General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Raychaudhuri Formalisms ADM NP BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Einstein–Rosen waves Wormhole Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Kantowski-Sachs Lemaître–Tolman Wahlquist Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Hartle–Thorne Vaidya Peres De Sitter-Schwarzschild McVittie Weyl Kerr-Newman-de-Sitter
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General relativity is atheory of gravitation that was developed byAlbert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping ofspace and time by those masses.

Before the advent of general relativity,Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses, even though Newton himself did not regard the theory as the final word on the nature of gravity. Within a century of Newton's formulation, careful astronomical observation revealed unexplainable differences between the theory and the observations. Under Newton's model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered by the unknown nature of that force, the basic framework was extremely successful at describing motion.

However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such asgravitational waves,gravitational lensing and an effect of gravity on time known asgravitational time dilation. Many of these predictions have been confirmed by experiment or observation, while others are the subject of ongoing research.

General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei ormicroquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.

The first relativistic theory of gravity was proposed byHenri Poincaré in 1905. He published a Lorentz invariant theory on four-dimensional spacetime, where gravity is transmitted bygravitational waves that travel at the speed of light.

As Einstein later said, the reason for the development of general relativity was the preference of inertial motion withinspecial relativity, while a theory which from the outset prefers no particular state of motion appeared more satisfactory to him.^[1] So, while still working at the patent office in 1907, Einstein had what he would call his "happiest thought". He realized that theprinciple of relativity could be extended to gravitational fields.

Consequently, in 1907 he wrote an article, published in 1908, on acceleration under special relativity.^[2]In that article, he argued thatfree fall is really inertial motion, and that for a freefalling observer the rules of special relativity must apply. This argument is called theequivalence principle. In the same article, Einstein also predicted the phenomenon of gravitational time dilation.

In 1911, Einstein published another article expanding on the 1907 article.^[3]There, he considered the case of a uniformly accelerated box not in a gravitational field, and noted that it would be indistinguishable from a box sitting still in an unchanging gravitational field. He used special relativity to show that clocks at the top of a box accelerating upward would run faster than clocks at the bottom. He concluded that the rate at which time passes depends on the position in a gravitational field, and that the difference in rate is proportional to the gravitational potential to a first approximation.

The article also predicted thedeflection of light by massive bodies, e.g., Jupiter, the Sun. Although the approximation was crude, it allowed him to calculate that the deflection is nonzero. Einstein urged astronomers to attempt direct observation of light deflection of fixed stars near the Sun during solar eclipses when they would be visible.^[3] German astronomerErwin Finlay-Freundlich publicized Einstein's challenge to scientists around the world.^[4]

In October 1911, Freundlich contacted astronomerCharles D. Perrine in Berlin to inquire as to the suitability of examining existing solar eclipse photographs to prove Einstein's prediction of light deflection. Perrine, the director of theArgentine National Observatory at Cordoba, had participated in four solar eclipse expeditions while at theLick Observatory, in 1900, 1901, 1905, and 1908. "...he had become, in the opinion of the director of the Lick Observatory,W. W. Campbell, an observer without peer in the field of solar eclipses."^[5] He did not believe existing eclipse photos would be useful in proving Einstein's claim. In 1912 Freundlich asked if Perrine would include observation of light deflection as part of his program for the solar eclipse of October 10, 1912, in Brazil. W. W. Campbell, director of the Lick Observatory, loaned Perrine its intramercurial camera lenses. Perrine and the Cordoba team were the only eclipse expedition to construct specialized equipment dedicated to observing light deflection. Unfortunately all the expeditions experienced heavy rain which prevented any observations. Nevertheless, Perrine was the first astronomer to make a dedicated attempt to observe light deflection to test Einstein's prediction.^[6]

Two years later, the three observatory directors, Perrine, Freundlich, and Campbell included light deflection in their expeditions to the Russian Empire for the solar eclipse of August 21, 1914. Unfortunately due to clouds and the outbreak ofWorld War I, no results were possible.^[6][7] However, Perrine was able to take the first photographs in an attempt to verify Einstein's prediction of light deflection. A light cloud cover prevented determining accurate star positions.^[8]

