Special and general relativity, in 3D
Relativity
Einstein's two theories change how time, length and gravity work, and every one of their strangest claims has been measured. Move the sliders and watch clocks slow down, ships shrink and the Sun bend starlight.
Above is the sky from a ship speeding up to 92% of light speed: the stars crowd forward, blue ahead and red behind. Every scene below runs on the real equations. Special relativity (1905) covers the first three, what happens at high speed. General relativity (1915) covers the rest: gravity as the shape of spacetime.
01 / 6
Moving clocks run slow
The idea
Light moves at the same speed for everyone, 299,792,458 metres a second, however fast they are going themselves1. Follow one pulse of light bouncing between two mirrors and that single fact is enough to slow a moving clock down.
In plain words
Seen from the side, the moving clock's light runs a slanted zigzag, a longer path than the straight up and down of the clock at rest. Light cannot speed up to cover the extra distance, so each tick takes longer. Everything travelling with the clock slows by the same amount, heartbeats included. Riding along, nothing seems odd: to the rider, it is our clock that runs slow.
The numbers
The stretch is γ = 1/√(1 − v²/c²). At 10% of light speed it is 1.005, at 80% it is 1.67 and at 99% it is 7.1. An airliner at 250 m/s loses about 30 nanoseconds a day against a clock on the ground, the size of effect Hafele and Keating measured2.
How we know
Muons made by cosmic rays high in the air live 2.2 microseconds at rest3, long enough to travel only about 660 m even at light speed, yet they reach the ground in large numbers. In 1963 Frisch and Smith counted them on Mount Washington and at sea level and found their clocks running about 9 times slow4. At CERN in 1977, muons circling at 99.94% of light speed lived 29 times longer than muons at rest5.
Try it
Slide the speed to 0.99 c. The moving clock's light barely climbs while the clock at rest ticks seven times.
02 / 6
Moving things get shorter
The idea
Anything moving past you is shorter along its direction of motion, by the same factor γ that slows its clocks. Height and width stay the same.
Both ways
The station measures the ship shortened, and the crew measures the station's posts closer together. Both are right, because length depends on who is measuring, as time does. Nothing squeezes the ship: no force acts on it, and the crew feel nothing at all.
The numbers
A ship 100 m long at rest measures 60 m when it passes at 80% of light speed and 14 m at 99%. A 200 m train at 300 km/h comes out shorter by about 8 picometres, under a tenth of the width of an atom.
How we know
Seen from a cosmic-ray muon, its own clock runs normally and the air between it and the ground is what shrinks, by the same factor that slows its clock in our view4. At Brookhaven's collider, gold nuclei travelling at 99.995% of light speed are flattened about 100 times into thin discs as they collide6.
Try it
Switch to the ship's view: the ship stops, the station rushes past, and now its posts are the ones squeezed together.
03 / 6
Observers disagree about “at the same time”
The idea
Lightning strikes both ends of a train at the same moment by the platform's clocks. For a passenger in the middle of the train, the front strike happened first. Both are right: whether two distant events happen at once depends on how you are moving.
Reading the diagram
This is a spacetime diagram: across is space, up is time, and light always runs at 45°. The platform's "same moment" is a flat line. The passenger's is tilted (the dashed lines). Switch to the train's frame and the diagram is redrawn the way the passenger slices it: their lines go flat, and the front strike sits lower, which means earlier.
The numbers
In the train's frame the front strike comes first by γvL/c², where L is the distance between the strikes. For a 200 m train at 300 km/h that is about 0.19 picoseconds, far too small to notice, which is why everyday life never shows it.
Try it
Press Train's frame and watch the strikes come apart in time. Slide the speed to zero and the two frames become one.
04 / 6
GPS has to correct for relativity
The idea
Clocks run slower in stronger gravity, and slower when they move. GPS satellites carry atomic clocks 20,200 km up9. There, the weaker gravity makes them gain time and their speed makes them lose a little. The gain wins.
The numbers
At GPS height a satellite clock gains about 45.7 microseconds a day from the weaker gravity and loses about 7.2 from its speed of 3.9 km/s, so it runs 38.5 microseconds a day fast8. Light covers 11.5 km in that time, so uncorrected fixes would drift by kilometres every day. The clocks are made to tick at 10.22999999543 MHz on the ground, which comes out at exactly 10.23 MHz in orbit8.
Lower down
Slide the orbit down. Below about 3,190 km the speed wins and an orbiting clock runs slow. At the space station's 420 km it loses about 24 microseconds a day, so over Scott Kelly's 340 days in orbit10 he came home about 8 milliseconds younger than if he had stayed on the ground.
How we know
In 1971 Hafele and Keating flew atomic clocks round the world on airliners. Flying east they lost 59 nanoseconds against clocks left at home, and flying west they gained 273, as relativity predicted2. In 2010 NIST measured the effect of gravity between two optical clocks with one raised by just 33 cm11.
See it in 3D
The viewer draws every active satellite where it is right now, GPS included.
05 / 6
The Sun bends starlight
The idea
Mass curves spacetime and light follows the curve. Starlight grazing the Sun is bent by 1.75 arcseconds, so during a total eclipse the stars round the Sun appear pushed slightly outward12.
The numbers
1.75 arcseconds is the width of a 1 cm coin seen from 1.2 km. Treating light as particles falling under Newton's gravity gives half as much, 0.87″. The bend falls off with distance: a star twice as far from the Sun's centre moves half as far. The Sun itself is about 960″ in radius13.
