the 15 km becomes about 950 metres, short enough for about one in four to make it.
Only lengths along the motion shrink, by the factor γ. Heights stay put, for the pencil reason above. The crew of a fast ship notices nothing odd about their ship, and to them it's your rulers that are short. Both are right, for the reason the clocks were. Measuring a moving thing's length means marking both ends at the same moment, and "the same moment" is what the two frames disagree on. (A photograph is another matter. It would show something fast looking rotated rather than squashed, the Terrell–Penrose effect, because light from its far side set off earlier.)
The twin "paradox"
One twin stays on Earth. The other flies at 0.8c to a star 4 light-years away and comes straight back. At 0.8c, γ = 5/3. Earth's calendar says the trip took 10 years (5 out, 5 back). The traveller's clock says 6 (3 out, 3 back). The traveller comes home four years younger.
The supposed paradox is that from the traveller's side Earth did the moving, so Earth's twin ought to be the younger one. Proper time answers it. The time a clock shows is the length of its path through spacetime, measured with a strange rule: between two events, the straight path is the longest. The Earth twin's path from goodbye to reunion is straight and the traveller's is bent, and everyone, in every frame, computes 10 and 6.
If that sounds like a trick of definitions, it can be checked with nothing but a telescope. Earth sends a flash of light every birthday. Scrub the traveller's clock and count what reaches them.
Flying away, the traveller catches Earth's flashes stretched out by the Doppler effect, one in three years. Flying home, they arrive bunched up, nine in three years. That makes ten Earth birthdays in six traveller years, counted from what reaches their eyes alone.
From Earth's end the count comes out the other way round. The traveller's own yearly flashes arrive at a third of the rate for 9 years, which is 3 traveller-years' worth, and at triple rate for the last year, 3 more, so Earth sees six in ten. The traveller spends equal time at each rate. Earth doesn't, because news of the turnaround, travelling at the speed of light, takes until year 9 to get home, and that delay is the whole asymmetry.
What they infer is the textbook bookkeeping: correct the telescope for light delay and you get the traveller's "now". Drag to year 3 and it swings across Earth's history, from Earth year 1.8 to 8.2, as the traveller turns. Nothing happens to Earth. The traveller starts slicing spacetime at a different tilt. The jolt of turning tells you who turned, but it isn't what causes the gap. Swap the turnaround for two passing ships, one calling its clock reading across to the other, and the total is still 6 years with nobody accelerating.
Hold on to the rule behind all this, because it turns out to explain why apples fall as well: a freely moving clock follows the path along which it ages most (compared with any nearby path).
4. E = mc²
It comes out of special relativity, in a three-page follow-up Einstein wrote in 1905, and it says that mass is a form of energy. A thing at rest with mass m contains energy mc², and if it loses energy (by emitting light, say), it loses mass in exact proportion.
c² is about 9 × 10¹⁶ in metres-and-seconds. That's the exchange rate, and it's ruinous.
Run the other way, the Hiroshima bomb, with 64 kg of uranium aboard, converted about 0.7 grams of its mass into energy. The Sun converts about 4 million tonnes every second, and has done roughly that for 4.6 billion years.
Two common misreadings. First, it isn't only about nuclear reactions. Burning a kilogram of petrol releases about 46 megajoules, and the products are lighter than the ingredients by half a microgram. Nuclear reactions shift about a million times more energy per atom, which is what makes their mass change measurable. Second, it doesn't mean you can "turn matter into energy" on demand. You only get out what some process releases. Converting a whole gram takes antimatter.
The same bookkeeping explains the speed limit. A mass m moving with Lorentz factor γ carries energy of motion (γ − 1)mc². That's the γ curve from section 2 again, and it goes to infinity at c, so reaching light speed would take infinite energy. A finite push only adds more nines. (Older books say a moving object's mass increases. Physicists have dropped that way of talking, so m stays put, and it's energy and momentum that grow without limit.)
One more consequence. Most of your mass is not the mass of the quarks and electrons you're made of. Protons and neutrons weigh about a hundred times more than the quarks inside them. Around 99% of their mass is the energy of those quarks and the gluons that bind them, moving and binding, weighed.
5. Gravity: general relativity
Special relativity covers steady motion. It took Einstein ten more years to fold in gravity and acceleration, starting from what he called the happiest thought of his life: a person falling freely doesn't feel their own weight.
The equivalence principle
You're in a closed box. Drop a ball. If the box sits on Earth's surface, the ball falls to the floor. If the box is out in deep space on a rocket accelerating at 9.8 m/s², the ball floats where you let it go and the floor rushes up to meet it, which from inside looks identical. Cut the cable on a box near Earth and everything inside floats, as it would drifting between the stars. No experiment in a small enough box can tell gravity from acceleration, or falling from floating.
