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| {{See introduction}}
| | My name is Joycelyn and I am studying Computing and Information Science and Architecture at Waytown / Great Britain.<br><br>Also visit my web-site [http://Indian-Tamil-Recipe.com/breakfast/aappam/ FIFA 15 Coin hack] |
| {{General relativity}}
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| [[File:Black Hole Milkyway.jpg|thumb|A simulated [[black hole]] of 10 [[solar mass]]es as seen from a distance of 600 kilometers with the [[Milky Way]] in the background.]]
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| '''General relativity''', or the '''general theory of relativity''', is the [[differential geometry|geometric]] [[Theoretical physics|theory]] of [[gravitation]] published by [[Albert Einstein]] in 1916<ref>{{cite web|title=Nobel Prize Biography|url=http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/einstein-bio.html|work=Nobel Prize Biography|publisher=Nobel Prize|accessdate=25 February 2011}}</ref> and the current description of gravitation in [[modern physics]]. General relativity generalizes [[special relativity]] and [[Newton's law of universal gravitation]], providing a unified description of gravity as a geometric property of [[space]] and [[Time in physics|time]], or [[spacetime]]. In particular, the [[curvature]] of spacetime is directly related to the [[energy]] and [[momentum]] of whatever [[matter]] and [[radiation]] are present. The relation is specified by the [[Einstein field equations]], a system of [[partial differential equation]]s.
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| Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in [[free fall]], and the propagation of [[light]]. Examples of such differences include [[gravitational time dilation]], [[gravitational lensing]], the [[gravitational redshift]] of light, and the [[Shapiro delay|gravitational time delay]]. The predictions of general relativity have been [[tests of general relativity|confirmed]] in all observations and experiments to date. Although general relativity is [[Alternatives to general relativity|not the only relativistic theory of gravity]], it is the [[Occam's razor|simplest theory]] that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of [[quantum mechanics|quantum physics]] to produce a complete and self-consistent theory of [[quantum gravity]].
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| Einstein's theory has important astrophysical implications. For example, it implies the existence of [[black hole]]s—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive [[star]]s. There is ample evidence that the intense [[radiation]] emitted by certain kinds of astronomical objects is due to black holes; for example, [[X-ray binary#Microquasar|microquasars]] and [[active galactic nucleus|active galactic nuclei]] result from the presence of [[stellar black hole]]s and black holes of a [[supermassive black hole|much more massive type]], respectively. The bending of light by gravity can lead to the phenomenon of [[gravitational lens]]ing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of [[gravitational wave]]s, which since have been observed indirectly; a direct measurement is the aim of projects such as [[LIGO]] and NASA/ESA [[Laser Interferometer Space Antenna]] and various [[pulsar timing array]]s. In addition, general relativity is the basis of current [[Physical cosmology|cosmological]] models of a consistently [[expanding universe]].
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| {{TOC limit|limit=3}}
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| ==History==
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| {{Main|History of general relativity|Classical theories of gravitation}}
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| [[File:Albert Einstein portrait.jpg|thumb|250px|right|upright|[[Albert Einstein]] developed the theories of special and general relativity. Picture from 1921.]]
