History of general relativity

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 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 thought about 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 see that the rate of clocks at the top of a box accelerating upward would be faster than the rate of clocks at the bottom. He concludes that the rates of clocks depend on their position in a gravitational field, and that the difference in rate is proportional to the gravitational potential to first approximation.

Also the deflection of light by massive bodies was predicted. Although the approximation was crude, it allowed him to calculate that the deflection is nonzero. German astronomer Erwin Finlay-Freundlich publicized Einstein's challenge to scientists around the world.[4] This urged astronomers to detect the deflection of light during a solar eclipse, and gave Einstein confidence that the scalar theory of gravity proposed by Gunnar Nordström was incorrect. But the actual value for the deflection that he calculated was too small by a factor of two, because the approximation he used doesn't work well for things moving at near the speed of light. When Einstein finished the full theory of general relativity, he would rectify this error and predict the correct amount of light deflection by the sun.

Another of Einstein's notable thought experiments about the nature of the gravitational field is that of the 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.

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 the Royal Society and the Royal Astronomical Society.[5] (Full text)

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.[6] This theory explains gravitation as 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 ended, 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.[7]

In 1917, several astronomers accepted Einstein's 1911 challenge from Prague. The Mount Wilson Observatory in California, U.S., published a solar spectroscopic analysis that showed no gravitational redshift.[8] In 1918, the Lick Observatory, also in California, announced that it too had disproved Einstein's prediction, although its findings were not published.[9]

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";[10] fellow laureate Paul Dirac was quoted saying it was "probably the greatest scientific discovery ever made".[11]

There have been claims that scrutiny of the specific photographs taken on the Eddington expedition showed the experimental uncertainty to be comparable to the same magnitude as the effect Eddington claimed to have demonstrated, and that a 1962 British expedition concluded that the method was inherently unreliable.[12] The deflection of light during a solar eclipse was confirmed by later, more accurate observations.[13] Some resented the newcomer's fame, notably among some nationalistic German physicists, who later started the Deutsche Physik (German Physics) movement.[14][15]

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.[16]

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 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:[17]

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

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.

Although Einstein is credited with finding the field equations, the German mathematician David Hilbert published them in an article before Einstein's article. This has resulted in accusations of plagiarism against Einstein, although not from Hilbert, and assertions that the field equations should be called the "Einstein–Hilbert field equations". However, Hilbert did not press his claim for priority and some have asserted that Einstein submitted the correct equations before Hilbert amended his own work to include them. This suggests that Einstein developed the correct field equations first, though Hilbert may have reached them later independently (or even learned of them afterwards through his correspondence with Einstein).[18] However, others have criticized those assertions.[19]

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,[20] 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."

Since the field equations are non-linear, Einstein assumed that they were unsolvable. However, Karl Schwarzschild discovered in 1915 and published in 1916[21] 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.

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 apparently 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 idea that the universe is 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).

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 in 1955), 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.