The satellites that tell your phone where you are carry atomic clocks. Left alone, those clocks would run about 38 microseconds a day ahead of clocks on the ground. That is 0.000038 of a second, and it would be fatal: a navigation system measures distance as light-travel time, and light covers 11 kilometres in 38 microseconds. Within a day, the blue dot on your map would be a long way from the road.
The clocks are not broken. Gravity is a little weaker 20,000 km up, and a clock in weaker gravity ticks faster. Engineers build that correction into the satellites before launch. This is general relativity, working quietly as engineering.
If you want the friendlier overview first, read Why Time Is Weirder Than You Think (13/03), Orbits: Why the Moon Keeps Falling (13/11) and The Big Bang and Our Expanding Universe (13/08). This piece goes further: it computes.
Stand in a lift whose cable has snapped (in thought only). You and a dropped coin fall together; the coin floats beside you, and no experiment inside the lift can tell you whether you are falling near a planet or drifting in empty space. Einstein called this the happiest thought of his life, in 1907. It is the equivalence principle: a small enough freely falling room feels no gravity at all.
Why is this so surprising? Because every object falls at the same rate, whatever its mass or material. In Newton's theory that is a coincidence: the mass that resists a push and the mass that gets pulled by gravity just happen to be equal. Einstein turned the coincidence into the foundation. If everything falls the same way, then falling is not a property of things. It is a property of the place.
The reverse picture helps too. In a rocket accelerating at 9.8 m/s² in empty space you feel your normal weight, and a ray of light crossing the cabin bends downward, because the floor rises to meet it. Equivalence says gravity must bend light as well.
Light climbing out of a gravity well loses energy, so its frequency drops (a redshift). Read the other way: clocks lower in a gravitational field run slow. To first order the fractional rate difference is the difference in gravitational potential divided by c². Near Earth it is a tiny number, but atomic clocks read it. Pound and Rebka measured the shift between the bottom and top of a 22.5 m tower at Harvard in 1959, and optical clocks today can see the difference for a height change of a few centimetres.
A GPS satellite gets both. Its altitude (about 20,200 km) speeds the clock up by about 45.8 µs a day; its speed of 3.9 km/s slows it by about 7.2 µs. The net is about +38.6 µs a day. A satellite in a low orbit like the International Space Station is dominated by the second term, so its clocks run slow. Somewhere in between the two effects cancel.
Circular orbit; Earth's geoid potential 6.2636856 × 10⁷ m²/s², GM = 3.986004418 × 10¹⁴ m³/s², c = 299,792,458 m/s. Weak-field formula; the satellite is compared with a clock at rest on the geoid. The "ground" preset is a hypothetical satellite just above the surface.
Slide the altitude: the clock is slowest overall at the surface, breaks even near 3,170 km, and runs steadily faster beyond. For GPS the uncorrected error would build up at 11.6 km a day. Notice what this means for everyday gravity: you age slightly faster at the top of a skyscraper than at its foot, by about one part in 1014 for every 100 m of height. Nobody notices, but precision clock networks have to account for it.
So what is gravity? Einstein's answer: spacetime has a shape, and freely falling things follow the straightest possible paths in it, called geodesics. The Moon is not pulled off a straight line; it is going straight, on a stage bent by the Earth. Think of two airliners flying due north from the equator. They start parallel and end up meeting at the pole. Neither turned, and no force pulled them together. The curvature of the surface did it.
That picture makes two predictions that differ from Newton's, and both have been measured.
1. Orbits do not close. In Newton's theory a single planet traces the same ellipse forever. In general relativity the ellipse slowly turns, so the closest point creeps forward each orbit. The weak-field result is a turn of Δφ = 6πGM ÷ (c²a(1 − e²)) per orbit. For Mercury it is 5.0 × 10−7 radians per orbit, which adds up to 43 arcseconds a century. Astronomers had measured an extra turn of that size since the 1850s; Newtonian tugs from other planets explained the rest, and general relativity explained the leftover without any adjustment, which Einstein found in 1915.