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| Soon after publishing the [[special relativity|special theory of relativity]] in 1905, Einstein started thinking about how to incorporate [[gravity]] into his new relativistic framework. In 1907, beginning with a simple [[thought experiment]] involving an observer in free fall, 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]]. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, and form the core of Einstein's general theory of relativity.<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}}. 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>
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| The Einstein field equations are [[nonlinear system#Nonlinear differential equations|nonlinear]] and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist [[Karl Schwarzschild]] found the first non-trivial exact solution to the Einstein field equations, the so-called [[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, which eventually resulted in the [[Reissner–Nordström metric|Reissner–Nordström solution]], now associated with 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 [[physical cosmology|cosmology]]. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the [[cosmological constant]]—to reproduce that "observation".<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 our [[Metric expansion of space|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 our 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 [[General relativity#Cosmology|Cosmology]], below</ref>
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| 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 himself had shown in 1915 how his theory explained the [[Tests of general relativity#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> Similarly, a 1919 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 May 29, 1919]],<ref>{{Harvnb|Kennefick|2005}}, {{Harvnb|Kennefick|2007}}</ref> making Einstein instantly famous.<ref>{{Harvnb|Pais|1982|loc=ch. 16}}</ref> Yet the theory entered the mainstream of [[theoretical physics]] and [[astrophysics]] only with the developments between approximately 1960 and 1975, now known as the [[History of general relativity#More about GR history|golden age of general relativity]].<ref>{{Cite book|title=The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday |chapter=Warping spacetime |first1=Kip |last1=Thorne |publisher=Cambridge University Press |year=2003 |isbn=0-521-82081-2 |page=74 |url=http://books.google.com/books?id=yLy4b61rfPwC|ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}, [http://books.google.com/books?id=yLy4b61rfPwC&pg=PA74 Extract of page 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 [[General relativity#Orbital effects and the relativity of direction|Orbital effects and the relativity of direction]], [[General relativity#Gravitational time dilation and frequency shift|Gravitational time dilation and frequency shift]] and [[General relativity#Light deflection and gravitational time delay|Light deflection and gravitational time delay]], and references therein</ref> and relativistic cosmology, too, became amenable to direct observational tests.<ref>Section [[General relativity#Cosmology|Cosmology]] and references therein; the historical development is in {{Harvnb|Overbye|1999}}</ref>
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| ==From classical mechanics to general relativity==
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| 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>
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| ===Geometry of Newtonian gravity===
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| [[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 provides the same relative force.]]
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| 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>
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| 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, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerating rocket generating a force equal to gravity.<ref>{{Harvnb|Wheeler|1990|loc=ch. 2}}</ref>
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| 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 experiment]]s 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 result 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>
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| ===Relativistic generalization===
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| [[File:Light cone.svg|thumb|left|upright|[[Light cone]]]]
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| As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a [[limiting case]] of [[special relativity|(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 also includes translations and rotations.) The differences between the two become significant when we are 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|2006a}}</ref>
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| With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see the image on the left). The light-cones define a causal structure: for each [[event (relativity)|event]] A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are 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 space–time'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>
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| Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, 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>
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| 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 frames. 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. [[General relativity#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 [[General relativity#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>
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| 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 [[General relativity#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>
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| ===Einstein's equations===
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| {{Main|Einstein field equations|Mathematics of general relativity}}
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| 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]] (that is, [[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 space-time. 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 set of equations are what are called Einstein's (field) equations:
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| :<math>G_{\mu\nu}\equiv R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu}.\,</math>
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| On the left-hand side is the [[Einstein tensor]], a specific divergence-free combination of the [[Ricci curvature|Ricci tensor]] <math>R_{\mu\nu}</math> and the metric. Where <math>G_{\mu\nu}</math> is symmetric. In particular,
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| :<math>R=g^{\mu\nu}R_{\mu\nu}\,</math>
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| is the curvature scalar. The Ricci tensor itself is related to the more general [[Riemann curvature tensor]] as
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| :<math>\quad R_{\mu\nu}={R^\alpha}_{\mu\alpha\nu}.\,</math>
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| On the right-hand side, ''<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 [[orbits]] (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8π''G''/''c''<sup>4</sup>, with ''G'' the [[gravitational constant]] and ''c'' the [[speed of light]].<ref>{{Harvnb|Kenyon|1990|loc=sec. 7.4}}</ref> When there is no matter present, so that the energy–momentum tensor vanishes, the result are the vacuum Einstein equations,
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| :<math>R_{\mu\nu}=0.\,</math>
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| 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 [[Brans–Dicke theory]], [[teleparallelism]], 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>
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| ==Definition and basic applications==
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| {{See also|Mathematics of general relativity|Physical theories modified by general relativity}}
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| 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.