In hindsight, the occluding weather and lack of results in 1912 and 1914 favored Einstein. If clear photographs and measurable results had been possible, Einstein's 1911 prediction might have been proven wrong. The amount of deflection that he calculated in 1911 was too small (0.83 seconds of arc) by a factor of two because the approximation he used does not work well for things moving at near the speed of light. When Einstein completed the full theory of general relativity in 1915, he rectified this error and predicted the correct amount of light deflection caused by the Sun (1.75 seconds of arc). Eddington and Dyson in 1919^[9] and W. W. Campbell in 1922^[10] were able to compare their results to Einstein's corrected prediction.

Another of Einstein's notable thought experiments about the nature of the gravitational field is that of a rotating disk (a variant of theEhrenfest paradox). He imagined an observer performing experiments on a rotating turntable. He noted that such an observer would find a different value for the mathematical constant π than the one predicted by Euclidean geometry. The reason is that the radius of a circle would be measured with an uncontracted ruler, but, according to special relativity, the circumference would seem to be longer because the ruler would be contracted. Since Einstein believed that the laws of physics were local, described by local fields, he concluded from this that spacetime could be locally curved. This led him to studyRiemannian geometry, and to formulate general relativity in this language.

Eddington's photograph of a solar eclipse, which confirmed Einstein's theory that light "bends".

The New York Times reported confirmation of "the Einstein theory" (specifically, the bending of light by gravitation) based on 29 May 1919 eclipse observations in Principe (Africa) and Sobral (Brazil), after the findings were presented on 6 November 1919 to a joint meeting in London of theRoyal Society and theRoyal Astronomical Society.^[11] (Full text)

In 1912, Einstein returned to Switzerland to accept a professorship at hisalma mater,ETH Zurich. Once back in Zurich, he immediately visited his old ETH classmateMarcel Grossmann, now a professor of mathematics, who introduced him toRiemannian geometry and, more generally, todifferential geometry. On the recommendation of Italian mathematicianTullio Levi-Civita, Einstein began exploring the usefulness ofgeneral covariance (essentially the use oftensors) for his gravitational theory. For a while, Einstein thought that there were problems with the approach, but he later returned to it and, by late 1915, had published hisgeneral theory of relativity in the form in which it is used today.^[12] This theory explains gravitation as the distortion of the structure ofspacetime by matter, affecting theinertial motion of other matter.

During World War I, the work ofCentral Powers scientists was available only to Central Powers academics, for national security reasons. Some of Einstein's work did reach the United Kingdom and the United States through the efforts of the AustrianPaul Ehrenfest and physicists in the Netherlands, especially 1902 Nobel Prize-winnerHendrik Lorentz andWillem de Sitter ofLeiden University. After the war, Einstein maintained his relationship with Leiden University, accepting a contract as anExtraordinary Professor; for ten years, from 1920 to 1930, he travelled to the Netherlands regularly to lecture.^[13]

In 1917, several astronomers accepted Einstein's 1911 challenge from Prague. TheMount Wilson Observatory in California, United States, published a solarspectroscopic analysis that showed no gravitational redshift.^[14] In 1918, theLick Observatory, also in California, announced that it too had disproved Einstein's prediction, although its findings were not published.^[15]

However, in May 1919, a team led by the British astronomerArthur Stanley Eddington claimed to have confirmed Einstein's prediction of gravitational deflection of starlight by the sun while photographing a solar eclipse with dual expeditions inSobral, northern Brazil, andPríncipe, a west African island.^[4] Nobel laureateMax Born praised general relativity as the "greatest feat of human thinking about nature";^[16] fellow laureatePaul Dirac was quoted saying it was "probably the greatest scientific discovery ever made".^[17]

There have been claims that scrutiny of the specific photographs taken on the Eddington expedition showed the experimental uncertainty to be comparable to the magnitude of the effect Eddington claimed to have demonstrated, and that a 1962 British expedition concluded that the method was inherently unreliable.^[18] The deflection of light during a solar eclipse was confirmed by later, more accurate observations.^[19] Some resented the newcomer's fame, notably some nationalistic German physicists, who later started theDeutsche Physik (German Physics) movement.^[20][21]

