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{{Short description|Theory of gravitation as curved spacetime}}
{{For|the graduate textbook by Robert Wald|General Relativity (book){{!}}''General Relativity'' (book)}}
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[[File:BBH gravitational lensing of gw150914.webm|266px |thumb|Slow motion computer simulation of the black hole [[binary system]] GW150914 as seen by a nearby observer, during 0.33 s of its final [[Binary black hole#Inspiral|inspiral]], [[Binary black hole#Merger|merge]], and [[Binary black hole#Ringdown|ringdown]]. The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extreme [[gravitational lens]]ing, as [[spacetime]] itself is distorted and dragged around by the rotating [[black hole]]s.<ref name="SXSproject">{{cite web |url=http://www.black-holes.org/gw150914 |title=GW150914: LIGO Detects Gravitational Waves |website=Black-holes.org |access-date=18 April 2016}}</ref>]]
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'''General relativity''', also known as the '''general theory of relativity''', and as '''Einstein's theory of gravity''', is the [[differential geometry|geometric]] [[scientific theory|theory]] of [[gravitation]] published by [[Albert Einstein]] in 1915 and is the current description of gravitation in [[modern physics]]. General [[theory of relativity|relativity]] generalizes [[special relativity]] and refines [[Newton's law of universal gravitation]], providing a unified description of gravity as a geometric property of [[space]] and [[time in physics|time]], or [[four-dimensional space|four-dimensional]] [[spacetime]]. In particular, the ''[[curvature]] of spacetime'' is directly related to the [[energy]] and [[momentum]] of whatever is 
present, including [[matter]] and [[radiation]]. The relation is specified by the [[Einstein field equations]], a system of second-order [[partial differential equation]]s.

[[Newton's law of universal gravitation]], which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond [[Newton's law of universal gravitation]] in [[classical physics]]. These predictions concern the passage of time, the [[geometry]] of space, the motion of bodies in [[free fall]], and the propagation of light, and include [[gravitational time dilation]], [[gravitational lens]]ing, the [[gravitational redshift]] of light, the [[Shapiro time delay]] and [[Gravitational singularity|singularities]]/[[black holes]]. So far, all [[tests of general relativity]] have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for [[cosmology]], thus leading to the discovery of the [[Big Bang]] and [[cosmic microwave background]] radiation. Despite the introduction of a number of [[Alternatives to general relativity|alternative theories]], general relativity continues to be the simplest theory consistent with [[experimental data]].

Reconciliation of general relativity with the laws of [[quantum mechanics|quantum physics]] remains a problem, however, as there is a lack of a self-consistent theory of [[quantum gravity]]. It is not yet known how gravity can be [[Theory of everything|unified]] with the three non-gravitational forces: [[strong interaction|strong]], [[weak interaction|weak]] and [[electromagnetism|electromagnetic]].

Einstein's theory has [[astrophysical]] implications, including the prediction of [[black holes]]—regions of space in which space and time are distorted in such a way that nothing, not even [[photon|light]], can escape from them. Black holes are the end-state for [[massive star]]s. [[Microquasar]]s and [[active galactic nucleus|active galactic nuclei]] are believed to be [[stellar black hole]]s and [[supermassive black hole]]s. It also predicts [[gravitational lensing]], where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of [[gravitational wave]]s, which have been [[List of gravitational wave observations|observed directly]] by the physics collaboration [[LIGO]] and other observatories. In addition, general relativity has provided the base of [[Physical cosmology|cosmological]] models of an [[Metric expansion of space|expanding universe]].

<!-- Before any further attempt of deleting the following claim, please, note its sources in the History section, and the conclusive consensus on the TP. -->
Widely acknowledged as a theory of extraordinary [[Mathematical beauty|beauty]], general relativity has often been described as the most beautiful of all existing physical theories.<ref name=":0" />

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== History ==
{{Main|History of general relativity|Classical theories of gravitation}}
[[Henri Poincaré]]'s 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity.  While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that [[gravitational waves]] propagate at the speed of light.<ref>{{Harvnb|Poincaré|1905}}</ref> Soon afterwards, Einstein started thinking about how to incorporate [[gravity]] into his relativistic framework. In 1907, beginning with a simple [[thought experiment]] involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the [[Prussian Academy of Science]] in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity.<ref>{{cite web|last1=O'Connor|first1=J.J.|last2=Robertson|first2=E.F.|date=May 1996|url= http://www-history.mcs.st-and.ac.uk/HistTopics/General_relativity.html|title=General relativity]}} {{Citation|url= http://www-history.mcs.st-and.ac.uk/Indexes/Math_Physics.html|title=History Topics: Mathematical Physics Index|archive-url= https://web.archive.org/web/20150204231934/http://www-history.mcs.st-and.ac.uk/Indexes/Math_Physics.html|archive-date=4 February 2015|publisher=School of Mathematics and Statistics, [[University of St. Andrews]]|location=Scotland|access-date=4 February 2015}}</ref> These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present.<ref>{{Harvnb|Pais|1982|loc=ch. 9 to 15}}, {{Harvnb|Janssen|2005}}; an up-to-date collection of current research, including reprints of many of the original articles, is {{Harvnb|Renn|2007}}; an accessible overview can be found in {{Harvnb|Renn|2005|pp=110ff}}. Einstein's original papers are found in [http://einsteinpapers.press.princeton.edu/ Digital Einstein], volumes 4 and 6. An early key article is {{Harvnb|Einstein|1907}}, cf. {{Harvnb|Pais|1982|loc=ch. 9}}. The publication featuring the field equations is {{Harvnb|Einstein|1915}}, cf. {{Harvnb|Pais|1982|loc=ch. 11–15}}</ref> A version of [[non-Euclidean geometry]], called [[Riemannian geometry]], enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.<ref>Moshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing</ref> This idea was pointed out by mathematician [[Marcel Grossmann]] and published by Grossmann and Einstein in 1913.<ref>Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261. [http://www.pitt.edu/~jdnorton/teaching/GR&Grav_2007/pdf/Einstein_Entwurf_1913.pdf English translate]</ref>

The Einstein field equations are [[Nonlinear differential equation|nonlinear]] and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist [[Karl Schwarzschild]] found the first non-trivial exact solution to the Einstein field equations, the [[Schwarzschild metric]]. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to [[electrical charge|electrically charged]] objects were taken, eventually resulting in the [[Reissner–Nordström metric|Reissner–Nordström solution]], which is now associated with [[charged black hole|electrically charged black holes]].<ref>{{Harvnb|Schwarzschild|1916a}}, {{Harvnb|Schwarzschild|1916b}} and {{Harvnb|Reissner|1916}} (later complemented in {{Harvnb|Nordström|1918}})</ref> In 1917, Einstein applied his theory to the [[universe]] as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the [[cosmological constant]]—to match that observational presumption.<ref>{{Harvnb|Einstein|1917}}, cf. {{Harvnb|Pais|1982|loc=ch. 15e}}</ref> By 1929, however, the work of [[Edwin Hubble|Hubble]] and others had shown that the universe is expanding. This is readily described by the expanding cosmological solutions found by [[Alexander Friedmann|Friedmann]] in 1922, which do not require a cosmological constant. [[Georges Lemaître|Lemaître]] used these solutions to formulate the earliest version of the [[Big Bang]] models, in which the universe has evolved from an extremely hot and dense earlier state.<ref>Hubble's original article is {{Harvnb|Hubble|1929}}; an accessible overview is given in {{Harvnb|Singh|2004|loc=ch. 2–4}}</ref> Einstein later declared the cosmological constant the biggest blunder of his life.<ref>As reported in {{Harvnb|Gamow|1970}}. Einstein's condemnation would prove to be premature, cf. the section [[#Cosmology|Cosmology]], below</ref>

During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to [[Newtonian gravity]], being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the [[Perihelion precession of Mercury|anomalous perihelion advance]] of the planet [[Mercury (planet)|Mercury]] without any arbitrary parameters ("[[wikt:fudge factor|fudge factors]]"),<ref>{{Harvnb|Pais|1982|pp=253–254}}</ref> and in 1919 an expedition led by [[Arthur Eddington|Eddington]] confirmed general relativity's prediction for the deflection of starlight by the Sun during the total [[solar eclipse of 29 May 1919]],<ref>{{Harvnb|Kennefick|2005}}, {{Harvnb|Kennefick|2007}}</ref> instantly making Einstein famous.<ref>{{Harvnb|Pais|1982|loc=ch. 16}}</ref> Yet the theory remained outside the mainstream of [[theoretical physics]] and astrophysics until developments between approximately 1960 and 1975, now known as the [[History of general relativity#Golden age|golden age of general relativity]].<ref>{{Harvnb|Thorne|2003|p=[https://books.google.com/books?id=yLy4b61rfPwC&pg=PA74 74]}}</ref> Physicists began to understand the concept of a black hole, and to identify [[quasar]]s as one of these objects' astrophysical manifestations.<ref>{{Harvnb|Israel|1987|loc=ch. 7.8–7.10}}, {{Harvnb|Thorne|1994|loc=ch. 3–9}}</ref> Ever more precise solar system tests confirmed the theory's predictive power,<ref>Sections [[#Orbital effects and the relativity of direction|Orbital effects and the relativity of direction]], [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]] and [[#Light deflection and gravitational time delay|Light deflection and gravitational time delay]], and references therein</ref> and relativistic cosmology also became amenable to direct observational tests.<ref>Section [[#Cosmology|Cosmology]] and references therein; the historical development is in {{Harvnb|Overbye|1999}}</ref>

General relativity has acquired a reputation as a theory of extraordinary beauty.<ref name=":0">{{Harvnb|Landau|Lifshitz|1975|loc=p. 228}} "...the ''general theory of relativity''...was established by Einstein, and represents probably the most beautiful of all existing physical theories."</ref><ref>{{Harvnb|Wald|1984|loc=p. 3}}</ref><ref>{{Harvnb|Rovelli|2015|loc=pp. 1–6}} "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more."</ref> [[Subrahmanyan Chandrasekhar]] has noted that at multiple levels, general relativity exhibits what [[Francis Bacon]] has termed a "strangeness in the proportion" (''i.e''. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time ''versus'' matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory.<ref>{{Harvnb|Chandrasekhar|1984|loc=p. 6}}</ref> Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.<ref>{{Harvnb|Engler|2002}}</ref>

In the preface to ''[[Relativity: The Special and the General Theory]]'', Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."<ref>{{cite book |title=Relativity – The Special and General Theory |author1=Albert Einstein |edition= |publisher=Read Books Ltd |year=2011 |isbn=978-1-4474-9358-7 |page=4 |url=https://books.google.com/books?id=yhN9CgAAQBAJ}} [https://books.google.com/books?id=yhN9CgAAQBAJ&pg=PT4 Extract of page 4]</ref>

== From classical mechanics to general relativity ==
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a [[heuristic]] derivation of general relativity.<ref>The following exposition re-traces that of {{Harvnb|Ehlers|1973|loc=sec. 1}}</ref><ref>{{Cite web|last=Al-Khalili|first=Jim|date=26 March 2021|title=Gravity and Me: The force that shapes our lives|url=https://www.bbc.co.uk/programmes/b08kgv7f|access-date=9 April 2021|website=www.bbc.co.uk}}</ref>