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| ===Definition and basic properties===
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| 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 [[energy–momentum]] contained in that spacetime.<ref>{{Harvnb|Wald|1984|loc=ch. 4}},
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| {{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]], [[orbit]]al motion, and [[spacecraft]] [[trajectories]]), correspond to inertial motion within a [[curvature|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|2004}}</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>
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| 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>
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| As it is constructed using [[tensor]]s, 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|2006b}}</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>
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| ===Model-building===
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| 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>
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| Einstein's equations are nonlinear [[partial differential equation]]s 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 metric|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 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>
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| 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 [[Parameterized post-Newtonian formalism|parametrized post-Newtonian]] (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and [[alternatives to general relativity|alternative theories]].<ref>{{Harvnb|Will|2006|loc=sec. 3.2}}, {{Harvnb|Will|1993|loc=ch. 4}}</ref>
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| ==Consequences of Einstein's theory==
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| 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 the ninety years of research that followed Einstein's initial publication.
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| ===Gravitational time dilation and frequency shift===
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| {{Main|Gravitational time dilation}}
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| [[File:Gravitational red-shifting.png|thumb|Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body]]
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| 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=§ 38.5}}</ref>
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| 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>
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| ===Light deflection and gravitational time delay===
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| {{Main|Kepler problem in general relativity|Gravitational lens|Shapiro delay}}
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| [[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)]]
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| General relativity predicts that the path of light is bent in a gravitational field; light passing a massive body is deflected towards that body. This effect has been confirmed by observing the light of stars or distant [[quasar]]s being deflected as it passes the [[Sun]].<ref>Cf. {{Harvnb|Kennefick|2005}} for the classic early measurements by the [[Arthur Eddington|Eddington]] 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>
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| This and related predictions follow from the fact that light follows what is called a light-like or 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]] 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>
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| Closely related to light deflection is the gravitational time delay (or Shapiro 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 (planet)|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>
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| {{clear}}
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| ===Gravitational waves===
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| {{Main|Gravitational waves}}
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| [[File:Gravwav.gif|thumb|Ring of test particles influenced by gravitational wave]]
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| One of several analogies between weak-field gravity and [[electromagnetism]] is that, analogous to [[electromagnetic wave]]s, there are [[gravitational waves]]: ripples in the metric of spacetime that propagate at the [[speed of light]].<ref>These have been indirectly observed through the loss of energy in binary pulsar systems such as the [[Hulse–Taylor binary]], the subject of the 1993 Nobel Prize in physics. A number of projects are underway to attempt to observe directly the effects of gravitational waves. For an overview, see {{Harvnb|Misner|Thorne|Wheeler|1973|loc=part VIII}}. Unlike electromagnetic waves, the dominant contribution for gravitational waves is not the [[dipole]], but the quadrupole; see {{Harvnb|Schutz|2001}}</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, for weak fields, a linear approximation can be made. Such linearized 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>
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| 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 so-called [[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 relativity|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>
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| ===Orbital effects and the relativity of direction===
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| {{Main|Kepler problem in general relativity}}
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| 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.