By 1912, Einstein was actively seeking a theory in whichgravitation was explained as ageometric phenomenon. At the urging of Tullio Levi-Civita, Einstein began by exploring the use of general covariance (which is essentially the use of curvaturetensors) to create a gravitational theory. However, in 1913 Einstein abandoned that approach, arguing that it is inconsistent based on the "hole argument". In 1914 and much of 1915, Einstein was trying to createfield equations based on another approach. When that approach was proven to be inconsistent, Einstein revisited the concept of general covariance and discovered that the hole argument was flawed.^[22]

When Einstein realized that general covariance was tenable, he quickly completed the development of the field equations that are named after him. However, he made a now-famous mistake. The field equations he published in October 1915 were

${\displaystyle R_{\mu \nu }=\kappa T_{\mu \nu }}$,

where${\displaystyle R_{\mu \nu }}$ is theRicci tensor,${\displaystyle T_{\mu \nu }}$ theenergy–momentum tensor and${\displaystyle \kappa }$ isEinstein gravitational constant. This predicted the non-Newtonianperihelion precession ofMercury, and so had Einstein very excited. However, it was soon realized^[23] that they were inconsistent with the localconservation of energy–momentum unless the universe had a constant density of mass–energy–momentum. In other words, air, rock and even a vacuum should all have the same density. This inconsistency with observation sent Einstein back to the drawing board and, on 25 November 1915, Einstein presented the updated Einstein field equations to thePrussian Academy of Sciences:^[24]

${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\kappa T_{\mu \nu }}$,

where${\displaystyle R}$ is theRicci scalar and${\displaystyle g_{\mu \nu }}$ themetric tensor. With the publication of the field equations, the issue became one of solving them for various cases and interpreting the solutions. This and experimental verification have dominated general relativity research ever since.

In the last year of Einstein's work on general relativity he met with and corresponded with the German mathematicianDavid Hilbert. Hilbert had been working on a unified field theory based on the ideas ofGustav Mie; he derived the theory of general relativity from an elegant variational principlealmost simultaneously with Einstein's discovery of the theory.^[25]: 170 The timing of the correspondence and publications has led to a number of in depth historical analyses.

In the early years after Einstein's theory was published,Sir Arthur Eddington lent his considerable prestige in the British scientific establishment in an effort to champion the work of this German scientist. Because the theory was so complex and abstruse (even today it is popularly considered the pinnacle of scientific thinking; in the early years it was even more so), it was rumored that only three people in the world understood it. There was an illuminating, though probably apocryphal, anecdote about this. As related byLudwik Silberstein,^[26] during one of Eddington's lectures he asked "Professor Eddington, you must be one of three persons in the world who understands general relativity." Eddington paused, unable to answer. Silberstein continued "Don't be modest, Eddington!" Finally, Eddington replied "On the contrary, I'm trying to think who the third person is."

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Since the field equations arenon-linear, Einstein assumed that they were unsolvable.^{[citation needed]} However,Karl Schwarzschild discovered in 1915 and published in 1916^[27] an exact solution for the case of a spherically symmetric spacetime surrounding a massive object inspherical coordinates. This is now known as theSchwarzschild solution. Since then, many other exact solutions have been found.

In 1922,Alexander Friedmann found a solution in which the universe may expand or contract, and laterGeorges Lemaître derived a solution for an expanding universe. However, Einstein believed that the universe was static, and since a static cosmology was not supported by the general relativistic field equations, he added acosmological constant${\displaystyle \mathrm{\Lambda }}$ to the field equations, which became:

${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$.

This permitted the creation ofsteady-state solutions, but they were unstable: the slightest perturbation of a static state would result in the universe expanding or contracting. In 1929,Edwin Hubble found evidence for the universe expanding. This resulted in Einstein dropping the cosmological constant, referring to it as "the biggest blunder in my career". At the time, it was anad hoc hypothesis to add in the cosmological constant, as it was only intended to justify one result (a static universe).