=== Geometry of Newtonian gravity<!--'Einstein's elevator experiment' redirects here--> ===
[[File:Elevator gravity.svg|thumb|According to general relativity, objects in a gravitational field behave similarly to objects within an accelerating enclosure. For example, an observer will see a ball fall the same way in a rocket (left) as it does on Earth (right), provided that the acceleration of the rocket is equal to 9.8&nbsp;m/s<sup>2</sup> (the acceleration due to gravity on the surface of the Earth).]]
At the base of [[classical mechanics]] is the notion that a [[physical body|body]]'s motion can be described as a combination of free (or [[inertia]]l) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second [[Newton's laws of motion|law of motion]], which states that the net [[force]] acting on a body is equal to that body's (inertial) [[mass]] multiplied by its [[acceleration]].<ref>{{Harvnb|Arnold|1989|loc=ch. 1}}</ref> The preferred inertial motions are related to the geometry of space and time: in the standard [[frame of reference|reference frames]] of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are [[geodesic]]s, straight [[world lines]] in [[curved spacetime]].<ref>{{Harvnb|Ehlers|1973|pp=5f}}</ref>

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as [[electromagnetism]] or [[friction]]), can be used to define the geometry of space, as well as a time [[coordinate]]. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of [[Loránd Eötvös|Eötvös]] and its successors (see [[Eötvös experiment]]), there is a universality of free fall (also known as the weak [[equivalence principle]], or the universal equality of inertial and passive-gravitational mass): the trajectory of a [[test body]] in free fall depends only on its position and initial speed, but not on any of its material properties.<ref>{{Harvnb|Will|1993|loc=sec. 2.4}}, {{Harvnb|Will|2006|loc=sec. 2}}</ref> A simplified version of this is embodied in '''Einstein's elevator experiment'''<!--boldface per WP:R#PLA-->, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.<ref>{{Harvnb|Wheeler|1990|loc=ch. 2}}</ref>

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific [[connection (mathematics)|connection]] which depends on the [[gradient]] of the [[gravitational potential]]. Space, in this construction, still has the ordinary [[Euclidean geometry]]. However, space''time'' as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not [[integrable systems|integrable]]. From this, one can deduce that spacetime is curved. The resulting [[Newton–Cartan theory]] is a geometric formulation of Newtonian gravity using only [[Covariance and contravariance of vectors#Informal usage|covariant]] concepts, i.e. a description which is valid in any desired coordinate system.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.2}}, {{Harvnb|Havas|1964}}, {{Harvnb|Künzle|1972}}. The simple thought experiment in question was first described in {{Harvnb|Heckmann|Schücking|1959}}</ref> In this geometric description, [[tidal effect]]s—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.<ref>{{Harvnb|Ehlers|1973|pp=10f}}</ref>

=== Relativistic generalization ===
[[File:Light cone.svg|thumb|left|upright|[[Light cone]]]]
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a [[limiting case (philosophy of science)|limiting case]] of (special) relativistic mechanics.<ref>Good introductions are, in order of increasing presupposed knowledge of mathematics, {{Harvnb|Giulini|2005}}, {{Harvnb|Mermin|2005}}, and {{Harvnb|Rindler|1991}}; for accounts of precision experiments, cf. part IV of {{Harvnb|Ehlers|Lämmerzahl|2006}}</ref> In the language of [[symmetry]]: where gravity can be neglected, physics is [[Lorentz invariance|Lorentz invariant]] as in special relativity rather than [[Galilean invariance|Galilei invariant]] as in classical mechanics. (The defining symmetry of special relativity is the [[Poincaré group]], which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the [[speed of light]], and with high-energy phenomena.<ref>An in-depth comparison between the two symmetry groups can be found in {{Harvnb|Giulini|2006}}</ref>

With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each [[event (relativity)|event]] {{math|A}}, there is a set of events that can, in principle, either influence or be influenced by {{math|A}} via signals or interactions that do not need to travel faster than light (such as event {{math|B}} in the image), and a set of events for which such an influence is impossible (such as event {{math|C}} in the image). These sets are [[frame of reference|observer]]-independent.<ref>{{Harvnb|Rindler|1991|loc=sec. 22}}, {{Harvnb|Synge|1972|loc=ch. 1 and 2}}</ref> In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a [[conformal structure]]<ref>{{Harvnb|Ehlers|1973|loc=sec. 2.3}}</ref> or conformal geometry.

Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global [[inertial frame]]s. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight [[time-like]] lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}}, {{Harvnb|Schutz|1985|loc=sec. 5.1}}</ref>

A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of [[preferred frame]]s. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. [[#Gravitational time dilation and frequency shift|below]]). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.<ref>{{Harvnb|Ehlers|1973|pp=17ff}}; a derivation can be found in {{Harvnb|Mermin|2005|loc=ch. 12}}. For the experimental evidence, cf. the section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]], below</ref> The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the [[Equivalence Principle#The Einstein equivalence principle|Einstein equivalence principle]], a crucial guiding principle for generalizing special-relativistic physics to include gravity.<ref>{{Harvnb|Rindler|2001|loc=sec. 1.13}}; for an elementary account, see {{Harvnb|Wheeler|1990|loc=ch. 2}}; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. {{Harvnb|Norton|1985}}</ref>

The same experimental data shows that time as measured by clocks in a gravitational field—[[proper time]], to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the [[Minkowski metric]]. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The [[metric tensor (general relativity)|metric tensor]] that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or [[pseudo-Riemannian]] metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the [[Levi-Civita connection]], and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable [[Local reference frame|locally inertial coordinates]], the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).<ref>{{Harvnb|Ehlers|1973|loc=sec. 1.4}} for the experimental evidence, see once more section [[#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]]. Choosing a different connection with non-zero [[torsion tensor|torsion]] leads to a modified theory known as [[Einstein–Cartan theory]]</ref>

=== Einstein's equations ===
{{Main|Einstein field equations|Mathematics of general relativity}}
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the [[energy–momentum tensor]], which includes both [[energy density|energy]] and momentum [[density|densities]] as well as [[stress (physics)|stress]]: [[pressure]] and shear.<ref>{{Harvnb|Ehlers|1973|p=16}}, {{Harvnb|Kenyon|1990|loc=sec. 7.2}}, {{Harvnb|Weinberg|1972|loc=sec. 2.8}}</ref> Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the [[field equation]] for gravity relates this tensor and the [[Ricci curvature|Ricci tensor]], which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, [[conservation of energy]]–momentum corresponds to the statement that the energy–momentum tensor is [[divergence]]-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-[[manifold]] counterparts, [[covariant derivative]]s studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations:
{{Equation box 1
|indent=:
|title='''Einstein's field equations'''
|equation=<math>G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = \kappa T_{\mu\nu}\,</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

On the left-hand side is the [[Einstein tensor]], <math>G_{\mu\nu}</math>, which is symmetric and a specific divergence-free combination of the Ricci tensor <math>R_{\mu\nu}</math> and the metric. In particular,
: <math>R=g^{\mu\nu}R_{\mu\nu}</math>
is the curvature scalar. The Ricci tensor itself is related to the more general [[Riemann curvature tensor]] as
: <math>R_{\mu\nu}={R^\alpha}_{\mu\alpha\nu}.</math>

On the right-hand side, <math>\kappa</math> is a constant and <math>T_{\mu\nu}</math> is the energy–momentum tensor. All tensors are written in [[abstract index notation]].<ref>{{Harvnb|Ehlers|1973|pp=19–22}}; for similar derivations, see sections 1 and 2 of ch. 7 in {{Harvnb|Weinberg|1972}}. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. {{Harvnb|Lovelock|1972}}. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as [[Bianchi identities]], the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. {{Harvnb|Schutz|1985|loc=sec. 8.3}}</ref> Matching the theory's prediction to observational results for [[planet]]ary [[orbit]]s or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant <math>\kappa</math> is found to be <math display="inline">\kappa=\frac{8\pi G}{c^4}</math>, where <math>G</math> is the [[Newtonian constant of gravitation]] and <math>c</math> the speed of light in vacuum.<ref>{{Harvnb|Kenyon|1990|loc=sec. 7.4}}</ref> When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
: <math>R_{\mu\nu}=0.</math>

In general relativity, the [[world line]] of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.

The [[Geodesics in general relativity|geodesic equation]] is:
: <math> {d^2 x^\mu \over ds^2}+\Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}=0,</math>
where <math>s</math> is a scalar parameter of motion (e.g. the [[proper time]]), and <math> \Gamma^\mu {}_{\alpha \beta}</math> are [[Christoffel symbols]] (sometimes called the [[affine connection]] coefficients or [[Levi-Civita connection]] coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the [[summation convention]] is used for repeated indices <math>\alpha</math> and <math>\beta</math>. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to [[Newton's laws of motion]] which likewise provide formulae for the acceleration of a particle. This equation of motion employs the [[Einstein notation]], meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a [[test particle]] whose motion is described by the geodesic equation.

=== Total force in general relativity ===
{{See also|Two-body problem in general relativity}}
In general relativity, the effective [[gravitational potential energy]] of an object of mass ''m'' revolving around a massive central body ''M'' is given by<ref>{{cite book |title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity|last=Weinberg, Steven|publisher=John Wiley|year=1972|isbn=978-0-471-92567-5}}</ref><ref>{{cite book |title=Relativity, Gravitation and Cosmology: a Basic Introduction|last=Cheng, Ta-Pei|publisher=Oxford and New York: Oxford University Press|year=2005|isbn=978-0-19-852957-6}}</ref>

:<math>U_f(r) =-\frac{GMm}{r}+\frac{L^{2}}{2mr^{2}}-\frac{GML^{2}}{mc^{2}r^{3}}</math>

A conservative total [[force]] can then be obtained as its [[Force#Potential_energy|negative gradient]]

:<math>F_f(r)=-\frac{GMm}{r^{2}}+\frac{L^{2}}{mr^{3}}-\frac{3GML^{2}}{mc^{2}r^{4}}</math>

where ''L'' is the [[angular momentum]]. The first term represents the [[Newton's law of universal gravitation|force of Newtonian gravity]], which is described by the inverse-square law. The second term represents the [[centrifugal force]] in the circular motion. The third term represents the relativistic effect.

=== Alternatives to general relativity ===
{{Main|Alternatives to general relativity}}
There are [[alternatives to general relativity]] built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are [[Whitehead's theory of gravitation|Whitehead's theory]], [[Brans–Dicke theory]], [[teleparallelism]], [[f(R) gravity|''f''(''R'') gravity]] and [[Einstein–Cartan theory]].<ref>{{Harvnb|Brans|Dicke|1961}}, {{Harvnb|Weinberg|1972|loc=sec. 3 in ch. 7}}, {{Harvnb|Goenner|2004|loc=sec. 7.2}}, and {{Harvnb|Trautman|2006}}, respectively</ref>

== Definition and basic applications ==
{{See also|Mathematics of general relativity|Physical theories modified by general relativity}}

The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.