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| ====Precession of apsides====
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| [[File:Relativistic precession.svg|thumb|Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star]]
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| 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 [[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 the [[Tests of general relativity#Perihelion precession of Mercury|anomalous perihelion shift]] of the planet [[Mercury (planet)|Mercury]], discovered earlier by [[Urbain Le Verrier]] in 1859, was important evidence that he had at last identified the correct form of the [[Einstein field equations|gravitational field equations]].<ref>{{Harvnb|Schutz|2003|pp=48–49}}, {{Harvnb|Pais|1982|pp=253–254}}</ref>
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| 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>
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| ====Orbital decay====
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| <!--This subsection is linked to from the subsection Gravitational Waves in Astrophysical Applications, please do not change its title -->
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| [[File:Psr1913+16-weisberg en.png|thumb|Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.<ref>A figure that includes error bars is fig. 7 in {{Harvnb|Will|2006|loc=sec. 5.1}}</ref>]]
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| 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 very 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>
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| 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]], in which both stars are pulsars.<ref>{{Harvnb|Kramer|2004}}</ref>
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| ====Geodetic precession and frame-dragging====
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| {{Main|Geodetic precession|Frame dragging}}
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| Several relativistic effects are directly related to the relativity of direction.<ref>{{Harvnb|Penrose|2004|loc=§14.5}}, {{Harvnb|Misner|Thorne|Wheeler|1973|loc=§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>
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| Near a rotating mass, there are so-called 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>{{Citation| author=Iorio L.|title=COMMENTS, REPLIES AND NOTES: A note on the evidence of the gravitomagnetic field of Mars |date=August 2006| journal=Classical Quantum Gravity|volume=23| issue=17| pages=5451–5454|doi=10.1088/0264-9381/23/17/N01|arxiv = gr-qc/0606092 |bibcode = 2006CQGra..23.5451I }}</ref><ref>{{Citation| author=Iorio L.|title=On the Lense–Thirring test with the Mars Global Surveyor in the gravitational field of Mars| journal=Central European Journal of Physics |date=June 2010| doi=10.2478/s11534-009-0117-6|volume= 8 |issue =3 |pages= 509–513|arxiv = gr-qc/0701146 |bibcode = 2010CEJPh...8..509I }}</ref>
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| ==Astrophysical applications==
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| ===Gravitational lensing===
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| {{Main|Gravitational lensing}}
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| [[File:Einstein cross.jpg|thumb|[[Einstein cross]]: four images of the same astronomical object, produced by a [[gravitational lens]]]]
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| 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 lens]]ing.<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>
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| 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>
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| 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>
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| ===Gravitational wave astronomy===
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| {{Main|Gravitational waves|Gravitational wave astronomy}}
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| [[File:LISA.jpg|thumb|180px|Artist's impression of the space-borne gravitational wave detector [[Laser Interferometer Space Antenna|LISA]]]]
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| Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see [[General relativity#Orbital decay|Orbital decay]], above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly. Such detection 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]] (three detectors), [[TAMA 300]] and [[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. | doi=10.1088/0264-9381/27/8/084013 | year=2010 | journal=Classical and Quantum Gravity | volume=27 | issue=8 | pages=084013 |arxiv=0911.5206 |bibcode = 2010CQGra..27h4013H }}</ref> 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]]) due for launch in 2014.<ref>{{cite web|url=http://www.esa.int/esaSC/120397_index_0_m.html|title=LISA pathfinder overview|publisher=ESA|accessdate=2012-04-23}}</ref>
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| 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>
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| ===Black holes and other compact objects===
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| {{Main|Black hole}}
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| 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 star]]s of around 1.4 [[solar mass]]es, and [[stellar black hole]]s 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>
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| [[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]]
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| 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 black hole|stellar]] or [[supermassive black hole]]s, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of [[Active galactic nucleus|active galactic nuclei]] on galactic scales and stellar-size objects such as [[microquasar]]s.<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 [[speed of light|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>
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| 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>
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| Black holes are also sought-after targets in the search for gravitational waves (cf. [[General relativity#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>
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| {{clear}}
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| ===Cosmology===
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| [[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]].]]
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| {{Main|Physical cosmology}}
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| The current models of cosmology are based on [[Einstein's field equations]], which include the [[cosmological constant]] Λ since it has important influence on the large-scale dynamics of the cosmos,
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| :<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>
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| where ''<math>g_{\mu\nu}</math>'' is the [[metric tensor (general relativity)|spacetime metric]].<ref>Originally {{Harvnb|Einstein|1917}}; cf. {{Harvnb|Pais|1982|pp=285–288}}</ref> [[Isotropic]] and [[homogeneity (physics)#Translation invariance|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 [[1000000000 (number)|billion]] 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>
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| 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 so-called [[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 [[supernova]]e 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|2007}}</ref>
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| A so-called [[cosmic inflation|inflationary phase]],<ref>A good introduction is {{Harvnb|Linde|1990}}; for a more recent review, see {{Harvnb|Linde|2005}}</ref> an additional phase of strongly accelerated expansion at cosmic times of around <math>10^{-33}</math> 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|2007|loc=sec. 2}}</ref> (cf. the section on [[General relativity#Quantum gravity|quantum gravity]], below).
