Early investigations

The first relativistic theory of gravity was proposed by Henri Poincaré in 1905. He published a Lorentz invariant theory on four-dimensional spacetime, where gravity is transmitted by gravitational 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 within special 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 the principle 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 that free fall is really inertial motion, and that for a freefalling observer the rules of special relativity must apply. This argument is called the equivalence 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 the deflection 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 astronomer Erwin Finlay-Freundlich publicized Einstein's challenge to scientists around the world.^[4]

In October 1911, Freundlich contacted astronomer Charles 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 the Argentine National Observatory at Cordoba, had participated in four solar eclipse expeditions while at the Lick 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 of World 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 the Ehrenfest 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 study Riemannian geometry, and to formulate general relativity in this language.

Developing general relativity

In 1912, Einstein returned to Switzerland to accept a professorship at his alma mater, ETH Zurich. Once back in Zurich, he immediately visited his old ETH classmate Marcel Grossmann, now a professor of mathematics, who introduced him to Riemannian geometry and, more generally, to differential geometry. On the recommendation of Italian mathematician Tullio Levi-Civita, Einstein began exploring the usefulness of general covariance (essentially the use of tensors) 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 his general theory of relativity in the form in which it is used today.^[12] This theory explains gravitation as the distortion of the structure of spacetime by matter, affecting the inertial motion of other matter.

During World War I, the work of Central 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 Austrian Paul Ehrenfest and physicists in the Netherlands, especially 1902 Nobel Prize-winner Hendrik Lorentz and Willem de Sitter of Leiden University. After the war, Einstein maintained his relationship with Leiden University, accepting a contract as an Extraordinary 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. The Mount Wilson Observatory in California, United States, published a solar spectroscopic analysis that showed no gravitational redshift.^[14] In 1918, the Lick 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 astronomer Arthur 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 in Sobral, northern Brazil, and Príncipe, a west African island.^[4] Nobel laureate Max Born praised general relativity as the "greatest feat of human thinking about nature";^[16] fellow laureate Paul 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 the Deutsche Physik (German Physics) movement.^[20][21]

General covariance and the hole argument

By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. At the urging of Tullio Levi-Civita, Einstein began by exploring the use of general covariance (which is essentially the use of curvature tensors) 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 create field 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]

The development of the Einstein field equations

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

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

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, and ${\displaystyle T_{\mu \nu }}$ the energy–momentum tensor. This predicted the non-Newtonian perihelion precession of Mercury, and so had Einstein very excited. However, it was soon realized^[23] that they were inconsistent with the local conservation 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 the Prussian Academy of Sciences:^[24]

${\displaystyle R_{\mu \nu }-{1 \over 2}Rg_{\mu \nu }=T_{\mu \nu }}$,

where ${\displaystyle R}$ is the Ricci scalar and ${\displaystyle g_{\mu \nu }}$ the metric 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.

Einstein and Hilbert

In the last year of Einstein's work on general relativity he met with and corresponded with the German mathematician David Hilbert. Hilbert had been working on a unified field theory based on the ideas of Gustav Mie; he derived the theory of general relativity from an elegant variational principle almost 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.

Sir Arthur Eddington

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 by Ludwik 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."

The Schwarzschild solution

Since the field equations are non-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 in spherical coordinates. This is now known as the Schwarzschild solution. Since then, many other exact solutions have been found.

The expanding universe and the cosmological constant

In 1922, Alexander Friedmann found a solution in which the universe may expand or contract, and later Georges 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 a cosmological constant Λ to the field equations, which became

${\displaystyle R_{\mu \nu }-{1 \over 2}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=T_{\mu \nu }}$.

This permitted the creation of steady-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 an ad hoc hypothesis to add in the cosmological constant, as it was only intended to justify one result (a static universe).

More exact solutions

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 the Reissner–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 by Roy Kerr in the 1960s and is called the Kerr solution. The Kerr–Newman solution for a rotating, charged massive object was published a few years later.