=== Definition and basic properties ===
General relativity is a [[metric (general relativity)|metric]] theory of gravitation. At its core are [[Einstein's equations]], which describe the relation between the geometry of a four-dimensional [[pseudo-Riemannian manifold]] representing spacetime, and the [[Stress–energy tensor|energy–momentum]] contained in that spacetime.<ref>{{Harvnb|Wald|1984|loc=ch. 4}}, {{Harvnb|Weinberg|1972|loc=ch. 7}} or, in fact, any other textbook on general relativity</ref> Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as [[free-fall]], orbital motion, and [[spacecraft]] [[Trajectory|trajectories]]), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.<ref>At least approximately, cf. {{Harvnb|Poisson|2004a}}</ref> The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist [[John Archibald Wheeler]], spacetime tells matter how to move; matter tells spacetime how to curve.<ref>{{Harvnb|Wheeler|1990|p=xi}}</ref>

While general relativity replaces the [[scalar field|scalar]] gravitational potential of classical physics by a symmetric [[Tensor#As multidimensional arrays|rank]]-two [[tensor]], the latter reduces to the former in certain [[Correspondence principle#Other scientific theories|limiting cases]]. For [[weak-field approximation|weak gravitational fields]] and [[slow-motion approximation|slow speed]] relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.<ref>{{Harvnb|Wald|1984|loc=sec. 4.4}}</ref>

As it is constructed using tensors, general relativity exhibits [[general covariance]]: its laws—and further laws formulated within the general relativistic framework—take on the same form in all [[coordinate system]]s.<ref>{{Harvnb|Wald|1984|loc=sec. 4.1}}</ref> Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is [[Background independence|background independent]]. It thus satisfies a more stringent [[general principle of relativity]], namely that the [[Physical law|laws of physics]] are the same for all observers.<ref>For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see {{Harvnb|Giulini|2007}}</ref> [[Local spacetime structure|Locally]], as expressed in the equivalence principle, spacetime is [[Minkowski space|Minkowskian]], and the laws of physics exhibit [[local Lorentz invariance]].<ref>section 5 in ch. 12 of {{Harvnb|Weinberg|1972}}</ref>

=== Model-building ===
The core concept of general-relativistic model-building is that of a [[solutions of the Einstein field equations|solution of Einstein's equations]]. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-[[Riemannian manifold]] (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.<ref>Introductory chapters of {{Harvnb|Stephani|Kramer|MacCallum|Hoenselaers|2003}}</ref>

Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.<ref>A review showing Einstein's equation in the broader context of other PDEs with physical significance is {{Harvnb|Geroch|1996}}</ref> Nevertheless, a number of [[exact solutions in general relativity|exact solutions]] are known, although only a few have direct physical applications.<ref>For background information and a list of solutions, cf. {{Harvnb|Stephani|Kramer|MacCallum|Hoenselaers|2003}}; a more recent review can be found in {{Harvnb|MacCallum|2006}}</ref> The best-known exact solutions, and also those most interesting from a physics point of view, are the [[Schwarzschild solution]], the [[Reissner–Nordström solution]] and the [[Kerr metric]], each corresponding to a certain type of black hole in an otherwise empty universe,<ref>{{Harvnb|Chandrasekhar|1983|loc=ch. 3,5,6}}</ref> and the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker]] and [[de Sitter universe]]s, each describing an expanding cosmos.<ref>{{Harvnb|Narlikar|1993|loc=ch. 4, sec. 3.3}}</ref> Exact solutions of great theoretical interest include the [[Gödel metric|Gödel universe]] (which opens up the intriguing possibility of [[time travel]] in curved spacetimes), the [[Taub–NUT space|Taub–NUT solution]] (a model universe that is [[Homogeneity (physics)|homogeneous]], but [[anisotropic]]), and [[anti-de Sitter space]] (which has recently come to prominence in the context of what is called the [[Maldacena conjecture]]).<ref>Brief descriptions of these and further interesting solutions can be found in {{Harvnb|Hawking|Ellis|1973|loc=ch. 5}}</ref>

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by [[numerical integration]] on a computer, or by considering small perturbations of exact solutions. In the field of [[numerical relativity]], powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.<ref>{{Harvnb|Lehner|2002}}</ref> In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as [[naked singularity|naked singularities]]. Approximate solutions may also be found by [[perturbation theory|perturbation theories]] such as [[linearized gravity]]<ref>For instance {{Harvnb|Wald|1984|loc=sec. 4.4}}</ref> and its generalization, the [[post-Newtonian expansion]], both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.<ref>{{Harvnb|Will|1993|loc=sec. 4.1 and 4.2}}</ref> An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.<ref>{{Harvnb|Will|2006|loc=sec. 3.2}}, {{Harvnb|Will|1993|loc=ch. 4}}</ref>

== Consequences of Einstein's theory ==
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.

=== Gravitational time dilation and frequency shift ===
{{Main|Gravitational time dilation}}
[[File:Gravitational red-shifting.png|thumb|Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body]]
Assuming that the equivalence principle holds,<ref>{{Harvnb|Rindler|2001|pp=24–26 vs. pp. 236–237}} and {{Harvnb|Ohanian|Ruffini|1994|pp=164–172}}. Einstein derived these effects using the equivalence principle as early as 1907, cf. {{Harvnb|Einstein|1907}} and the description in {{Harvnb|Pais|1982|pp=196–198}}</ref> gravity influences the passage of time. Light sent down into a [[gravity well]] is [[blueshift]]ed, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is [[redshift]]ed; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.<ref>{{Harvnb|Rindler|2001|pp=24–26}}; {{Harvnb|Misner|Thorne|Wheeler|1973 |loc=§&nbsp;38.5}}</ref>

Gravitational redshift has been measured in the laboratory<ref>[[Pound–Rebka experiment]], see {{Harvnb|Pound|Rebka|1959}}, {{Harvnb|Pound|Rebka|1960}}; {{Harvnb|Pound|Snider|1964}}; a list of further experiments is given in {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.1 on p. 186}}</ref> and using astronomical observations.<ref>{{Harvnb|Greenstein|Oke|Shipman|1971}}; the most recent and most accurate Sirius B measurements are published in {{Harvnb|Barstow, Bond et al.|2005}}.</ref> Gravitational time dilation in the Earth's gravitational field has been measured numerous times using [[atomic clocks]],<ref>Starting with the [[Hafele–Keating experiment]], {{Harvnb|Hafele|Keating|1972a}} and {{Harvnb|Hafele|Keating|1972b}}, and culminating in the [[Gravity Probe A]] experiment; an overview of experiments can be found in {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.1 on p. 186}}</ref> while ongoing validation is provided as a side effect of the operation of the [[Global Positioning System]] (GPS).<ref>GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see {{Harvnb|Ashby|2002}} and {{Harvnb|Ashby|2003}}</ref> Tests in stronger gravitational fields are provided by the observation of [[binary pulsar]]s.<ref>{{Harvnb|Stairs|2003}} and {{Harvnb|Kramer|2004}}</ref> All results are in agreement with general relativity.<ref>General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; {{Harvnb|Ohanian|Ruffini|1994|loc=sec. 4.2}}</ref> However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.<ref>{{Harvnb|Ohanian|Ruffini|1994|pp=164–172}}</ref>

=== Light deflection and gravitational time delay ===
{{Main|Schwarzschild geodesics|Kepler problem in general relativity|Gravitational lens|Shapiro delay}}
[[File:Light deflection.png|thumb|left|upright|Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)]]
General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a massive object. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the [[Sun]].<ref>Cf. {{Harvnb|Kennefick|2005}} for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see {{Harvnb|Ohanian|Ruffini|1994|loc=ch. 4.3}}. For the most precise direct modern observations using quasars, cf. {{Harvnb|Shapiro|Davis|Lebach|Gregory|2004}}</ref>

This and related predictions follow from the fact that light follows what is called a light-like or [[Geodesic (general relativity)|null geodesic]]—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the [[Invariant (mathematics)|invariance]] of lightspeed in special relativity.<ref>This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell [[Lagrangian (field theory)|Lagrangian]] using a [[WKB approximation]], cf. {{Harvnb|Ehlers|1973|loc=sec. 5}}</ref> As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),<ref>{{Harvnb|Blanchet|2006|loc=sec. 1.3}}</ref> several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,<ref>{{Harvnb|Rindler|2001|loc=sec. 1.16}}; for the historical examples, {{Harvnb|Israel|1987|pp=202–204}}; in fact, Einstein published one such derivation as {{Harvnb|Einstein|1907}}. Such calculations tacitly assume that the geometry of space is [[Euclidean space|Euclidean]], cf. {{Harvnb|Ehlers|Rindler|1997}}</ref> the angle of deflection resulting from such calculations is only half the value given by general relativity.<ref>From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. {{Harvnb|Rindler|2001|loc=sec. 11.11}}</ref>

Closely related to light deflection is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.<ref>For the Sun's gravitational field using radar signals reflected from planets such as [[Venus]] and Mercury, cf. {{Harvnb|Shapiro|1964}}, {{Harvnb|Weinberg|1972|loc=ch. 8, sec. 7}}; for signals actively sent back by space probes ([[transponder]] measurements), cf. {{Harvnb|Bertotti|Iess|Tortora|2003}}; for an overview, see {{Harvnb|Ohanian|Ruffini|1994|loc=table 4.4 on p. 200}}; for more recent measurements using signals received from a [[pulsar]] that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. {{Harvnb|Stairs|2003|loc=sec. 4.4}}</ref> In the [[parameterized post-Newtonian formalism]] (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.<ref>{{Harvnb|Will|1993|loc=sec. 7.1 and 7.2}}</ref>
{{clear}}

=== Gravitational waves ===
{{Main|Gravitational wave}}
[[File:Gravwav.gif|thumb|Ring of test particles deformed by a passing (linearized, amplified for better visibility) gravitational wave]]
Predicted in 1916<ref>{{cite journal|author=Einstein, A|title=Näherungsweise Integration der Feldgleichungen der Gravitation|date=22 June 1916|url=http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|journal=[[Prussian Academy of Sciences|Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin]]|issue=part&nbsp;1|pages=688–696|bibcode=1916SPAW.......688E|access-date=12 February 2016|archive-url=https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|archive-date=21 March 2019}}</ref><ref>{{cite journal|author=Einstein, A|title=Über Gravitationswellen|date=31 January 1918|url=http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin|issue=part&nbsp;1|pages=154–167|bibcode=1918SPAW.......154E|access-date=12 February 2016|archive-url=https://web.archive.org/web/20190321062928/http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte|archive-date=21 March 2019}}</ref> by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to [[electromagnetic wave]]s. On 11 February 2016, the Advanced LIGO team announced that they had [[Gravitational wave observation|directly detected gravitational waves]] from a [[Binary black hole|pair]] of black holes [[Stellar collision|merging]].<ref name="Discovery 2016">{{cite journal |title=Einstein's gravitational waves found at last |journal=Nature News| url=http://www.nature.com/news/einstein-s-gravitational-waves-found-at-last-1.19361 |date=11 February 2016 |last1= Castelvecchi |first1=Davide |last2=Witze |first2=Witze |doi=10.1038/nature.2016.19361 |s2cid=182916902|access-date= 11 February 2016 }}</ref><ref name="Abbot">{{cite journal |title=Observation of Gravitational Waves from a Binary Black Hole Merger| author1=B. P. Abbott |collaboration=LIGO Scientific Collaboration and Virgo Collaboration| journal=Physical Review Letters| year=2016| volume=116|issue=6| doi= 10.1103/PhysRevLett.116.061102 | pmid=26918975| page=061102|arxiv = 1602.03837 |bibcode = 2016PhRvL.116f1102A | s2cid=124959784 }}</ref><ref name="NSF">{{cite web|title = Gravitational waves detected 100 years after Einstein's prediction |website= NSF – National Science Foundation|url = https://www.nsf.gov/news/news_summ.jsp?cntn_id=137628 |date = 11 February 2016}}</ref>

The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).<ref>Most advanced textbooks on general relativity contain a description of these properties, e.g. {{Harvnb|Schutz|1985|loc=ch. 9}}</ref> Since Einstein's equations are [[non-linear]], arbitrarily strong gravitational waves do not obey [[linear superposition]], making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by <math>10^{-21}</math> or less. Data analysis methods routinely make use of the fact that these linearized waves can be [[Fourier decomposition|Fourier decomposed]].<ref>For example {{Harvnb|Jaranowski|Królak|2005}}</ref>

Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space<ref>{{Harvnb|Rindler|2001|loc=ch. 13}}</ref> or [[Gowdy universe]]s, varieties of an expanding cosmos filled with gravitational waves.<ref>{{Harvnb|Gowdy|1971}}, {{Harvnb|Gowdy|1974}}</ref> But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.<ref>See {{Harvnb|Lehner|2002}} for a brief introduction to the methods of numerical relativity, and {{Harvnb|Seidel|1998}} for the connection with gravitational wave astronomy</ref>

=== Orbital effects and the relativity of direction ===
{{Main|Two-body problem in general relativity}}
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ([[precession]]) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.

