Field · Emerged 1907 – 1915

General Relativity

What is gravity, if it is not a force?

6 chapters5 min read6 turning points 3 open problems

Branched from: Special Relativity + Classical Mechanics
Branched into: Compact Objects + Physical Cosmology
Figures: Albert Einstein, David Hilbert, Marcel Grossmann, Karl Schwarzschild, Arthur Eddington, Frank Dyson, Rainer Weiss, Barry Barish, Kip Thorne

In brief

General relativity is Einstein's theory of gravity. Mass and energy curve the four-dimensional spacetime around them, and objects in free fall simply follow the straightest possible paths through that curved geometry. The Earth orbits the Sun not because a force pulls it, but because the Sun has bent the spacetime it moves through.

It explains what Newton's gravity got slightly wrong, like Mercury's orbit and the bending of starlight, and predicts things Newton's never could: black holes, gravitational waves, and clocks that tick at different rates at different heights. It is also where physics' map most plainly runs out, at the centre of a black hole and at the beginning of time.

Key ideas

Equivalence principleEnters 1907: Locally, gravity cannot be told apart from acceleration. In a freely falling lift you float as if gravity were switched off. Gravity is therefore a property of spacetime, not a force acting within it.
Curved spacetimeEnters 1915: Spacetime is a four-dimensional curved (pseudo-Riemannian) manifold whose metric $g_{\mu\nu}$ is shaped by matter and energy. Its curvature is gravity.
Einstein field equationsEnters 1915: $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ : the curvature of spacetime (left) equals its content of mass, energy and momentum (right). "Spacetime tells matter how to move; matter tells spacetime how to curve" (Wheeler).
Event horizonEnters 1916: The boundary of a black hole: a surface in spacetime from inside which not even light can escape. For a non-rotating mass $M$ it lies at the Schwarzschild radius $r_s = 2GM/c^2$ .
Gravitational wavesEnters 2015 – 2016: Ripples in the curvature of spacetime itself, emitted by accelerating masses and travelling at the speed of light. They stretch and squeeze space as they pass.

Draws on other domains

↙ Mathematics
Numerical Methods for PDEs
Templates for LIGO
↙ Mathematics
Riemannian Geometry
Gravity is curvature, and satellites notice

Chapter I

The Happiest Thought

Special relativity left gravity out. Newton's gravity acts instantly across space, but after 1905 nothing, not even an influence, could outrun light. In 1907, writing a review article, Einstein had what he later called the happiest thought of his life. A person falling off a roof would not feel their own weight. In free fall, gravity disappears.

That makes gravity unlike any other force. Every object falls with the same acceleration regardless of what it is made of, as Galileo had found, so gravity cannot be a force that acts on some property of the object. It must be a property of the space and time the object moves through. Einstein immediately drew consequences: light must bend near massive bodies, and clocks lower in a gravitational field must run slower.

Chapter II

Gravity Is Geometry

Turning the idea into a theory took eight years and mathematics Einstein did not know. His friend Marcel Grossmann pointed him to Riemannian geometry and the tensor calculus of Ricci and Levi-Civita: the mathematics of curved spaces described entirely from the inside. Spacetime would be a four-dimensional curved manifold, and objects in free fall would follow its geodesics, the straightest possible paths.

After a false start in 1913 and a frantic November of 1915, he had it:

G_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu} .

The left side measures the curvature of spacetime. The right side measures the mass, energy and momentum present. In John Wheeler's summary, spacetime tells matter how to move, and matter tells spacetime how to curve. The week before, Einstein had used his equations to calculate Mercury's orbit and found exactly the 43 arcseconds per century that had defeated Newtonian astronomy since 1859. He wrote that he was beside himself with joy for days.

David Hilbert, working in parallel, submitted a derivation of gravitational field equations five days earlier, and the question of who reached the final equations first has been argued ever since.

Chapter III

Tests from the Sky

The prediction that starlight bends near the Sun could be checked only during a total eclipse. In 1919 Arthur Eddington and Frank Dyson sent expeditions to Brazil and West Africa, and the measured shifts favoured Einstein's value over the Newtonian one. The announcement made Einstein a household name. Whether the eclipse data alone justified such confidence has been debated by historians since, but decades of later tests, from radar echoes off planets to clocks in towers and satellites, have confirmed the theory to high precision.