The principle looks harmless until you shine a torch horizontally across the accelerating box. Seen from outside, the beam goes straight, but the box is speeding up underneath it, so the beam hits the far wall lower than it started. Inside, you see it curve downward. By equivalence the same has to happen in the box on Earth, which means that gravity bends light.
Gravity is curved spacetime
Einstein stopped treating gravity as a force. Mass and energy curve spacetime, and objects with no forces on them, falling freely, follow the straightest possible paths through it. From the ground those paths look like orbits and falling apples. John Wheeler's summary is the one physicists quote: spacetime tells matter how to move; matter tells spacetime how to curve.
You've probably seen the rubber-sheet picture: a heavy ball dents a stretched sheet, and a marble circles in the dent. It's worth knowing what it gets wrong. It uses gravity (the marble rolls down into the dent) to explain gravity, which is circular. And it shows only curved space. Near Earth, for anything moving slowly, curved space barely matters. What makes an apple fall is overwhelmingly curved time. Curved space only becomes important for very fast things, like light.
Lower clocks run slow
"Curved time" means something concrete. Clocks lower down in a gravity well run slow compared with clocks higher up. Near the ground the fractional rate difference is g·h / c², about 1.1 × 10⁻¹⁶ per metre, or roughly 9 picoseconds a day for each metre of height. Your head ages faster than your feet, by about half a microsecond over an 80-year life (for a 1.7 m person). In 2010, NIST measured the difference between two optical clocks with one lifted by 33 centimetres.
Why things fall
Take the rule from the twins, that a freely moving clock follows the path through spacetime along which it ages most. Throw a ball up and catch it two seconds later. Of all the arcs it could take from your hand back to your hand in those two seconds, which one ages it most? (The toy only tries arcs of one shape; allowing every possible path changes nothing.)
Climbing helps, because higher clocks run faster. But climbing means moving, and moving clocks run slow, and the higher you go in a fixed two seconds, the faster you have to move. So there's a best compromise. Drag the top of the arc and find it. The lower graph keeps a trace of the heights you try.
Once you've found it, the winner is exactly the parabola a real ball thrown for two seconds follows, the one Newton would compute with g = 9.8 m/s². The time gained is absurdly small, a few times 10⁻¹⁶ seconds, but the shape of the best path is the whole of everyday gravity. Switch gravity off and the best path is to sit still in your hand, which is the twin rule again, where the straight line ages most. Switch it on and "straight" bends into a parabola. Nothing pulls the ball down. It follows the most-ageing path through a time that runs at different rates at different heights.
Why your phone needs Einstein
For GPS, 20,200 km up, the effect is large. The satellites sit much less deep in Earth's gravity well, so their clocks gain about 45.7 µs a day, against the 7.2 µs a day they lose to their speed.
Net, about 38 microseconds a day fast, and light travels 11.5 km in that time. Your receiver works out its position by timing signals from several satellites at once, solving for its own clock error as it goes, so an offset every satellite shared equally would cancel out. But satellites launched or reset on different dates would disagree with each other by kilometres' worth, and that doesn't cancel. Uncorrected, the system would need large daily fixes from the ground, and GPS time, which banks and phone networks set their clocks by, would drift away. The fix is mundane. Before launch the satellite clocks are set to tick at 10.22999999543 MHz instead of 10.23 MHz, so that they come out right in orbit. Every phone finding its way across a city depends on both of Einstein's theories.
Try some heights. Held still, higher always gains. In orbit, low satellites move fast enough that the speed loss wins; high ones gain. The chart shows where they break even.
You don't need a satellite to measure both effects. In 1971 Joseph Hafele and Richard Keating flew caesium clocks around the world on ordinary airliners, once eastward and once westward, and compared them with clocks left at the US Naval Observatory. The eastbound clocks lost about 59 nanoseconds and the westbound ones gained about 273, in line with predictions that combine the height effect with the speed effect (Earth's spin makes the two directions differ).
Light bending, and the 1919 eclipse
General relativity predicts that starlight grazing the Sun gets bent by 1.75 arcseconds, about the width of a coin seen from 2.5 km. Treating light as Newtonian particles gives half that, 0.87″, and so does the box argument above, which captures only the curved-time part. The other half is the curvature of space. You can only see stars near the Sun during a total eclipse, so on 29 May 1919, expeditions led by Arthur Eddington (on Príncipe, off West Africa) and Andrew Crommelin (at Sobral, Brazil) photographed the stars around the eclipsed Sun and compared them with their usual positions.