==== Precession of apsides ====
[[File:Relativistic precession.svg|thumb|upright=1.05|Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star. The influence of other planets is ignored.]]
{{Main|Apsidal precession}}

In general relativity, the [[apsis|apsides]] of any orbit (the point of the orbiting body's closest approach to the system's [[center of mass]]) will [[apsidal precession|precess]]; the orbit is not an [[ellipse]], but akin to an ellipse that rotates on its focus, resulting in a [[rose (mathematics)|rose curve]]-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a [[test particle]]. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by [[Urbain Le Verrier]] in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.<ref>{{Harvnb|Schutz|2003|pp=48–49}}, {{Harvnb|Pais|1982|pp=253–254}}</ref>

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)<ref>{{Harvnb|Rindler|2001|loc=sec. 11.9}}</ref> or the much more general [[post-Newtonian formalism]].<ref>{{Harvnb|Will|1993|pp=177–181}}</ref> It is due to the influence of gravity on the geometry of space and to the contribution of [[self-energy]] to a body's gravity (encoded in the [[nonlinearity]] of Einstein's equations).<ref>In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. {{Harvnb|Will|2006|loc=sec. 3.5}} and {{Harvnb|Will|1993|loc=sec. 7.3}}</ref> Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),<ref>The most precise measurements are [[VLBI]] measurements of planetary positions; see {{Harvnb|Will|1993|loc=ch. 5}}, {{Harvnb|Will|2006|loc=sec. 3.5}}, {{Harvnb|Anderson|Campbell|Jurgens|Lau|1992}}; for an overview, {{Harvnb|Ohanian|Ruffini|1994|pp=406–407}}</ref> as well as in binary pulsar systems, where it is larger by five [[order of magnitude|orders of magnitude]].<ref>{{Harvnb|Kramer|Stairs|Manchester|McLaughlin|2006}}</ref>

In general relativity the perihelion shift <math>\sigma</math>, expressed in radians per revolution, is approximately given by:{{sfn|Dediu|Magdalena|Martín-Vide|2015|p=[https://books.google.com/books?id=XmwiCwAAQBAJ&pg=PA141 141]}}

:<math>\sigma=\frac {24\pi^3L^2} {T^2c^2(1-e^2)} \ ,</math>

where:
*<math>L</math> is the [[semi-major axis]]
*<math>T</math> is the [[orbital period]]
*<math>c</math> is the speed of light in vacuum
*<math>e</math> is the [[orbital eccentricity]]

==== Orbital decay ====
<!--This subsection is linked to from the subsection Gravitational Waves in Astrophysical Applications, please do not change its title -->

[[File:PSRJ0737−3039shift2021.png|thumb|upright=0.8|Orbital decay for PSR J0737−3039: time shift (in [[second|s]]), tracked over 16 years (2021).<ref name=":1">{{Cite journal|last1=Kramer|first1=M.|last2=Stairs|first2=I. H.|last3=Manchester|first3=R. N.|last4=Wex|first4=N.|last5=Deller|first5=A. T.|last6=Coles|first6=W. A.|last7=Ali|first7=M.|last8=Burgay|first8=M.|last9=Camilo|first9=F.|last10=Cognard|first10=I.|last11=Damour|first11=T.|date=13 December 2021|title=Strong-Field Gravity Tests with the Double Pulsar|url=https://link.aps.org/doi/10.1103/PhysRevX.11.041050|journal=Physical Review X|language=en|volume=11|issue=4|page=041050|doi=10.1103/PhysRevX.11.041050|arxiv=2112.06795|bibcode=2021PhRvX..11d1050K|s2cid=245124502|issn=2160-3308}}</ref>]]

According to general relativity, a [[Binary system (astronomy)|binary system]] will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the [[Solar System]] or for ordinary [[double star]]s, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting [[neutron star]]s, one of which is a [[pulsar]]: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.<ref>{{Harvnb|Stairs|2003}}, {{Harvnb|Schutz|2003|pp=317–321}}, {{Harvnb|Bartusiak|2000|pp=70–86}}</ref>

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by [[Russell Alan Hulse|Hulse]] and [[Joseph Hooton Taylor, Jr.|Taylor]], using the binary pulsar [[PSR1913+16]] they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 [[Nobel Prize]] in physics.<ref>{{Harvnb|Weisberg|Taylor|2003}}; for the pulsar discovery, see {{Harvnb|Hulse|Taylor|1975}}; for the initial evidence for gravitational radiation, see {{Harvnb|Taylor|1994}}</ref> Since then, several other binary pulsars have been found, in particular the double pulsar [[PSR J0737−3039]], where both stars are pulsars<ref>{{Harvnb|Kramer|2004}}</ref> and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.<ref name=":1" />

==== Geodetic precession and frame-dragging ====
{{Main|Geodetic precession|Frame dragging}}
Several relativistic effects are directly related to the relativity of direction.<ref>{{Harvnb|Penrose|2004|loc=§&nbsp;14.5}}, {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;11.4}}</ref> One is [[geodetic effect|geodetic precession]]: the axis direction of a [[gyroscope]] in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("[[parallel transport]]").<ref>{{Harvnb|Weinberg|1972|loc=sec. 9.6}}, {{Harvnb|Ohanian|Ruffini|1994|loc=sec. 7.8}}</ref> For the Moon–Earth system, this effect has been measured with the help of [[lunar laser ranging]].<ref>{{Harvnb|Bertotti|Ciufolini|Bender|1987}}, {{Harvnb|Nordtvedt|2003}}</ref> More recently, it has been measured for test masses aboard the satellite [[Gravity Probe B]] to a precision of better than 0.3%.<ref>{{Harvnb|Kahn|2007}}</ref><ref>A mission description can be found in {{Harvnb|Everitt|Buchman|DeBra|Keiser|2001}}; a first post-flight evaluation is given in {{Harvnb|Everitt|Parkinson|Kahn|2007}}; further updates will be available on the mission website {{Harvnb|Kahn|1996–2012}}.</ref>

Near a rotating mass, there are gravitomagnetic or [[frame-dragging]] effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for [[Kerr solution|rotating black holes]] where, for any object entering a zone known as the [[ergosphere]], rotation is inevitable.<ref>{{Harvnb|Townsend|1997|loc=sec. 4.2.1}}, {{Harvnb|Ohanian|Ruffini|1994|pp=469–471}}</ref> Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.<ref>{{Harvnb|Ohanian|Ruffini|1994|loc=sec. 4.7}}, {{Harvnb|Weinberg|1972|loc=sec. 9.7}}; for a more recent review, see {{Harvnb|Schäfer|2004}}</ref> Somewhat controversial tests have been performed using the [[LAGEOS]] satellites, confirming the relativistic prediction.<ref>{{Harvnb|Ciufolini|Pavlis|2004}}, {{Harvnb|Ciufolini|Pavlis|Peron|2006}}, {{Harvnb|Iorio|2009}}</ref> Also the [[Mars Global Surveyor]] probe around Mars has been used.<ref>{{Harvnb|Iorio|2006}}, {{Harvnb|Iorio|2010}}</ref>

== Astrophysical applications ==
=== Gravitational lensing ===
{{Main|Gravitational lensing}}
[[File:Einstein cross (cropped).jpg|thumb|upright=0.88|[[Einstein cross]]: four images of the same astronomical object, produced by a gravitational lens]]
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.<ref>For overviews of gravitational lensing and its applications, see {{Harvnb|Ehlers|Falco|Schneider|1992}} and {{Harvnb|Wambsganss|1998}}</ref> Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an [[Einstein ring]], or partial rings called arcs.<ref>For a simple derivation, see {{Harvnb|Schutz|2003|loc=ch. 23}}; cf. {{Harvnb|Narayan|Bartelmann|1997|loc=sec. 3}}</ref>
The [[Twin Quasar|earliest example]] was discovered in 1979;<ref>{{Harvnb|Walsh|Carswell|Weymann|1979}}</ref> since then, more than a hundred gravitational lenses have been observed.<ref>Images of all the known lenses can be found on the pages of the CASTLES project, {{Harvnb|Kochanek|Falco|Impey|Lehar|2007}}</ref> Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "[[microlensing]] events" have been observed.<ref>{{Harvnb|Roulet|Mollerach|1997}}</ref>

Gravitational lensing has developed into a tool of [[observational astronomy]]. It is used to detect the presence and distribution of [[dark matter]], provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the [[Hubble constant]]. Statistical evaluations of lensing data provide valuable insight into the structural evolution of [[galaxy|galaxies]].<ref>{{Harvnb|Narayan|Bartelmann|1997|loc=sec. 3.7}}</ref>

=== Gravitational-wave astronomy ===
{{Main|Gravitational wave|Gravitational-wave astronomy}}
[[File:LISA.jpg|thumb|upright=0.8|Artist's impression of the space-borne gravitational wave detector [[Laser Interferometer Space Antenna|LISA]]]]
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see [[#Orbital decay|Orbital decay]], above). Detection of these waves is a major goal of current relativity-related research.<ref>{{Harvnb|Barish|2005}}, {{Harvnb|Bartusiak|2000}}, {{Harvnb|Blair|McNamara|1997}}</ref> Several land-based [[gravitational wave detector]]s are currently in operation, most notably the [[Interferometric gravitational wave detector|interferometric detectors]] [[GEO 600]], [[LIGO]] (two detectors), [[TAMA 300]] and [[Virgo interferometer|VIRGO]].<ref>{{Harvnb|Hough|Rowan|2000}}</ref> Various [[pulsar timing array]]s are using [[millisecond pulsar]]s to detect gravitational waves in the 10<sup>−9</sup> to 10<sup>−6</sup> [[hertz]] frequency range, which originate from binary supermassive blackholes.<ref>{{Citation | last1=Hobbs | first1=George |title=The international pulsar timing array project: using pulsars as a gravitational wave detector | last2=Archibald | first2=A. | last3=Arzoumanian | first3=Z. | last4=Backer | first4=D. | last5=Bailes | first5=M. | last6=Bhat | first6=N. D. R. | last7=Burgay | first7=M. | last8=Burke-Spolaor | first8=S. | last9=Champion | first9=D. | display-authors = 8| doi=10.1088/0264-9381/27/8/084013 | date=2010 | journal=Classical and Quantum Gravity | volume=27 | issue=8 | page=084013 |arxiv=0911.5206 |bibcode = 2010CQGra..27h4013H | s2cid=56073764 }}</ref> A European space-based detector, [[Laser Interferometer Space Antenna|eLISA / NGO]], is currently under development,<ref>{{Harvnb|Danzmann|Rüdiger|2003}}</ref> with a precursor mission ([[LISA Pathfinder]]) having launched in December 2015.<ref>{{cite web|url=http://www.esa.int/esaSC/120397_index_0_m.html|title=LISA pathfinder overview|publisher=ESA|access-date=23 April 2012}}</ref>