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| ==Advanced concepts==
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| ===Causal structure and global geometry===
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| {{Main|Causal structure}}
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| [[File:Penrose.svg|thumb|Penrose–Carter diagram of an infinite [[Minkowski space|Minkowski universe]]]]
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| 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 ([[Geodesic (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>
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| Aware of the importance of causal structure, [[Roger Penrose]] and others developed what is known as [[global geometry]]. In global geometry, the object of study is not one particular [[Solutions of the Einstein field equations|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 so-called [[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>
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| ===Horizons===
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| {{Main|Horizon (general relativity)|No hair theorem|Black hole mechanics}}
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| 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 horizon]]s 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>
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| [[File:Ergosphere.svg|thumb|left|The [[ergosphere]] of a [[rotating black hole]], which plays a key role when it comes to extracting energy from such a black hole]]
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| Early studies of black holes relied on [[Exact solutions in general relativity|explicit solutions]] of [[Einstein field equations|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. In the long run, they are rather simple objects characterized by eleven parameters specifying [[energy]], [[linear momentum]], [[angular momentum]], location at a specified time and [[electric charge]]. This is stated by the [[no hair theorem|black hole uniqueness theorems]]: "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>
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| 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 black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature, black holes should emit [[thermal radiation]]. Semi-classical 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 [[General relativity#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>
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| There are 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 semi-classical 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>
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| ===Singularities===
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| {{Main|Spacetime singularity}}
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| Another general—and quite disturbing—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 [[worldline]]s 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 metric|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>
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| 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 so-called [[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>
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| ===Evolution equations===
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| {{Main|Initial value formulation (general relativity)}}
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| Each [[Solutions of the Einstein field equations|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>
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| To understand Einstein's equations as [[partial differential equation]]s, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in so-called "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=§21.4–§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>
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| ===Global and quasi-local quantities===
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| {{Main|Mass in general relativity}}
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| 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=§20.4}}</ref>
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| 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 so-called [[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 [[Roger Penrose|Penrose's]] quasi-local energy–momentum based on [[twistor]] methods; cf. the review article {{Harvnb|Szabados|2004}}</ref>
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| ==Relationship with quantum theory==
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| If general relativity is considered one of the two pillars of modern physics, [[quantum mechanics|quantum theory]], the basis of understanding matter from [[elementary particle]]s to [[solid state physics]], is 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, it is still an open question as to how the concepts of quantum theory can be reconciled with those of general relativity.