Computed live by integrating the Schwarzschild orbit equation u″ + u = 1/L² + 3u² (units G = M = c = 1) with a fourth-order Runge–Kutta step, over four orbits. The Newtonian case drops the 3u² term. Mercury's own orbit is about 650,000 times wider than the smallest shown here, so its turn is far too small to see.
Compare the simulated turn with the weak-field formula. For wide orbits they agree to a few per cent; for tight ones the formula underestimates, because the curvature is no longer weak.
2. Light bends twice as much as Newton would say. Light follows geodesics too, and a ray passing a mass of M at distance b is deflected by an angle α = 4GM ÷ (c²b). That is twice the value you would get by treating light as fast Newtonian particles. For rays grazing the Sun the angle is 1.75 arcseconds. Arthur Eddington's expeditions photographed stars near the Sun during the total eclipse of 29 May 1919, at Príncipe (off West Africa) and Sobral (Brazil), and reported shifts close to Einstein's number (roughly 1.6 and 2.0 arcseconds, with uncertainties of a few tenths; historians still debate how the plates were chosen and reduced). Modern radio measurements agree with Einstein to a small fraction of a percent.
α = 4GM/(c²b), valid when b is much larger than 2GM/c². The cluster is 1014 solar masses with distance in kiloparsecs. The bend in the drawing is exaggerated and not to scale.
The cluster preset is why the effect matters to astronomers. A cluster of galaxies bends background light by tens of arcseconds, enough to smear distant galaxies into arcs and to weigh the cluster, including the mass we cannot see.
Squeeze a mass into a small enough space and no path leads outward. The critical radius is the Schwarzschild radius, found in 1916 within months of Einstein's equations:
The Sun would have to be squeezed to 3 km and the Earth to under 9 mm. The strange part is that the horizon of a heavy hole is not a place of extreme conditions. The average density inside rs falls as 1/M², so the hole in M87, at about 6.5 billion Suns, is less dense than air. And the tidal force, the difference in pull between your head and feet, at the horizon falls as 1/M². A stellar black hole stretches you violently long before you reach it; a supermassive one lets you cross in comfort.
rs = 2GM/c²; density = M/(4/3 π rs³); tidal stretch across 2 m at rs = 2GM·(2 m)/rs³ (Newtonian estimate, indicative only). The bar marks rs on a log scale from 1 km to 100 AU.
The Event Horizon Telescope imaged the shadow of the black hole in the galaxy M87 in 2019, with a mass of about 6.5 billion Suns, and of the one at the centre of our own galaxy, Sagittarius A*, in 2022, near 4 million Suns. Both ring sizes matched general relativity's prediction within the uncertainties.
If mass shapes spacetime, moving mass makes the shape change, and the change spreads at the speed of light. Two black holes in orbit stir spacetime like two spoons in a pond. The waves carry energy away, so the orbit shrinks, the pair spins faster and the wave frequency climbs: a chirp. At leading order the climb depends only on one combination of the masses, the chirp mass Mc = (m₁m₂)3/5 ÷ (m₁+m₂)1/5, and the time left before merger is
On 14 September 2015 the two LIGO detectors recorded the first such signal, GW150914 (announced 11 February 2016): black holes of about 36 and 29 Suns merging into one of about 62, with roughly 3 solar masses of energy radiated in gravitational waves. Chirp mass: about 28 Suns. From 35 Hz the formula gives 0.19 seconds to merger, matching the roughly 0.2 s the signal lasted in the band. The distortion of space at Earth was about 10−21: a 4 km arm changes length by far less than a proton's width. The 2017 Nobel Prize in Physics went to Rainer Weiss, Barry Barish and Kip Thorne for the detector.