Observations of gravitational waves promise to complement observations in the [[electromagnetic spectrum]].<ref>{{Harvnb|Thorne|1995}}</ref> They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of [[supernova]] implosions, and about processes in the very early universe, including the signature of certain types of hypothetical [[cosmic string]].<ref>{{Harvnb|Cutler|Thorne|2002}}</ref> In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.<ref name="Discovery 2016" /><ref name="Abbot" /><ref name="NSF" />

=== Black holes and other compact objects ===
{{Main|Black hole}}
[[File:Star collapse to black hole.png|thumb|left|Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves]]
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of [[stellar evolution]], neutron stars of around 1.4 [[solar mass]]es, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars.<ref>{{Harvnb|Miller|2002|loc=lectures 19 and 21}}</ref> Usually a galaxy has one [[supermassive black hole]] with a few million to a few [[1000000000 (number)|billion]] solar masses in its center,<ref>{{Harvnb|Celotti|Miller|Sciama|1999|loc=sec. 3}}</ref> and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.<ref>{{Harvnb|Springel|White|Jenkins|Frenk|2005}} and the accompanying summary {{Harvnb|Gnedin|2005}}</ref>

Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.<ref>{{Harvnb|Blandford|1987|loc=sec. 8.2.4}}</ref> [[Accretion (astrophysics)|Accretion]], the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.<ref>For the basic mechanism, see {{Harvnb|Carroll|Ostlie|1996|loc=sec. 17.2}}; for more about the different types of astronomical objects associated with this, cf. {{Harvnb|Robson|1996}}</ref> In particular, accretion can lead to [[relativistic jet]]s, focused beams of highly energetic particles that are being flung into space at almost light speed.<ref>For a review, see {{Harvnb|Begelman|Blandford|Rees|1984}}. To a distant observer, some of these jets even appear to move [[superluminal motion|faster than light]]; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see {{Harvnb|Rees|1966}}</ref>
General relativity plays a central role in modelling all these phenomena,<ref>For stellar end states, cf. {{Harvnb|Oppenheimer|Snyder|1939}} or, for more recent numerical work, {{Harvnb|Font|2003|loc=sec. 4.1}}; for supernovae, there are still major problems to be solved, cf. {{Harvnb|Buras|Rampp|Janka|Kifonidis|2003}}; for simulating accretion and the formation of jets, cf. {{Harvnb|Font|2003|loc=sec. 4.2}}. Also, relativistic lensing effects are thought to play a role for the signals received from [[X-ray pulsar]]s, cf. {{Harvnb|Kraus|1998}}</ref> and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.<ref>The evidence includes limits on compactness from the observation of accretion-driven phenomena ("[[Eddington luminosity]]"), see {{Harvnb|Celotti|Miller|Sciama|1999}}, observations of stellar dynamics in the center of our own [[Milky Way]] galaxy, cf. {{Harvnb|Schödel|Ott|Genzel|Eckart|2003}}, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of [[X-ray burst]]s for which the central compact object is either a neutron star or a black hole; cf. {{Harvnb|Remillard|Lin|Cooper|Narayan|2006}} for an overview, {{Harvnb|Narayan|2006|loc=sec. 5}}. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. {{Harvnb|Falcke|Melia|Agol|2000}}</ref>

Black holes are also sought-after targets in the search for gravitational waves (cf. [[#Gravitational waves|Gravitational waves]], above). Merging [[binary black hole|black hole binaries]] should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "[[standard candle]]" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.<ref>{{Harvnb|Dalal|Holz|Hughes|Jain|2006}}</ref> The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.<ref>{{Harvnb|Barack|Cutler|2004}}</ref>

=== Cosmology ===
{{Main|Physical cosmology}}
[[File:Lensshoe hubble.jpg|thumb|This blue horseshoe is a distant galaxy that has been magnified and warped into a nearly complete ring by the strong gravitational pull of the massive foreground luminous red galaxy.]]
The current models of cosmology are based on [[Einstein's field equations]], which include the cosmological constant <math>\Lambda</math> since it has important influence on the large-scale dynamics of the cosmos,
:<math> R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} + \Lambda\ g_{\mu\nu} = \frac{8\pi G}{c^{4}}\, T_{\mu\nu} </math>
where ''<math>g_{\mu\nu}</math>'' is the spacetime metric.<ref>{{Harvnb|Einstein|1917}}; cf. {{Harvnb|Pais|1982|pp=285–288}}</ref> [[Isotropic]] and homogeneous solutions of these enhanced equations, the [[Friedmann–Lemaître–Robertson–Walker metric|Friedmann–Lemaître–Robertson–Walker solutions]],<ref>{{Harvnb|Carroll|2001|loc=ch. 2}}</ref> allow physicists to model a universe that has evolved over the past 14&nbsp;[[1000000000 (number)|billion]]&nbsp;years from a hot, early Big Bang phase.<ref>{{Harvnb|Bergström|Goobar|2003|loc=ch. 9–11}}; use of these models is justified by the fact that, at large scales of around hundred million [[light-year]]s and more, our own universe indeed appears to be isotropic and homogeneous, cf. {{Harvnb|Peebles|Schramm|Turner|Kron|1991}}</ref> Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,<ref>E.g. with [[WMAP]] data, see {{Harvnb|Spergel|Verde|Peiris|Komatsu|2003}}</ref> further observational data can be used to put the models to the test.<ref>These tests involve the separate observations detailed further on, see, e.g., fig. 2 in {{Harvnb|Bridle|Lahav|Ostriker|Steinhardt|2003}}</ref> Predictions, all successful, include the initial abundance of chemical elements formed in a period of [[Big Bang nucleosynthesis|primordial nucleosynthesis]],<ref>{{Harvnb|Peebles|1966}}; for a recent account of predictions, see {{Harvnb|Coc, Vangioni‐Flam et al.|2004}}; an accessible account can be found in {{Harvnb|Weiss|2006}}; compare with the observations in {{Harvnb|Olive|Skillman|2004}}, {{Harvnb|Bania|Rood|Balser|2002}}, {{Harvnb|O'Meara|Tytler|Kirkman|Suzuki|2001}}, and {{Harvnb|Charbonnel|Primas|2005}}</ref> the large-scale structure of the universe,<ref>{{Harvnb|Lahav|Suto|2004}}, {{Harvnb|Bertschinger|1998}}, {{Harvnb|Springel|White|Jenkins|Frenk|2005}}</ref> and the existence and properties of a "[[thermal radiation|thermal]] echo" from the early cosmos, the [[cosmic background radiation]].<ref>{{Harvnb|Alpher|Herman|1948}}, for a pedagogical introduction, see {{Harvnb|Bergström|Goobar|2003|loc=ch. 11}}; for the initial detection, see {{Harvnb|Penzias|Wilson|1965}} and, for precision measurements by satellite observatories, {{Harvnb|Mather|Cheng|Cottingham|Eplee|1994}} ([[Cosmic Background Explorer|COBE]]) and {{Harvnb|Bennett|Halpern|Hinshaw|Jarosik|2003}} (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's [[polarized light|polarization]], cf. {{Harvnb|Kamionkowski|Kosowsky|Stebbins|1997}} and {{Harvnb|Seljak|Zaldarriaga|1997}}</ref>

Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.<ref>Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. {{Harvnb|Peebles|1993|loc=ch. 18}}, evidence from gravitational lensing, cf. {{Harvnb|Peacock|1999|loc=sec. 4.6}}, and simulations of large-scale structure formation, see {{Harvnb|Springel|White|Jenkins|Frenk|2005}}</ref> There is no generally accepted description of this new kind of matter, within the framework of known [[particle physics]]<ref>{{Harvnb|Peacock|1999|loc=ch. 12}}, {{Harvnb|Peskin|2007}}; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual [[elementary particle]]s ("non-[[baryon]]ic matter"), cf. {{Harvnb|Peacock|1999|loc=ch. 12}}</ref> or otherwise.<ref>Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in {{Harvnb|Mannheim|2006|loc=sec. 9}}</ref> Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual [[equation of state]], known as [[dark energy]], the nature of which remains unclear.<ref>{{Harvnb|Carroll|2001}}; an accessible overview is given in {{Harvnb|Caldwell|2004}}. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. {{Harvnb|Mannheim|2006|loc=sec. 10}}; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. {{Harvnb|Buchert|2008}}</ref>

An [[cosmic inflation|inflationary phase]],<ref>A good introduction is {{Harvnb|Linde|2005}}; for a more recent review, see {{Harvnb|Linde|2006}}</ref> an additional phase of strongly accelerated expansion at cosmic times of around 10<sup>−33</sup> seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.<ref>More precisely, these are the [[flatness problem]], the [[horizon problem]], and the [[monopole problem]]; a pedagogical introduction can be found in {{Harvnb|Narlikar|1993|loc=sec. 6.4}}, see also {{Harvnb|Börner|1993|loc=sec. 9.1}}</ref> Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.<ref>{{Harvnb|Spergel|Bean|Doré|Nolta|2007|loc=sec. 5,6}}</ref> However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.<ref>More concretely, the [[potential]] function that is crucial to determining the dynamics of the [[inflaton]] is simply postulated, but not derived from an underlying physical theory</ref> An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang [[Gravitational singularity|singularity]]. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed<ref>{{Harvnb|Brandenberger|2008|loc=sec. 2}}</ref> (cf. the section on [[#Quantum gravity|quantum gravity]], below).

=== Exotic solutions: time travel, warp drives ===
[[Kurt Gödel]] showed<ref>{{harvnb|Gödel|1949}}</ref> that solutions to Einstein's equations exist that contain [[closed timelike curve]]s (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the [[Tipler cylinder]] and [[Wormhole#Traversable wormholes|traversable wormholes]]. [[Stephen Hawking]] introduced [[chronology protection conjecture]], which is an assumption beyond those of standard general relativity to prevent [[time travel]].

Some [[exact solutions in general relativity]] such as [[Alcubierre drive]] present examples of [[warp drive]] but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability.
<ref>{{Cite journal
|last1=Finazzi|first1=Stefano
|last2= Liberati|first2=Stefano
|last3=Barceló|first3=Carlos
|date=15 June 2009
|title=Semiclassical instability of dynamical warp drives
|journal=Physical Review D
|language=en-US
|volume=79|issue=12|page=124017
|doi=10.1103/PhysRevD.79.124017
|arxiv=0904.0141
|bibcode=2009PhRvD..79l4017F
|s2cid=59575856
}}</ref>

== Advanced concepts ==
=== Asymptotic symmetries ===
{{Main|Bondi–Metzner–Sachs group}}
The spacetime symmetry group for [[special relativity]] is the [[Poincaré group]], which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries, if any, might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, ''viz.'', the Poincaré group.