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| ===Quantum field theory in curved spacetime===
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| {{Main|Quantum field theory in curved spacetime}}
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| 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 [[General relativity#Horizons|above]], this radiation plays an important role for the thermodynamics of black holes.<ref>{{Harvnb|Wald|2001|loc=ch. 3}}</ref>
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| ===Quantum gravity===
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| {{Main|Quantum gravity}}
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| {{See also|String theory|Canonical general relativity|Loop quantum gravity|Causal sets}}
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| 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 [[spacetime singularity|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>A timeline and overview can be found in {{Harvnb|Rovelli|2000}}</ref>
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| [[Image:Calabi yau.jpg|left|thumb|Projection of a [[Calabi–Yau manifold]], one of the ways of [[compactification (physics)|compactifying]] the extra dimensions posited by string theory]]
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| 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. 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 result are models devoid of all predictive power ("[[non-renormalizable|non-renormalizability]]").<ref>In particular, a 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|Goroff|Sagnotti|1985}}</ref>
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| [[File:Spin network.svg|thumb|Simple [[spin network]] of the type used in loop quantum gravity]]
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| 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]] 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>
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| Another approach starts with the [[canonical quantization]] procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. [[General relativity#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.<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|2006}}; 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>
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| 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 [[dynamical triangulation]]s,<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 formulation|path-integral]] based models of [[quantum cosmology]].<ref>{{Harvnb|Hawking|1987}}</ref>
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| 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>
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| ==Current status==
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| General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous [[Tests of general relativity|observational and experimental tests]]. However, there are strong indications 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 [[General relativity#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 [[General relativity#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> and increasingly powerful computer simulations (such as those describing merging black holes) are run.<ref>A review of the various problems and the techniques being developed to overcome them, see {{Harvnb|Lehner|2002}}</ref> The race for the first direct detection of gravitational waves continues,<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 [http://geo600.aei.mpg.de GEO 600] and [http://www.ligo.caltech.edu/ LIGO]</ref> in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.<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|2007}}; 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> More than ninety years after its publication, general relativity remains a highly active area of research.<ref>See, e.g., the electronic review journal [http://relativity.livingreviews.org Living Reviews in Relativity]</ref>
| |
| | |
| ==See also==
| |
| {{Col-begin}}
| |
| {{Col-1-of-3}}
| |
| * [[Center of mass (relativistic)]]
| |
| * [[Contributors to general relativity]]
| |
| * [[Derivations of the Lorentz transformations]]
| |
| * [[Ehrenfest paradox]]
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| * [[Einstein–Hilbert action]]
| |
| {{Col-2-of-3}}
| |
| * [[Introduction to mathematics of general relativity]]
| |
| * [[Relativity priority dispute]]
| |
| * [[Ricci calculus]]
| |
| {{Col-3-of-3}}
| |
| * [[Tests of general relativity]]
| |
| * [[Timeline of gravitational physics and relativity]]
| |
| * [[Two-body problem in general relativity]]
| |
| {{col-end}}
| |
| | |
| ==Notes==
| |
| {{Reflist|2}}
| |
| | |
| ==References==
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| {{refend}}
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| | |
| ==Further reading==
| |
| ;Popular books
| |