Progress in solving the field equations and understanding the solutions has been ongoing. The solution for a spherically symmetric charged object was discovered by Reissner and later rediscovered by Nordström, and is called theReissner–Nordström solution. The black hole aspect of the Schwarzschild solution was very controversial, and Einstein did not believe that singularities could be real. However, in 1957 (two years after Einstein's death),Martin Kruskal published a proof that black holes are called for by the Schwarzschild solution. Additionally, the solution for a rotating massive object was obtained byRoy Kerr in the 1960s and is called theKerr solution. TheKerr–Newman solution for a rotating, charged massive object was published a few years later.

The first piece of evidence in support of general relativity came from its correct prediction of the anomalous rate of precession of Mercury's orbit. Subsequently, Arthur Stanley Eddington's 1919 expedition confirmed Einstein's prediction of the deflection of light by the Sun during the totalsolar eclipse of 29 May 1919, which helped to cement the status of general relativity as a viable theory. Since then, many observations have shown agreement with the predictions of general relativity. These include studies ofbinary pulsars, observations of radio signals passing the limb of the Sun, and even theglobal positioning system.

The theory predictsgravitational waves, which are ripples in thecurvature of spacetime that propagate aswaves, travelling outward from the source. Thefirst observation of gravitational waves, which came from the merger of twoblack holes, was made on 14 September 2015 by theAdvanced LIGO team, corroborating another prediction of the theory 100 years after it was published.^[31][32][33]

The first image of a black hole, the supermassive one at the center of galaxyMessier 87, was published by theEvent Horizon Telescope Collaboration on 10 April 2019.^[34]

There have been various attempts to find modifications to general relativity. The most famous of these are theBrans–Dicke theory (also known asscalar–tensor theory), andRosen's bimetric theory. Both of these theories proposed changes to the field equations of general relativity, and both suffer from these changes permitting the presence of bipolar gravitational radiation. As a result, Rosen's original theory has been refuted by observations of binary pulsars. As for Brans–Dicke (which has a tunable parameterω such that ω = ∞ is the same as general relativity), the amount by which it can differ from general relativity has been severely constrained by these observations. Many other alternatives to general relativity have also been ruled out by analyses of the neutron-star mergerGW170817.^{[35][36][37][38]}

In addition, general relativity is inconsistent withquantum mechanics, the physical theory that describes the wave–particle duality of matter, and quantum mechanics does not currently describe gravitational attraction at relevant (microscopic) scales. There is a great deal of speculation in the physics community as to the modifications that might be needed to both general relativity and quantum mechanics in order to unite them consistently. The speculative theory that unites general relativity and quantum mechanics is usually calledquantum gravity, prominent examples of which includestring theory andloop quantum gravity.

Kip Thorne identifies the "golden age of black hole research" as the period roughly from 1960 to 1975, during which the study ofgeneral relativity,^[39] which had previously been regarded as something of a curiosity, entered the mainstream oftheoretical physics.^[40]: 258 During this period, many of the concepts and terms which continue to inspire the imaginations of gravitation researchers and the general public were introduced, including black holes andgravitational singularities. At the same time, in a closely related development, the study ofphysical cosmology entered the mainstream and theBig Bang became well established.

Fulvio Melia refers frequently to the "golden age of relativity" in his bookCracking the Einstein Code.Andrzej Trautman hosted a relativity conference inWarsaw in 1962 to which Melia refers:

General relativity moved very successfully from that meeting in Warsaw, hot on the heels of thePound–Rebka experiment, and entered its golden age of discovery that lasted into the mid-1970s.^[41]

Roy Kerr, protagonist of the book, contributed an Afterword, saying of the book: "It is a remarkable piece of writing capturing beautifully the period we now refer to as the golden age of relativity."^[42]

• List of contributors to general relativity
• Golden age of cosmology
• Golden age of physics
• Mach's principle
• Timeline of gravitational physics and relativity
• W. K. Clifford#Premonition of relativity

• Works related toThe Foundation of the Generalised Theory of Relativity at Wikisource

• General relativity
• History of physics
• Albert Einstein

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