In 1962 [[Hermann Bondi]], M. G. van der Burg, A. W. Metzner<ref name="bondi etal 1962">{{cite journal|title=Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems|journal= Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences|volume=269|pages=21–52|doi=10.1098/rspa.1962.0161|year=1962|last1=Bondi|first1=H.|last2=Van der Burg|first2=M.G.J.|last3=Metzner|first3=A.|issue=1336|bibcode=1962RSPSA.269...21B|s2cid=120125096}}</ref> and [[Rainer K. Sachs]]<ref name=sachs1962>{{cite journal|title=Asymptotic symmetries in gravitational theory|journal=Physical Review|volume=128|pages=2851–2864|doi=10.1103/PhysRev.128.2851|year=1962|last1=Sachs|first1=R.|issue=6|bibcode=1962PhRv..128.2851S}}</ref> addressed this [[Bondi–Metzner–Sachs group|asymptotic symmetry]] problem in order to investigate the flow of energy at infinity due to propagating [[gravitational wave]]s. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no ''a priori'' assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as ''supertranslations''. This implies the conclusion that General Relativity (GR) does ''not'' reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal [[soft graviton theorem]] in [[quantum field theory]] (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.<ref name=strominger2017>{{cite arXiv|title=Lectures on the Infrared Structure of Gravity and Gauge Theory|eprint=1703.05448|year=2017|last1=Strominger|first1=Andrew|class=hep-th|quote=...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.}}</ref>

=== Causal structure and global geometry ===
{{Main|Causal structure}}
[[File:Penrose.svg|thumb|Penrose–Carter diagram of an infinite [[Minkowski space|Minkowski universe]]]]
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event ''A'' can reach any other location ''X'' before light sent out at ''A'' to ''X''. In consequence, an exploration of all light worldlines ([[Geodesics in general relativity|null geodesics]]) yields key information about the spacetime's causal structure. This structure can be displayed using [[Penrose diagram|Penrose–Carter diagrams]] in which infinitely large regions of space and infinite time intervals are shrunk ("[[Compactification (mathematics)|compactified]]") so as to fit onto a finite map, while light still travels along diagonals as in standard [[spacetime diagram]]s.<ref>{{Harvnb|Frauendiener|2004}}, {{Harvnb|Wald|1984|loc=sec. 11.1}}, {{Harvnb|Hawking|Ellis|1973|loc=sec. 6.8, 6.9}}</ref>

Aware of the importance of causal structure, [[Roger Penrose]] and others developed what is known as [[Spacetime topology|global geometry]]. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the [[Raychaudhuri equation]], and additional non-specific assumptions about the nature of matter (usually in the form of [[energy conditions]]) are used to derive general results.<ref>{{Harvnb|Wald|1984|loc=sec. 9.2–9.4}} and {{Harvnb|Hawking|Ellis|1973|loc=ch. 6}}</ref>

=== Horizons ===
{{Main|Horizon (general relativity)|No hair theorem|Black hole mechanics}}
Using global geometry, some spacetimes can be shown to contain boundaries called [[event horizon|horizons]], which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the [[hoop conjecture]], the relevant length scale is the [[Schwarzschild radius]]<ref>{{Harvnb|Thorne|1972}}; for more recent numerical studies, see {{Harvnb|Berger|2002|loc=sec. 2.1}}</ref>), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's ''horizon'', is not a physical barrier.<ref>{{Harvnb|Israel|1987}}. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and [[apparent horizon]]s cf. {{Harvnb|Hawking|Ellis|1973|pp=312–320}} or {{Harvnb|Wald|1984|loc=sec. 12.2}}; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. {{Harvnb|Ashtekar|Krishnan|2004}}</ref>

[[File:Ergosphere of a rotating black hole.svg|thumb|The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole]]
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a [[Static spacetime|static]] black hole) and the axisymmetric [[Kerr solution]] (used to describe a rotating, [[Stationary spacetime|stationary]] black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, [[linear momentum]], [[angular momentum]], and location at a specified time. This is stated by the [[no hair theorem|black hole uniqueness theorem]]: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.<ref>For first steps, cf. {{Harvnb|Israel|1971}}; see {{Harvnb|Hawking|Ellis|1973|loc=sec. 9.3}} or {{Harvnb|Heusler|1996|loc=ch. 9 and 10}} for a derivation, and {{Harvnb|Heusler|1998}} as well as {{Harvnb|Beig|Chruściel|2006}} as overviews of more recent results</ref>

Even more remarkably, there is a general set of laws known as [[black hole mechanics]], which is analogous to the [[laws of thermodynamics]]. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the [[entropy]] of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the [[Penrose process]]).<ref>The laws of black hole mechanics were first described in {{Harvnb|Bardeen|Carter|Hawking|1973}}; a more pedagogical presentation can be found in {{Harvnb|Carter|1979}}; for a more recent review, see {{Harvnb|Wald|2001|loc=ch. 2}}. A thorough, book-length introduction including an introduction to the necessary mathematics {{Harvnb|Poisson|2004}}. For the Penrose process, see {{Harvnb|Penrose|1969}}</ref> There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.<ref>{{Harvnb|Bekenstein|1973}}, {{Harvnb|Bekenstein|1974}}</ref> This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for the black hole area to decrease as long as other processes ensure that entropy increases overall. As thermodynamical objects with nonzero temperature, black holes should emit [[thermal radiation]]. Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in [[Planck's law]]. This radiation is known as [[Hawking radiation]] (cf. the [[#Quantum field theory in curved spacetime|quantum theory section]], below).<ref>The fact that black holes radiate, quantum mechanically, was first derived in {{Harvnb|Hawking|1975}}; a more thorough derivation can be found in {{Harvnb|Wald|1975}}. A review is given in {{Harvnb|Wald|2001|loc=ch. 3}}</ref>

There are many other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("[[particle horizon]]"), and some regions of the future cannot be influenced (event horizon).<ref>{{Harvnb|Narlikar|1993|loc=sec. 4.4.4, 4.4.5}}</ref> Even in flat Minkowski space, when described by an accelerated observer ([[Rindler space]]), there will be horizons associated with a semiclassical radiation known as [[Unruh effect|Unruh radiation]].<ref>Horizons: cf. {{Harvnb|Rindler|2001|loc=sec. 12.4}}. Unruh effect: {{Harvnb|Unruh|1976}}, cf. {{Harvnb|Wald|2001|loc=ch. 3}}</ref>

=== Singularities ===
{{Main|Spacetime singularity}}
Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as [[spacetime singularity|spacetime singularities]], where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the [[Ricci scalar]], take on infinite values.<ref>{{Harvnb|Hawking|Ellis|1973|loc=sec. 8.1}}, {{Harvnb|Wald|1984|loc=sec. 9.1}}</ref> Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,<ref>{{Harvnb|Townsend|1997|loc=ch. 2}}; a more extensive treatment of this solution can be found in {{Harvnb|Chandrasekhar|1983|loc=ch. 3}}</ref> or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.<ref>{{Harvnb|Townsend|1997|loc=ch. 4}}; for a more extensive treatment, cf. {{Harvnb|Chandrasekhar|1983|loc=ch. 6}}</ref> The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities ([[Big Crunch]]) as well.<ref>{{Harvnb|Ellis|Van Elst|1999}}; a closer look at the singularity itself is taken in {{Harvnb|Börner|1993|loc=sec. 1.2}}</ref>

Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.<ref>Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called [[eikonal approximation]]s of many wave equations, namely the "[[caustic (mathematics)|caustics]]", are resolved into finite peaks beyond that approximation.</ref> The famous [[singularity theorems]], proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage<ref>Namely when there are [[trapped null surface]]s, cf. {{Harvnb|Penrose|1965}}</ref> and also at the beginning of a wide class of expanding universes.<ref>{{Harvnb|Hawking|1966}}</ref> However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the [[BKL singularity|BKL conjecture]]).<ref>The conjecture was made in {{Harvnb|Belinskii|Khalatnikov|Lifschitz|1971}}; for a more recent review, see {{Harvnb|Berger|2002}}. An accessible exposition is given by {{Harvnb|Garfinkle|2007}}</ref> The [[cosmic censorship hypothesis]] states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.<ref>The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in {{Harvnb|Penrose|1969}}; a textbook-level account is given in {{Harvnb|Wald|1984|pp=302–305}}. For numerical results, see the review {{Harvnb|Berger|2002|loc=sec. 2.1}}</ref>

=== Evolution equations ===
{{Main|Initial value formulation (general relativity)}}
Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the [[time evolution]] of the metric tensor. It must be combined with a [[coordinate condition]], which is analogous to [[gauge fixing]] in other field theories.<ref>{{Harvnb|Hawking|Ellis|1973|loc=sec. 7.1}}</ref>

To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the [[ADM formalism]].<ref>{{Harvnb|Arnowitt|Deser|Misner|1962}}; for a pedagogical introduction, see {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;21.4–§&nbsp;21.7}}</ref> These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always [[existence theorem|exist]], and are uniquely defined, once suitable initial conditions have been specified.<ref>{{Harvnb|Fourès-Bruhat|1952}} and {{Harvnb|Bruhat|1962}}; for a pedagogical introduction, see {{Harvnb|Wald|1984|loc=ch. 10}}; an online review can be found in {{Harvnb|Reula|1998}}</ref> Such formulations of Einstein's field equations are the basis of numerical relativity.<ref>{{Harvnb|Gourgoulhon|2007}}; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see {{Harvnb|Lehner|2001}}</ref>

=== Global and quasi-local quantities ===
{{Main|Mass in general relativity}}
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.<ref>{{Harvnb|Misner|Thorne|Wheeler|1973|loc=§&nbsp;20.4}}</ref>

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" ([[ADM mass]])<ref>{{Harvnb|Arnowitt|Deser|Misner|1962}}</ref> or suitable symmetries ([[Komar mass]]).<ref>{{Harvnb|Komar|1959}}; for a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}}; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. {{Harvnb|Ashtekar|Magnon-Ashtekar|1979}}</ref> If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the [[Mass in general relativity#ADM and Bondi masses in asymptotically flat space-times|Bondi mass]] at null infinity.<ref>For a pedagogical introduction, see {{Harvnb|Wald|1984|loc=sec. 11.2}}</ref> Just as in [[Physics in the Classical Limit|classical physics]], it can be shown that these masses are positive.<ref>{{Harvnb|Wald|1984|p=295 and refs therein}}; this is important for questions of stability—if there were [[negative mass]] states, then flat, empty Minkowski space, which has mass zero, could evolve into these states</ref> Corresponding global definitions exist for momentum and angular momentum.<ref>{{Harvnb|Townsend|1997|loc=ch. 5}}</ref> There have also been a number of attempts to define ''quasi-local'' quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about [[isolated system]]s, such as a more precise formulation of the hoop conjecture.<ref>Such quasi-local mass–energy definitions are the [[Hawking energy]], [[Geroch energy]], or Penrose's quasi-local energy–momentum based on [[Twistor theory|twistor]] methods; cf. the review article {{Harvnb|Szabados|2004}}</ref>

== Relationship with quantum theory ==
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to [[solid-state physics]], would be the other.<ref>An overview of quantum theory can be found in standard textbooks such as {{Harvnb|Messiah|1999}}; a more elementary account is given in {{Harvnb|Hey|Walters|2003}}</ref> However, how to reconcile quantum theory with general relativity is still an open question.

