General relativity

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}




Equations


Formalisms


Advanced theory






General relativity (GR) is a theory of gravitation that was developed by Albert 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 of space 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 variations 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 as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment, while others are the subject of ongoing research. For example, although there is indirect evidence for gravitational waves, direct evidence of their existence is still being sought by several teams of scientists in experiments such as the LIGO and GEO 600 projects.
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 or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Contents

Creation of general relativity 1

Early investigations 1.1

Developing general relativity 1.2

General covariance and the hole argument 1.3

The development of the Einstein field equations 1.4

Einstein and Hilbert 1.5

Sir Arthur Eddington 1.6

Solutions 2

The Schwarzschild solution 2.1

The expanding universe and the cosmological constant 2.2

More exact solutions 2.3

Testing the theory 3

Alternative theories 4

More about GR history 5

See also 6

Notes 7

References 8
Creation of general relativity
Albert Einstein developed the theories of special and general relativity. Picture from 1921.
Early investigations
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 state of motion (even accelerated ones) 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 (published 1908) he wrote an article 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 FinlayFreundlich 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.
Developing general relativity
Eddington's photograph of a solar eclipse, which confirmed Einstein's theory that light "bends".
In 1912, Einstein returned to Switzerland to accept a professorship at his alma mater, the ETH. 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 LeviCivita, 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.^{[5]} 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 Prizewinner 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 Holland regularly to lecture.^{[6]}
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.^{[7]} In 1918, the Lick Observatory, also in California, announced that it too had disproved Einstein's prediction, although its findings were not published.^{[8]}
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";^{[9]} fellow laureate Paul Dirac was quoted saying it was "probably the greatest scientific discovery ever made".^{[10]} The international media guaranteed Einstein's global renown.
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.^{[11]} The deflection of light during a solar eclipse was confirmed by later, more accurate observations.^{[12]} Some resented the newcomer's fame, notably among some German physicists, who later started the Deutsche Physik (German Physics) movement.^{[13]}^{[14]}
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 LeviCivita, 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.
The development of the Einstein field equations
When Einstein realized that general covariance was actually tenable, he quickly completed the development of the field equations that are named after him. However, he made a nowfamous mistake. The field equations he published in October 1915 were

R_{\mu\nu} = T_{\mu\nu}\,,
where R_{\mu\nu} is the Ricci tensor, and T_{\mu\nu} the energy–momentum tensor. This predicted the nonNewtonian 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. However, the solution was all but obvious, and on November 25, 1915 Einstein presented the actual Einstein field equations to the Prussian Academy of Sciences:^{[15]}

R_{\mu\nu}  {1\over 2}R g_{\mu\nu} = T_{\mu\nu},
where R is the Ricci scalar and 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
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).^{[16]} However, others have criticized those assertions.^{[17]}
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,^{[18]} 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."
Solutions
The Schwarzschild solution
Since the field equations are nonlinear, Einstein assumed that they were unsolvable. However, in 1915 Karl Schwarzschild discovered 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, cosmological constant Λ to the field equations, which became

R_{\mu\nu}  {1\over 2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\mu\nu}.
This permitted the creation of steadystate 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).
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 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 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.
Testing the theory
The perihelion precession of Mercury was the first evidence that general relativity is correct. Sir Arthur Stanley Eddington's 1919 expedition in which he confirmed Einstein's prediction for the deflection of light by the Sun during the total solar eclipse of 29 May 1919 helped to cement the status of general relativity as a likely true theory. Since then many observations have confirmed the correctness of general relativity. These include studies of binary pulsars, observations of radio signals passing the limb of the Sun, and even the GPS system.
Alternative theories
There have been various attempts to find modifications to general relativity. The most famous of these are the Brans–Dicke theory (also known as scalartensor theory), and Rosen'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 parameter ω 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.
In addition, general relativity is inconsistent with quantum 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 called quantum gravity, prominent examples of which include String Theory and Loop Quantum Gravity.
More about GR history
Kip Thorne identifies the "golden age of general relativity" as the period roughly from 1960 to 1975 during which the study of general relativity,^{[19]} which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics. During this period, many of the concepts and terms which continue to inspire the imagination of gravitation researchers and the general public were introduced, including black holes and 'gravitational singularity'. At the same time, in a closely related development, the study of physical cosmology entered the mainstream and the Big Bang became well established.
See also
Notes

^ Albert Einstein, Nobel lecture in 1921

^ Einstein, A., "Relativitätsprinzip und die aus demselben gezogenen Folgerungen (On the Relativity Principle and the Conclusions Drawn from It)", Jahrbuch der Radioaktivität (Yearbook of Radioactivity) 4: 411–462 page 454 (Wir betrachen zwei Bewegung systeme ...)

^ Einstein, Albert (1911), "Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravity on the Propagation of Light)", Annalen der Physik 35: 898–908, (also in Collected Papers Vol. 3, document 23)

^ ^{a} ^{b} Crelinsten, Jeffrey. "Einstein's Jury: The Race to Test Relativity". Princeton University Press. 2006. Retrieved on 13 March 2007. ISBN 9780691123103

^ O'Connor, J.J. and E.F. Robertson (1996), "General relativity". Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland, May, 1996. Retrieved 20150204.

^ Two friends in Leiden, retrieved 11 June 2007

^ Crelinsten, Jeffrey (2006), Einstein's Jury: The Race to Test Relativity, Princeton University Press, pp. 103–108,

^ Crelinsten, Jeffrey (2006), Einstein's Jury: The Race to Test Relativity, Princeton University Press, pp. 114–119,

^ Smith, PD (17 September 2005), The genius of space and time, London: The Guardian, retrieved 31 March 2007

^ Jürgen Schmidhuber. "Albert Einstein (1879–1955) and the 'Greatest Scientific Discovery Ever'". 2006. Retrieved on 4 October 2006.

^ Andrzej, Stasiak (2003), "Myths in science", EMBO Reports 4 (3): 236,

^ See the table in MathPages Bending Light

^ Hentschel, Klaus and Ann M. (1996), Physics and National Socialism: An Anthology of Primary Sources, Birkhaeuser Verlag, xxi,

^ For a discussion of astronomers' attitudes and debates about relativity, see Crelinsten, Jeffrey (2006), Einstein's Jury: The Race to Test Relativity, Princeton University Press, , especially chapters 6, 9, 10 and 11.

^

^ Leo Corry, Jürgen Renn, John Stachel: "Belated Decision in the HilbertEinstein Priority Dispute", SCIENCE, Vol. 278, 14 November 1997  article text

^ Friedwart Winterberg's response to the CoryRennStachel paper as printed in "Zeitschrift für Naturforschung" 59a, 715719.

^ John Waller (2002), Einstein's Luck, Oxford University Press, ISBN 0198607199

^ Thorne, Kip (2003). "Warping spacetime". The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday. Cambridge University Press. p. 74. , Extract of page 74
References

Pais, Abraham (1982). Subtle is the lord: the science and life of Albert Einstein. Oxford: Oxford University Press.

Einstein, A.;

Einstein and the Changing Worldviews of Physics (editors—Lehner C., Renn J., Schemmel M.) 2012 (Birkhäuser).

Genesis of general relativity series
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