Leading-order (Newtonian-quadrupole) inspiral, valid until the innermost stable orbit at f = 4,400 Hz ÷ (total mass in Suns). The plot shows the last 0.1 s of that stage, amplitude ∝ f2/3. The real merger and ringdown, up to about 250 Hz for GW150914, need numerical relativity.
Gravitational waves were predicted by Einstein in 1916, and indirectly confirmed earlier by the Hulse–Taylor binary pulsar, discovered in 1974, whose orbit shrinks as general relativity says (Nobel Prize 1993). LIGO and its partners turned that into an astronomy, and in 2017 they heard two neutron stars merge at the same time as light was seen from the same event.
Everything above comes from one tensor equation, in words: matter and energy tell spacetime how to curve; curved spacetime tells matter how to move (John Wheeler's summary).
It is ten coupled equations, hard to solve in general, which is why exact solutions (Schwarzschild 1916, Kerr 1963, the expanding universe of Friedmann and Lemaître in the 1920s) are famous. Its record is remarkable: Mercury, starlight, the tower, GPS, binary pulsars, black hole images and gravitational waves.
| Test | When | Result |
|---|---|---|
| Mercury's perihelion turn | 1915 | 43″/century explained |
| Starlight bending | 1919 | about 1.75″ at the limb |
| Clock rates (Pound–Rebka) | 1959 | shift seen, as predicted |
| Binary pulsar | 1974 on | orbit shrinks as predicted |
| Gravitational waves | 2015 | GW150914 seen |
| Black hole shadows | 2019, 2022 | M87*, Sgr A* match |
Where it stops: general relativity is a classical theory, and at a black hole's centre or at the Big Bang it predicts infinite curvature, which is a signal that something else takes over. Nobody yet has a tested theory of quantum gravity. On the largest scales it works only if most of the universe is something we have not identified (dark matter and dark energy), and its models disagree at present about how fast the universe expands (the Hubble tension). Both are the business of Astrophysics & Cosmology (34/88).
Gravity is not a pull but the shape of spacetime. Free fall is the straight path; clocks, light and orbits all follow from the geometry, and it is precise enough to correct a satellite's clock by 38 microseconds a day. The same geometry gives black holes with a size of 3 km per Sun and ripples that shake detectors by a thousandth of a proton.
Try this: this week, when you check a maps app, remember that its blue dot needs two relativity corrections, one for speed and one for gravity, and that without them it would drift by about 11 km a day.
Sources. Equivalence principle, geodesics, the perihelion formula, light deflection, Schwarzschild radius and the leading-order chirp are standard (Hartle; Carroll). Constants from CODATA 2018 and IAU nominal values (GMsun = 1.32712440018 × 10²⁰ m³/s²). GPS: about 45 µs a day from gravity, about 7 µs a day from speed, net about 38 µs a day (Ashby, Living Reviews in Relativity, as reported in GPS World). GW150914: LIGO–Virgo, 36 and 29 solar masses to about 62, about 3 solar masses radiated (Phys. Rev. Lett. 116, 061102, 2016, and the follow-up properties paper). Eddington's 1919 results: Dyson, Eddington and Davidson, 1920; values of about 1.98″ (Sobral) and 1.61″ (Príncipe), with uncertainties of a few tenths of an arcsecond reported differently in secondary sources. EHT: M87* mass (6.5 ± 0.7) × 10⁹ solar masses (2019), Sgr A* about 4 × 10⁶ solar masses (2022). Pound–Rebka 1959, Hulse–Taylor 1974 and Nobel 1993, Mercury 1915, Schwarzschild 1916 and the 2017 Nobel Prize are cited from the literature and not re-checked for this piece. Every number an interactive shows was computed in node: for example net clock shift +38.59 µs/day at 20,200 km, break-even at 3,167 km, 43.0″ a century for Mercury, 1.751″ at the solar limb, 0.016″ for Jupiter, 39″ for a 1014 solar mass cluster at 100 kpc, rs = 2.95 km for the Sun, and 0.19 s from 35 Hz for GW150914.