=== Quantum field theory in curved spacetime ===
{{Main|Quantum field theory in curved spacetime}}
Ordinary [[quantum field theory|quantum field theories]], which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.<ref>{{Harvnb|Ramond|1990}}, {{Harvnb|Weinberg|1995}}, {{Harvnb|Peskin|Schroeder|1995}}; a more accessible overview is {{Harvnb|Auyang|1995}}</ref> In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.<ref>{{Harvnb|Wald|1994}}, {{Harvnb|Birrell|Davies|1984}}</ref> Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as [[Hawking radiation]] leading to the possibility that they [[Black hole evaporation|evaporate]] over time.<ref>For Hawking radiation {{Harvnb|Hawking|1975}}, {{Harvnb|Wald|1975}}; an accessible introduction to black hole evaporation can be found in {{Harvnb|Traschen|2000}}</ref> As briefly mentioned [[#Horizons|above]], this radiation plays an important role for the thermodynamics of black holes.<ref>{{Harvnb|Wald|2001|loc=ch. 3}}</ref>

=== Quantum gravity ===
{{Main|Quantum gravity}}
{{See also|String theory|Canonical general relativity|Loop quantum gravity|Causal dynamical triangulation|Causal sets}}
[[File:Calabi yau.jpg|left|thumb|upright|Projection of a [[Calabi–Yau manifold]], one of the ways of [[compactification (physics)|compactifying]] the extra dimensions posited by string theory]]

The demand for consistency between a quantum description of matter and a geometric description of spacetime,<ref>Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. {{Harvnb|Carlip|2001|loc=sec. 2}}</ref> as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.<ref>{{Harvnb|Schutz|2003|p=407}}</ref> Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.<ref name="Hamber 2009">{{Harvnb|Hamber|2009}}</ref><ref>A timeline and overview can be found in {{Harvnb|Rovelli|2000}}</ref>

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.<ref>{{Harvnb|'t Hooft|Veltman|1974}}</ref> Some have argued that at low energies, this approach proves successful, in that it results in an acceptable [[effective field theory|effective (quantum) field theory]] of gravity.<ref>{{Harvnb|Donoghue|1995}}</ref> At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative [[non-renormalizable|non-renormalizability]]").<ref>In particular, a perturbative technique known as [[renormalization]], an integral part of deriving predictions which take into account higher-energy contributions, cf. {{Harvnb|Weinberg|1996|loc=ch. 17, 18}}, fails in this case; cf. {{Harvnb|Veltman|1975}}, {{Harvnb|Goroff|Sagnotti|1985}}; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see {{Harvnb|Hamber|2009}}</ref>

[[File:Spin network.svg|thumb|upright|Simple [[spin network]] of the type used in loop quantum gravity]]
One attempt to overcome these limitations is [[string theory]], a quantum theory not of [[point particle]]s, but of minute one-dimensional extended objects.<ref>An accessible introduction at the undergraduate level can be found in {{Harvnb|Zwiebach|2004}}; more complete overviews can be found in {{Harvnb|Polchinski|1998a}} and {{Harvnb|Polchinski|1998b}}</ref> The theory promises to be a [[theory of everything|unified description]] of all particles and interactions, including gravity;<ref>At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different [[normal mode|modes]] of oscillation of one and the same type of fundamental string appear as particles with different ([[Electric charge|electric]] and other) [[Charge (physics)|charges]], e.g. {{Harvnb|Ibanez|2000}}. The theory is successful in that one mode will always correspond to a [[graviton]], the [[messenger particle]] of gravity, e.g. {{Harvnb|Green|Schwarz|Witten|1987|loc=sec. 2.3, 5.3}}</ref> the price to pay is unusual features such as six [[Superstring theory#Extra dimensions|extra dimensions]] of space in addition to the usual three.<ref>{{Harvnb|Green|Schwarz|Witten|1987|loc=sec. 4.2}}</ref> In what is called the [[second superstring revolution]], it was conjectured that both string theory and a unification of general relativity and [[supersymmetry]] known as [[supergravity]]<ref>{{Harvnb|Weinberg|2000|loc=ch. 31}}</ref> form part of a hypothesized eleven-dimensional model known as [[M-theory]], which would constitute a uniquely defined and consistent theory of quantum gravity.<ref>{{Harvnb|Townsend|1996}}, {{Harvnb|Duff|1996}}</ref>

Another approach starts with the [[canonical quantization]] procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. [[#Evolution equations|evolution equations]] above), the result is the [[Wheeler–deWitt equation]] (an analogue of the [[Schrödinger equation]]) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.<ref>{{Harvnb|Kuchař|1973|loc=sec. 3}}</ref> However, with the introduction of what are now known as [[Ashtekar variables]],<ref>These variables represent geometric gravity using mathematical analogues of [[electric field|electric]] and [[magnetic field]]s; cf. {{Harvnb|Ashtekar|1986}}, {{Harvnb|Ashtekar|1987}}</ref> this leads to a promising model known as [[loop quantum gravity]]. Space is represented by a web-like structure called a [[spin network]], evolving over time in discrete steps.<ref>For a review, see {{Harvnb|Thiemann|2007}}; more extensive accounts can be found in {{Harvnb|Rovelli|1998}}, {{Harvnb|Ashtekar|Lewandowski|2004}} as well as in the lecture notes {{Harvnb|Thiemann|2003}}</ref>

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,<ref>{{Harvnb|Isham|1994}}, {{Harvnb|Sorkin|1997}}</ref> there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman [[Path integral formulation|Path Integral]] approach and [[Regge calculus]],<ref name="Hamber 2009" /> [[Causal dynamical triangulation|dynamical triangulations]],<ref>{{Harvnb|Loll|1998}}</ref> [[causal sets]],<ref>{{Harvnb|Sorkin|2005}}</ref> twistor models<ref>{{Harvnb|Penrose|2004|loc=ch. 33 and refs therein}}</ref> or the path integral based models of [[quantum cosmology]].<ref>{{Harvnb|Hawking|1987}}</ref>

[[File:LIGO measurement of gravitational waves.svg|thumb|Observation of gravitational waves from binary black hole merger GW150914]]
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.<ref>{{Harvnb|Ashtekar|2007}}, {{Harvnb|Schwarz|2007}}</ref>

== Current status ==
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete.<ref>{{Harvnb|Maddox|1998|pp=52–59, 98–122}}; {{Harvnb|Penrose|2004|loc=sec. 34.1, ch. 30}}</ref> The problem of quantum gravity and the question of the reality of spacetime singularities remain open.<ref>section [[#Quantum gravity|Quantum gravity]], above</ref> Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.<ref>section [[#Cosmology|Cosmology]], above</ref>

Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,<ref>{{Harvnb|Friedrich|2005}}</ref> while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes).<ref>A review of the various problems and the techniques being developed to overcome them, see {{Harvnb|Lehner|2002}}</ref> In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015.<ref name="NSF" /><ref>See {{Harvnb|Bartusiak|2000}} for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as [[GEO600]] and [[LIGO]]</ref><ref>For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see {{Harvnb|Blanchet|Faye|Iyer|Sinha|2008}}, and {{Harvnb|Arun|Blanchet|Iyer|Qusailah|2008}}; for a review of work on compact binaries, see {{Harvnb|Blanchet|2006}} and {{Harvnb|Futamase|Itoh|2006}}; for a general review of experimental tests of general relativity, see {{Harvnb|Will|2006}}</ref> A century after its introduction, general relativity remains a highly active area of research.<ref>See, e.g., the ''[[Living Reviews in Relativity]]'' journal.</ref>
{{clear}}

== See also ==
* {{Annotated link|Alcubierre drive}} (warp drive)
* {{Annotated link|Alternatives to general relativity}}
* {{Annotated link|Contributors to general relativity}}
* {{Annotated link|Derivations of the Lorentz transformations}}
* {{Annotated link|Ehrenfest paradox}}
* {{Annotated link|Einstein–Hilbert action}}
* {{Annotated link|Einstein's thought experiments}}
* {{Annotated link|General relativity priority dispute}}
* {{Annotated link|Introduction to the mathematics of&nbsp;general relativity}}
* {{Annotated link|Nordström's theory of gravitation}}
* {{Annotated link|Problems with Einstein's general theory of relativity}}
* {{Annotated link|Ricci calculus}}
* {{Annotated link|Timeline of gravitational physics and relativity}}

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|title=Simulations of the formation, evolution and clustering of galaxies and quasars
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{{refend}}

== Further reading ==
=== Popular books ===
* {{Citation|ref=none|last=Einstein|first=A.|author-link=Albert Einstein|title=Relativity: The Special and the General Theory|title-link=Relativity: The Special and the General Theory|location=Berlin|date=1916|isbn=978-3-528-06059-6}}{{clarify|reason=Surely Einstein 1916 had a German title published without an ISBN; pick one edition and cite that; don't jam multiple editions into a singe template; name the publisher;|date=December 2024}}
* {{Citation|ref=none|last=Geroch|first= R.|author-link=Robert Geroch| title=General Relativity from A to B|location=Chicago|publisher=University of Chicago Press|date=1981|isbn=978-0-226-28864-2}}
* {{Citation|ref=none|author=Lieber, Lillian|author-link=Lillian Lieber| title=The Einstein Theory of Relativity: A Trip to the Fourth Dimension|location=Philadelphia|publisher=Paul Dry Books, Inc.|date=2008|isbn=978-1-58988-044-3}}
* {{Citation |ref=none |author-link=Bernard F. Schutz |last=Schutz |first=Bernard F. |contribution=Gravitational radiation |editor-last=Murdin |editor-first=Paul |title=Encyclopedia of Astronomy and Astrophysics |date=2001 |publisher=Institute of Physics Pub. |isbn=978-1-56159-268-5}}
*{{Cite book|ref=none|title=Black Holes and Time Warps: Einstein's Outrageous Legacy|title-link=Black Holes and Time Warps|last1=Thorne|first1=Kip|last2=Hawking|first2=Stephen|publisher=W. W. Norton|year=1994|isbn=0-393-03505-0|location=New York|author-link=Kip Thorne}}
* {{Citation|ref=none|author=Wald, Robert M.|author-link=Robert Wald|title=Space, Time, and Gravity: the Theory of the Big Bang and Black Holes|location=Chicago|publisher=University of Chicago Press|date=1992|isbn=978-0-226-87029-8}}
* {{Citation|ref=none|author-link=John Archibald Wheeler|last1=Wheeler|first1=John|last2=Ford|first2=Kenneth|date=1998|title=Geons, Black Holes, & Quantum Foam: a life in physics|isbn=978-0-393-31991-0|location=New York|publisher=W. W. Norton}}

=== Beginning undergraduate textbooks ===
*{{Citation|ref=none|author=Callahan, James J.|title=The Geometry of Spacetime: an Introduction to Special and General Relativity| location=New York|publisher=Springer|date=2000|isbn=978-0-387-98641-8}}
*{{Citation|ref=none|author1=Taylor, Edwin F.|author2=Wheeler, John Archibald|title=Exploring Black Holes: Introduction to General Relativity|publisher=Addison Wesley|date=2000|isbn=978-0-201-38423-9}}