Chapter IV

Black Holes and Ripples

Within weeks of the final equations, Karl Schwarzschild, writing from the First World War's eastern front, found their exact solution around a single mass. It contained a spherical surface where the equations misbehaved. For decades most physicists, Einstein included, believed nature would never produce such a thing. By the 1960s it was clear that sufficiently massive collapsing stars must, and Wheeler popularised the name black hole.

The theory also predicts that accelerating masses shake spacetime itself, sending out gravitational waves. Einstein doubted they could ever be detected. In 2015 the two LIGO observatories, whose four-kilometre laser arms change length by a small fraction of a proton's width, caught the waves from two black holes merging over a billion light years away. In 2019 the Event Horizon Telescope photographed the glowing ring around a black hole's shadow.

Chapter V

A Closer Look: Forty-Three Seconds of Arc

Newton's gravity predicts that a single planet orbiting the Sun traces the same ellipse forever. In reality Mercury's ellipse slowly turns, its closest point to the Sun advancing by 574 arcseconds per century. The pulls of the other planets account for 531. The remaining 43 arcseconds per century, about a hundredth of a degree, was unexplained from 1859 until 1915.

General relativity predicts that any orbit around a mass $M$ turns a little each lap, by the angle

\Delta\phi = \frac{6\pi G M}{c^2 a (1 - e^2)} ,

where $a$ is the orbit's average radius and $e$ its eccentricity. For Mercury, $a = 5.79 \times 10^{10}$ m and $e = 0.206$ . With the Sun's $GM = 1.327 \times 10^{20}$ m³/s² and $c = 3.00 \times 10^8$ m/s:

\Delta\phi = \frac{6\pi \times 1.327 \times 10^{20}}{(3.00 \times 10^8)^2 \times 5.79 \times 10^{10} \times (1 - 0.206^2)} \approx 5.0 \times 10^{-7} \text{ radians per orbit} .

Mercury completes an orbit every 88 days, about 415 times a century. Converting radians to arcseconds (one radian is 206,265 arcseconds):

5.0 \times 10^{-7} \times 415 \times 206{,}265 \approx 43 \text{ arcseconds per century} .

Einstein found this number in November 1915 with no adjustable constants, and wrote that he was beside himself with joy for days.

The same theory predicts how much starlight bends grazing the Sun: $4GM/(c^2 R)$ , where $R$ is the Sun's radius, about 1.75 arcseconds. That is twice what a Newtonian argument gives, which is why the 1919 eclipse could tell the two apart. Both effects are tiny because $GM/c^2$ for the Sun is only 1.5 km, compared with distances of tens of millions of kilometres. Where that ratio is not small, near black holes, the theory's effects dominate.

Chapter VI

Where the Map Runs Out

General relativity predicts its own breakdown. In 1965 Roger Penrose proved that under very general conditions collapse produces a singularity, where curvature becomes infinite and the equations stop making sense. The universe's own beginning is another. At those points gravity and quantum physics must be combined, and no one knows how. Whether singularities always hide behind horizons is Penrose's cosmic censorship conjecture, a problem now pursued as much by geometric analysts as by physicists.

Applied to the universe as a whole, the theory gave birth to a new science. That is physical cosmology.