How we know
Eddington and Dyson sent two expeditions to the total eclipse of 29 May 1919. The plates from Sobral in Brazil gave 1.98″ ± 0.12″ and those from Príncipe 1.61″ ± 0.30″, closer to Einstein's value than to Newton's14. Radio telescopes now measure the bending of quasar light to better than 0.1%15.
Much stronger elsewhere
The same bending, far stronger, makes the Einstein rings round distant galaxies and the warped sky round a black hole. Close enough to a black hole, light can travel in a circle.
See it in 3D
Our viewer's black hole bends the light of the real sky behind it, and the wormhole near Saturn lenses the stars into rings.
06 / 6
Mercury's orbit turns
The idea
Mercury's oval orbit slowly turns, so its closest point to the Sun creeps round a little each time. The other planets' pulls explain most of it. The last 43 arcseconds a century puzzled astronomers for 56 years, until Einstein's theory gave exactly that12.
The numbers
Astronomers measure the closest point turning about 575″ a century. Newton's gravity with every planet pulling gives about 532″, and relativity adds the missing 43″16. Mercury goes round 415 times a century17, so that is about 0.1″ an orbit.
How we know
Urbain Le Verrier found the unexplained part in 1859 and proposed an unseen planet inside Mercury's orbit, Vulcan. Nobody ever found it. In November 1915 Einstein worked out Mercury's orbit with general relativity and got 43″ a century, with nothing adjusted to fit12.
Why Mercury
Spacetime is curved most strongly close to the Sun, and an orbit there does not quite close on itself. The effect is largest for the innermost planet; for the Earth it is under 4″ a century.
See it in 3D
The viewer places Mercury on its real orbit for any date you choose.
Tested every day
Relativity is easy to state and hard to believe, which is why it has been checked so many times: with muons, atomic clocks on airliners, satellites, eclipses, planets and gravitational waves. The map on your phone uses it every time it works out where you are8.
Questions
What is the theory of relativity, in simple terms?
Two theories by Albert Einstein. Special relativity (1905) says the laws of physics and the speed of light are the same for everyone moving steadily, which makes moving clocks run slow and moving objects shorter7. General relativity (1915) says gravity is the curving of space and time by mass and energy12.
Why can nothing go faster than light?
As anything with mass speeds up, the energy needed for more speed grows without limit: at 99% of light speed γ is 7.1, at 99.99% it is 71, and reaching light speed would take infinite energy. Light has no mass and always moves at that speed.
Does time really slow down, or do the clocks just run slow?
Time itself. Every process slows by the same factor: atomic clocks, the decay of muons, chemistry and biology4. No clock keeps a "true" time for the others to lag behind.
Is E = mc² part of relativity?
Yes. In a short follow-up paper in 1905 Einstein showed that mass is a form of energy, E = mc²18. The Sun shines by it, turning about 4 million tonnes of mass into light every second13.
How does GPS use relativity?
Satellite clocks gain about 38 microseconds a day on clocks on the ground: 45.7 from the weaker gravity minus 7.2 from their speed. They are tuned slightly slow before launch so that in orbit they keep time with the ground8.
Has relativity ever failed a test?
Not yet. Every test agrees with it, from clocks 33 cm apart in height11 to black holes colliding 1.3 billion light-years away19. Physicists still expect it to need changing somewhere, at the centre of a black hole or the start of the Universe, where it has to meet quantum mechanics.
Sources
- NIST CODATA: the speed of light in vacuum, 299,792,458 m/s (exact)
- Hafele, J. C. and Keating, R. E. (1972), Around-the-World Atomic Clocks: Observed Relativistic Time Gains, Science 177, 168
- Particle Data Group (2024): the muon, mean life 2.197 microseconds
- Frisch, D. H. and Smith, J. H. (1963), Measurement of the Relativistic Time Dilation Using μ-Mesons, American Journal of Physics 31, 342
- Bailey, J. et al. (1977), Measurements of relativistic time dilatation for positive and negative muons in a circular orbit, Nature 268, 301
- The Relativistic Heavy Ion Collider, Brookhaven National Laboratory
- Einstein, A. (1905), On the Electrodynamics of Moving Bodies, Annalen der Physik 17, 891
- Ashby, N. (2003), Relativity in the Global Positioning System, Living Reviews in Relativity 6, 1
- GPS.gov: the space segment (orbits 20,200 km up, planes at 55°)
- NASA: the Twins Study (Scott Kelly's 340 days on the space station)
- Chou, C. W., Hume, D. B., Rosenband, T. and Wineland, D. J. (2010), Optical Clocks and Relativity, Science 329, 1630
- Will, C. M. (2014), The Confrontation between General Relativity and Experiment, Living Reviews in Relativity 17, 4
- NASA Sun fact sheet (mass, radius, luminosity, apparent size)
- Dyson, F. W., Eddington, A. S. and Davidson, C. (1920), A Determination of the Deflection of Light by the Sun's Gravitational Field, Philosophical Transactions of the Royal Society A 220, 291
- Shapiro, S. S. et al. (2004), Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic VLBI Data, Physical Review Letters 92, 121101
- Tests of general relativity: the perihelion precession of Mercury
- NASA Mercury fact sheet (orbital period 87.969 days, eccentricity 0.2056)
- Einstein, A. (1905), Does the Inertia of a Body Depend upon its Energy Content?, Annalen der Physik 18, 639
- Abbott, B. P. et al. (2016), Observation of Gravitational Waves from a Binary Black Hole Merger, Physical Review Letters 116, 061102
The scenes run on the formulas in their readouts. Speeds of light, distances and sizes are scaled to fit the screen, and the readouts say whenever a shift is drawn larger than life. Clock rates for orbits leave out the ground's own spin.