=== Advanced undergraduate textbooks ===
*{{Citation|ref=none|last=Cheng|first=Ta-Pei|title=Relativity, Gravitation and Cosmology: a Basic Introduction|location=Oxford and New York| publisher=Oxford University Press|date=2005|isbn=978-0-19-852957-6}}
*{{Citation|ref=none|last= Dirac|first=Paul|author-link=Paul Dirac|title=General Theory of Relativity|publisher=Princeton University Press|date=1996|isbn=978-0-691-01146-2}}
*{{Citation|ref=none|last1=Gron|first1=O.|last2=Hervik|first2=S.| title=Einstein's General theory of Relativity|publisher=Springer|date=2007|isbn=978-0-387-69199-2}}
*{{Citation|ref=none|author=Hartle, James B.|author-link=James Hartle|title=Gravity: an Introduction to Einstein's General Relativity|location=San Francisco|publisher=Addison-Wesley|date=2003|isbn=978-0-8053-8662-2}}
*{{Citation|ref=none|author=[[Lane P. Hughston|Hughston, L. P.]]|author2=Tod, K. P.|title=Introduction to General Relativity|location=Cambridge|publisher=Cambridge University Press|date=1991|isbn=978-0-521-33943-8}}
*{{Citation|ref=none|author=d'Inverno, Ray|title=Introducing Einstein's Relativity|location=Oxford|publisher=Oxford University Press|date=1992|isbn=978-0-19-859686-8|url-access=registration|url=https://archive.org/details/introducingeinst0000dinv}}
*{{cite book|ref=none|last=Ludyk|first=Günter|title=Einstein in Matrix Form|date=2013|publisher=Springer|location=Berlin|isbn= 978-3-642-35797-8 |edition=1st}}
*{{Citation|ref=none|first=Christian|last=Møller|title=The Theory of Relativity|publisher=Oxford University Press|year=1955|orig-date=1952|url=https://archive.org/details/theoryofrelativi029229mbp|oclc=7644624}}
*{{Citation|ref=none|author=Moore, Thomas A|title=A General Relativity Workbook|publisher=University Science Books|date=2012|isbn=978-1-891389-82-5}}
*{{Citation |ref=none |author-link=Bernard F. Schutz |author=Schutz, B. F. |title=A First Course in General Relativity |edition=Second |publisher=Cambridge University Press |date=2009 |isbn=978-0-521-88705-2 |url=https://archive.org/details/firstcourseingen00bern_0}}

=== Graduate textbooks ===
*{{Citation|ref=none|author=Carroll, Sean M. |author-link=Sean M. Carroll |title=Spacetime and Geometry: An Introduction to General Relativity |location=San Francisco |publisher=Addison-Wesley |date=2004 |isbn=978-0-8053-8732-2}}
*{{Citation|ref=none|last=Grøn|first=Øyvind |author-link=Øyvind Grøn| author2=Hervik, Sigbjørn|title=Einstein's General Theory of Relativity|location=New York|publisher=Springer|date=2007|isbn=978-0-387-69199-2}}
*{{Citation|ref=none|author=Landau, Lev D.|author-link=Lev Landau|author2= Lifshitz, Evgeny F.| author2-link=Evgeny Lifshitz|title=The Classical Theory of Fields (4th ed.)|location=London|publisher=Butterworth-Heinemann|date=1980|isbn=978-0-7506-2768-9}}
*{{Cite book|title=Foundations of General Relativity: From Einstein to Black Holes|last=Landsman|first=Klaas|publisher=Radboud University Press|year=2021|isbn=9789083178929|url=https://books.radbouduniversitypress.nl/index.php/rup/catalog/book/foundations_of_general_relativity}}
*{{Citation|ref=none|author=Stephani, Hans|title=General Relativity: An Introduction to the Theory of the Gravitational Field|location=Cambridge|publisher=Cambridge University Press|date=1990|isbn=978-0-521-37941-0}}
*{{Citation|ref=none|author=Charles W. Misner |author2=Kip S. Thorne |author3=John Archibald Wheeler |author3-link=John Archibald Wheeler |author2-link=Kip S. Thorne |author-link=Charles W. Misner| title=Gravitation | title-link=Gravitation (book) |publisher=W. H. Freeman, Princeton University Press |date=1973 |isbn=0-7167-0344-0}}
* {{Citation|ref=none|author = R.K. Sachs | author2= H. Wu | title = General Relativity for Mathematicians | publisher=Springer-Verlag | date = 1977 | isbn=1-4612-9905-5 | author-link= Rainer K. Sachs}}
* {{cite book |last=Wald |first=Robert M. |title=General Relativity |title-link=General Relativity (book) |publisher=University of Chicago Press |publication-place=Chicago |date=1984 |isbn=0-226-87032-4 |oclc=10018614 |author-link=Robert Wald |ref=none}}

=== Specialists' books ===
*{{Cite book|ref=none|title=The Large Scale Structure of Space-time|title-link=The Large Scale Structure of Space-Time|last1=Hawking|first1=Stephen|last2=Ellis|first2=George|publisher=Cambridge University Press|year=1975|isbn=978-0-521-09906-6|author-link=Stephen Hawking|author-link2=George F. R. Ellis}}
*{{Cite book|ref=none|title=A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics|last=Poisson|first=Eric|publisher=Cambridge University Press|year=2007|isbn=978-0-521-53780-3|author-link=Eric Poisson}}

=== Journal articles ===
*{{Citation|ref=none|last=Einstein|first=Albert|author-link=Albert Einstein|title=Die Grundlage der allgemeinen Relativitätstheorie|journal=Annalen der Physik|volume=49|issue=7|date=1916|pages=769–822|doi= 10.1002/andp.19163540702|bibcode= 1916AnP...354..769E|url= https://einsteinpapers.press.princeton.edu/vol6-doc/311}} See also [https://einsteinpapers.press.princeton.edu/vol6-trans/158 English translation at Einstein Papers Project]
*{{Citation|ref=none|last1=Flanagan|first1=Éanna É.|first2=Scott A.|last2=Hughes|title=The basics of gravitational wave theory|journal=New J. Phys.|volume=7|issue=1|date=2005|page=204|doi= 10.1088/1367-2630/7/1/204|doi-access=free|arxiv=gr-qc/0501041|bibcode= 2005NJPh....7..204F}}
*{{Citation|ref=none|last1=Landgraf|first1=M.|first2=M.|last2=Hechler|first3=S.|last3=Kemble|date=2005|title=Mission design for LISA Pathfinder|journal=Class. Quantum Grav.|volume=22|issue=10|pages=S487–S492|arxiv= gr-qc/0411071|doi= 10.1088/0264-9381/22/10/048|bibcode= 2005CQGra..22S.487L|s2cid=119476595}}
*{{Citation|ref=none|last=Nieto|first=Michael Martin|title=The quest to understand the Pioneer anomaly|journal=Europhysics News|volume=37|issue=6|date=2006|pages=30–34|arxiv= gr-qc/0702017|doi= 10.1051/epn:2006604|doi-access=free|bibcode= 2006ENews..37f..30N|url= http://www.europhysicsnews.org/articles/epn/pdf/2006/06/epn06604.pdf |archive-url=https://web.archive.org/web/20150924004350/http://www.europhysicsnews.org/articles/epn/pdf/2006/06/epn06604.pdf |archive-date=2015-09-24 |url-status=live}}
*{{Citation|ref=none|author-link=Irwin I. Shapiro|last1=Shapiro|first1= I. I.|last2=Pettengill|first2=Gordon|last3=Ash|first3=Michael|last4=Stone|first4=Melvin|last5=Smith|first5=William|last6=Ingalls|first6=Richard|last7=Brockelman|first7=Richard|title=Fourth test of general relativity: preliminary results|journal=Phys. Rev. Lett.|volume=20|issue=22|pages=1265–1269|date=1968|doi= 10.1103/PhysRevLett.20.1265|bibcode= 1968PhRvL..20.1265S}}
*{{Citation|ref=none|last1=Valtonen|first1= M. J.|last2=Lehto|first2=H. J.|last3=Nilsson|first3=K.|last4=Heidt|first4=J.|last5=Takalo|first5=L. O.|last6=Sillanpää|first6=A.|last7=Villforth|first7=C.|last8=Kidger|first8=M.|display-authors=etal|title=A massive binary black-hole system in OJ 287 and a test of general relativity|journal=Nature|volume=452|issue=7189|pages=851–853|date=2008|doi= 10.1038/nature06896|pmid= 18421348|bibcode= 2008Natur.452..851V|arxiv= 0809.1280|s2cid=4412396}}

== External links ==
{{Commons category}}
{{Wikibooks}}
{{Wikiquote}}
{{Wikiversity}}
{{Wikisource portal|Relativity}}
{{Wikisource|Relativity: The Special and General Theory}}
*[http://www.einstein-online.info/ Einstein Online] {{Webarchive|url=https://web.archive.org/web/20140601190245/http://www.einstein-online.info/ |date=1 June 2014 }}&nbsp;– Articles on a variety of aspects of relativistic physics for a general audience; hosted by the [[Max Planck Institute for Gravitational Physics]]
*[http://www.geo600.de/ GEO600 home page], the official website of the GEO600 project.
*[http://www.ligo.caltech.edu/ LIGO Laboratory]
*[https://web.archive.org/web/20150327100311/http://archive.ncsa.illinois.edu/Cyberia/NumRel/NumRelHome.html NCSA Spacetime Wrinkles]&nbsp;– produced by the numerical relativity group at the [[National Center for Supercomputing Applications|NCSA]], with an elementary introduction to general relativity
'''{{hlist|Courses|Lectures|Tutorials}}'''
* {{YouTube |id=hbmf0bB38h0&list=EC6C8BDEEBA6BDC78D |title=Einstein's General Theory of Relativity}} (lecture by [[Leonard Susskind]] recorded 22 September 2008 at [[Stanford University]]).
*[https://web.archive.org/web/20160303175347/http://www.luth.obspm.fr/IHP06/ Series of lectures on General Relativity] given in 2006 at the [[Institut Henri Poincaré]] (introductory/advanced).
*[https://web.archive.org/web/20070707011024/http://math.ucr.edu/home/baez/gr/ General Relativity Tutorials] by [[John Baez]].
*{{cite web|ref=none |author=Brown, Kevin |title=Reflections on relativity |work=Mathpages.com |url=http://www.mathpages.com/rr/rrtoc.htm |access-date=29 May 2005 |archive-url=https://web.archive.org/web/20151218205507/http://mathpages.com/rr/rrtoc.htm |archive-date=18 December 2015 }}
*{{cite arXiv|ref=none |author=Carroll, Sean M. |title=Lecture Notes on General Relativity |eprint=gr-qc/9712019 |year=1997 }}
*{{cite web|ref=none |author=Moor, Rafi |title=Understanding General Relativity |url=http://www.rafimoor.com/english/GRE.htm |access-date=11 July 2006}}
*{{cite web|ref=none |author=Waner, Stefan |title=Introduction to Differential Geometry and General Relativity |url=http://www.zweigmedia.com/diff_geom/tc.html |access-date=5 April 2015}}
*[https://feynmanlectures.caltech.edu/II_42.html The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space]

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