The results favoured Einstein, and within days his name was in the London and New York papers. In fairness, the 1919 plates were rough, and the analysis involved some judgement calls. Later measurements, especially with radio telescopes, confirm the GR value to better than 0.1%.
Scale up the lens and you get gravitational lensing, in which a galaxy or cluster bends the light of something far behind it into arcs, multiple images or, when the two line up perfectly, a complete ring.
Mercury's wandering orbit
Planets don't retrace their ellipses lap after lap. The ellipse slowly rotates, mostly because the other planets tug on it, and for Mercury Newtonian gravity accounts for nearly all of the observed drift, about 532 of the 575 arcseconds per century. But by 1859 Urbain Le Verrier had found a leftover, now pinned at 43 arcseconds per century, that nobody could explain. (He proposed an unseen inner planet, Vulcan. People looked for decades.) In 1915, general relativity produced the missing 43″ with no adjustable parts. Einstein said the result gave him heart palpitations.
Here is the relativistic part on its own. At its true size nothing visibly happens, which is the point. Exaggerate it and follow where each lap's closest approach to the Sun lands.
Black holes
Squeeze enough mass into a small enough space and you get a region nothing can leave. Its boundary is the event horizon, at the Schwarzschild radius 2GM/c²: about 3 km for the Sun's mass, 9 millimetres for Earth's.
"Even light can't escape" makes it sound as if light is too slow, like a ball thrown too gently. A better picture is that inside the horizon the future itself points inward. Here is light aimed straight outward from different distances, drawn so that light falling inward always goes at 45°.
The horizon isn't a surface, and a falling astronaut feels nothing special there. For the black hole in M87, the stretch from head to foot at the horizon is about a twenty-billionth of a g, and for the one at the centre of the Milky Way about a ten-thousandth. Small black holes are another matter. One of ten solar masses would pull you apart long before the horizon, with the stretch already at a hundred g some 1,700 km out; at the horizon itself it is about twenty million g. The horizon is a point of no return, set by where the light cones point. From outside you never see anyone cross. Their light arrives ever redder and fainter, and fades out.
Inside, the singularity lies in your future the way next Tuesday does, and there is no steering round it. It is also where general relativity stops making sense, predicting infinite curvature, which is usually read as a sign that the theory needs replacing there by a quantum theory of gravity nobody has yet. In 2019 the Event Horizon Telescope published the first image of a black hole's shadow, in galaxy M87, and in 2022 an image of the one at the centre of the Milky Way.
Gravitational waves
If mass curves spacetime, moving masses should make ripples in it, travelling at the speed of light. A passing wave alternately stretches space one way and squeezes it the other, across the direction it's travelling.
Einstein predicted them in 1916 and doubted they'd ever be measured. On 14 September 2015, both LIGO detectors (Louisiana and Washington) caught one: two black holes of about 36 and 29 solar masses, 1.3 billion light-years away, spiralling together into one of about 62. The missing three solar masses left as gravitational waves, in a fraction of a second.
The signal was a strain of about 10⁻²¹: each 4 km arm changed length by about 4 × 10⁻¹⁸ metres. That's the equivalent of measuring the distance to the nearest star to within the width of a human hair. It earned the 2017 Nobel Prize, and LIGO and its partners have since logged hundreds of mergers.
What to remember
- Light's speed is the same for everyone. Almost everything else follows from taking that seriously.
- Moving clocks run slow, by γ = 1/√(1 − v²/c²), as measured by anyone they move past. Negligible at everyday speeds, enormous near c. Muons and GPS both depend on it.
- "At the same time" depends on who's asking, and that's why two observers can each find the other's clock slow. Moving things shrink along their motion. The order of events that light could connect never changes.
- Some things are absolute: the spacetime interval, and the time a clock itself shows between two events.
- Between two events, the straight path through spacetime ages you most. The twin who took the bent path comes home younger, and a telescope shows it too.
- E = mc²: mass is energy, at an exchange rate where one gram is about 21 kilotons. The same accounting puts c out of reach for anything with mass.
- Gravity is curved spacetime, and near Earth it's mostly curved time: lower clocks run slow, and falling is the most-ageing path through that.
- GPS clocks gain about 38 µs a day (+45.7 from height, −7.2 from speed), about 11 km of light, so the satellites are built to correct for it.
- It's all been measured: light bending (1919 onward), Mercury's 43″ per century, clocks on airliners (1971), black hole images (2019), gravitational waves (2015).