Turning points6

1907DISCOVERY
Einstein's "happiest thought"
Writing a review of relativity, Einstein realises that a person falling freely would not feel their own weight. Gravity and acceleration are locally the same thing. He deduces at once that light should bend in a gravitational field and that clocks lower down should run slower.
Albert Einstein
› Sources (1)
- Einstein, A. (1907). Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jahrbuch der Radioaktivität und Elektronik 4: 411–462.
1915THEORY-REPLACEDContested
The field equations of gravitation
After eight years, a false start, and the help of his friend Marcel Grossmann in learning Riemannian geometry and tensor calculus, Einstein presents the final field equations to the Prussian Academy on 25 November 1915. A week earlier he had shown that his theory accounts exactly for the 43 arcseconds per century of Mercury's orbit that Newton could not. Newton's gravity is replaced.
What is disputed
David Hilbert submitted a paper on 20 November 1915, five days before Einstein's final paper, deriving gravitational field equations from a variational principle, and priority was long attributed to him or shared. In 1997 Corry, Renn and Stachel reported printer's proofs of Hilbert's paper, dated 6 December, that do not show the explicit field equations, which suggests he added them after seeing Einstein's. Others point out that part of the proofs has been cut away and dispute that conclusion. That Einstein created the physical theory is not in doubt.
David Hilbert, Albert Einstein, Marcel Grossmann
› Sources (2)
- Einstein, A. (1915). Die Feldgleichungen der Gravitation. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften: 844–847.
- Corry, L., Renn, J. & Stachel, J. (1997). Belated decision in the Hilbert–Einstein priority dispute. Science 278(5341): 1270–1273.
1916DISCOVERY
Schwarzschild's exact solution
Serving on the Russian front, Karl Schwarzschild finds the exact spacetime around a spherical mass within weeks of Einstein's paper. It contains a radius at which the mathematics breaks down. Only decades later was that recognised as the event horizon of what John Wheeler would popularise as a "black hole". Schwarzschild died in 1916 of an illness contracted at the front.
Karl Schwarzschild
› Sources (1)
- Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften: 189–196.
1919EXPERIMENTContested
The eclipse expeditions measure starlight bending
British expeditions to Sobral in Brazil and the island of Príncipe photograph stars near the Sun during a total eclipse. The shifts favour Einstein's predicted deflection of 1.75 arcseconds over the Newtonian half-value. The announcement in November makes Einstein world famous overnight.
What is disputed
Whether the data justified the conclusion is disputed. Earman and Glymour (1980) argued that Eddington and Dyson discarded unfavourable plates from one instrument in a way that suggested bias. Kennefick (2009) and others have since argued that the choices were reasonable on technical grounds, and a 1979 re-analysis of the plates supported Einstein's value. Later radio measurements confirmed the prediction to high precision.
Arthur Eddington, Frank Dyson
› Sources (3)
- Dyson, F. W., Eddington, A. S. & Davidson, C. (1920). A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of May 29, 1919. Philosophical Transactions of the Royal Society A 220: 291–333.
- Earman, J. & Glymour, C. (1980). Relativity and eclipses: the British eclipse expeditions of 1919 and their predecessors. Historical Studies in the Physical Sciences 11(1): 49–85.
- Kennefick, D. (2009). Testing relativity from the 1919 eclipse — a question of bias. Physics Today 62(3): 37–42.
2015 – 2016EXPERIMENT
Gravitational waves are detected directly
On 14 September 2015 the two LIGO detectors in the United States register a passing ripple in spacetime from two black holes merging about 1.3 billion light years away. It changed their four-kilometre arms by a small fraction of a proton's width. It was the first direct detection of gravitational waves and the first observation of a black-hole pair. Rainer Weiss, Barry Barish and Kip Thorne shared the 2017 Nobel Prize.
Rainer Weiss, Barry Barish, Kip Thorne
› Sources (1)
- Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). Observation of gravitational waves from a binary black hole merger. Physical Review Letters 116: 061102.
2019EXPERIMENT
The first image of a black hole's shadow
The Event Horizon Telescope, a planet-sized array of radio dishes, images the ring of light around the black hole at the centre of the galaxy M87. Its size matches general relativity's prediction for a mass of about 6.5 billion Suns.
› Sources (1)
- Event Horizon Telescope Collaboration (2019). First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole. Astrophysical Journal Letters 875: L1.

Applications

Where it is used

Navigation
GPS clocks correct for gravity
Clocks on GPS satellites, higher up in Earth's gravity, run about 45 microseconds a day fast compared with the ground, partly offset by 7 microseconds of special-relativistic slowing. Without the correction, position fixes would drift by kilometres a day.
› Sources (1)
- Ashby, N. (2003). Relativity in the Global Positioning System. Living Reviews in Relativity 6: 1.
Geodesy
Measuring height with clocks
Today's best optical atomic clocks are so precise that raising one by about 30 centimetres measurably speeds it up, as general relativity predicts. Comparing clocks can map Earth's gravity field and heights, a technique called relativistic geodesy.
› Sources (1)
- Chou, C. W., Hume, D. B., Rosenband, T. & Wineland, D. J. (2010). Optical clocks and relativity. Science 329(5999): 1630–1633.
Astronomy
Gravitational lenses as cosmic telescopes
Massive galaxy clusters bend light from galaxies behind them, magnifying objects too faint to see otherwise and revealing the invisible mass doing the bending. Lensing is one of the main ways dark matter is mapped, and small-scale lensing finds planets around other stars.
› Sources (1)
- Clowe, D. et al. (2006). A direct empirical proof of the existence of dark matter. Astrophysical Journal Letters 648: L109–L113.
Astrophysics
Where gold comes from
In 2017 gravitational waves from two merging neutron stars were caught together with light from the explosion. The spectra showed freshly made heavy elements, confirming that such mergers forge much of the universe's gold and platinum.
› Sources (1)
- Abbott, B. P. et al. (2017). GW170817: Observation of gravitational waves from a binary neutron star inspiral. Physical Review Letters 119: 161101.

Open problems

Where the map runs out

Open

Quantum gravity

No accepted theory as of 2026; candidates include string theory and loop quantum gravity.

General relativity treats spacetime as smooth and definite. Quantum theory, which describes everything else, treats all physical quantities as uncertain and fluctuating. Near a black hole's singularity or at the Big Bang both must apply, and no one knows how to combine them.

Why it is hard: Applying the standard quantum methods to gravity produces infinities that cannot be tamed. The candidate theories are mathematically demanding, differ on what spacetime fundamentally is, and make predictions at the Planck scale, roughly $10^{-35}$ m, far beyond any foreseeable experiment.
What resolving it unlocks: It would explain what happens inside black holes and at the beginning of the universe, and might reveal what space and time are made of.

› Sources (2)

Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.
Polchinski, J. (1998). String Theory (2 vols.). Cambridge University Press.

Open

The black hole information paradox

Recent calculations suggest information does escape, but how is still debated as of 2026.

Hawking showed in 1974–76 that black holes slowly evaporate by emitting radiation that appears perfectly random. If a black hole evaporates completely, the information about what fell in seems to be destroyed. Quantum mechanics forbids that.

Why it is hard: The paradox sits exactly where general relativity and quantum theory meet, and each proposed resolution gives up something cherished: locality, smooth horizons, or quantum unitarity. Calculations since 2019 reproduce the expected release of information, but without a full theory of quantum gravity the mechanism is unclear.
What resolving it unlocks: Resolving it is widely seen as the best route to quantum gravity, and to understanding how spacetime might emerge from quantum information.

› Sources (2)

Hawking, S. W. (1976). Breakdown of predictability in gravitational collapse. Physical Review D 14: 2460–2473.
Almheiri, A., Hartman, T., Maldacena, J., Shaghoulian, E. & Tajdini, A. (2021). The entropy of Hawking radiation. Reviews of Modern Physics 93: 035002.

Conjectured, unproven

Cosmic censorship

Proposed by Penrose in 1969; unproven in general.

General relativity predicts singularities, points where curvature becomes infinite and the theory breaks down. Roger Penrose conjectured that realistic collapse always hides them behind event horizons, so no "naked" singularity is ever visible to the outside universe.

Why it is hard: It is a statement about all possible solutions of nonlinear equations with generic starting conditions. Special counterexamples exist, so any proof must show that they are unstable and non-generic. The mathematics is at the frontier of geometric analysis.
What resolving it unlocks: It would guarantee that general relativity remains predictive outside black holes. Its sharper cousin, the Penrose inequality, is an open problem in the mathematics survey.

› Sources (1)

Penrose, R. (1969). Gravitational collapse: the role of general relativity. Rivista del Nuovo Cimento 1: 252–276.