| * {{Citation|last=Geroch|first= R|authorlink=Robert Geroch| title=General Relativity from A to B|location=Chicago|publisher=University of Chicago Press|year=1981|isbn=0-226-28864-1}}
| |
| * {{Citation|author=Lieber, Lillian|authorlink=Lillian Lieber| title=The Einstein Theory of Relativity: A Trip to the Fourth Dimension|location=Philadelphia|publisher=Paul Dry Books, Inc.|year=2008|isbn=978-1-58988-044-3}}
| |
| * {{Citation|author=Wald, Robert M.|authorlink=Robert Wald|title=Space, Time, and Gravity: the Theory of the Big Bang and Black Holes|location=Chicago|publisher=University of Chicago Press|year=1992|isbn=0-226-87029-4}}
| |
| * {{citation|authorlink=John Archibald Wheeler|last1=Wheeler|first1=John|last2=Ford|first2=Kenneth|year=1998|title=Geons, Black Holes, & Quantum Foam: a life in physics|isbn=0-393- 31991-1|location=New York |publisher=W. W. Norton }}
| |
| ;Beginning undergraduate textbooks
| |
| * {{Citation|author=Callahan, James J.|title=The Geometry of Spacetime: an Introduction to Special and General Relativity| location=New York|publisher=Springer|year=2000|isbn=0-387-98641-3}}
| |
| * {{Citation|author=Taylor, Edwin F.; [[John Archibald Wheeler|Wheeler, John Archibald]]|title=Exploring Black Holes: Introduction to General Relativity|publisher=Addison Wesley|year=2000|isbn=0-201-38423-X}}
| |
| | |
| ;Advanced undergraduate textbooks
| |
| * {{Citation|author=B. F. Schutz|title=A First Course in General Relativity (Second Edition)|publisher=Cambridge University Press| year=2009|isbn=978-0-521-88705-2}}
| |
| * {{Citation|author=Cheng, Ta-Pei|title=Relativity, Gravitation and Cosmology: a Basic Introduction|location=Oxford and New York| publisher=Oxford University Press| year=2005|isbn=0-19-852957-0}}
| |
| * {{Citation|last=Gron|first=O.|last2=Hervik|first2=S.| title=Einstein's General theory of Relativity|publisher=Springer|year=2007|isbn=978-0-387-69199-2}}
| |
| * {{Citation|author=Hartle, James B.|authorlink=James Hartle|title=Gravity: an Introduction to Einstein's General Relativity|location=San Francisco|publisher=Addison-Wesley|year=2003|isbn=0-8053-8662-9}}
| |
| * {{Citation|author=[[Lane P. Hughston|Hughston, L. P.]] & Tod, K. P.|title=Introduction to General Relativity|location=Cambridge|publisher=Cambridge University Press|year=1991|isbn=0-521-33943-X}}
| |
| * {{Citation|author=d'Inverno, Ray|title=Introducing Einstein's Relativity|location=Oxford|publisher=Oxford University Press|year=1992|isbn=0-19-859686-3}}
| |
| *{{cite book|last=Ludyk|first=Günter|title=Einstein in Matrix Form|year=2013|publisher=Springer|location=Berlin|isbn= 9783642357978 |edition=1st ed.}}
| |
| | |
| ;Graduate-level textbooks
| |
| * {{Citation|author=Carroll, Sean M.|authorlink=Sean M. Carroll|title=Spacetime and Geometry: An Introduction to General Relativity|location=San Francisco|publisher=Addison-Wesley|year=2004|isbn=0-8053-8732-3|url=http://spacetimeandgeometry.net/}}
| |
| * {{Citation|last=Grøn|first=Øyvind |authorlink=Øyvind Grøn| coauthors=Hervik, Sigbjørn|title=Einstein's General Theory of Relativity|location=New York|publisher=Springer|year=2007|isbn=978-0-387-69199-2}}
| |
| * {{Citation|author=Landau, Lev D.|authorlink=Lev Landau|author2= Lifshitz, Evgeny F.| author2-link=Evgeny Lifshitz|title=The Classical Theory of Fields (4th ed.)|location=London|publisher=Butterworth-Heinemann|year=1980|isbn=0-7506-2768-9}}
| |
| * {{Citation|first=Charles W.|last=Misner|authorlink=Charles W. Misner|first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=[[Gravitation (book)|Gravitation]]|publisher= W. H. Freeman|year=1973|isbn=0-7167-0344-0}}
| |
| * {{Citation|author=Stephani, Hans|title=General Relativity: An Introduction to the Theory of the Gravitational Field,|location=Cambridge|publisher=Cambridge University Press|year=1990|isbn=0-521-37941-5}}
| |
| * {{Citation|last=Wald|first=Robert M.|authorlink=Robert Wald|title=[[General Relativity (book)|General Relativity]]|publisher=University of Chicago Press|year=1984|isbn=0-226-87033-2}}
| |
| | |
| ==External links==
| |
| {{Commons category}}
| |
| {{Wikibooks}}
| |
| {{Wikisource|Relativity: The Special and General Theory}}
| |
| {{Wikisource|The Foundation of the Generalised Theory of Relativity}}
| |
| * [http://publicliterature.org/books/relativity/xaa.php ''Relativity: The special and general theory''] ([http://publicliterature.org/pdf/relat10.pdf PDF])
| |
| * [http://www.einstein-online.info Einstein Online] – Articles on a variety of aspects of relativistic physics for a general audience; hosted by the [[Max Planck Institute for Gravitational Physics]]
| |
| * [http://archive.ncsa.uiuc.edu/Cyberia/NumRel/NumRelHome.html NCSA Spacetime Wrinkles] – produced by the [[numerical relativity]] group at the [[National Center for Supercomputing Applications|NCSA]], with an elementary introduction to general relativity
| |
| | |
| ;Courses/Lectures/Tutorials
| |
| * [http://www.youtube.com/watch?v=hbmf0bB38h0&feature=BFa&list=EC6C8BDEEBA6BDC78D Einstein's General Theory of Relativity] by [[Leonard Susskind]]'s Modern Physics lectures. Recorded September 22, 2008 at [[Stanford University]]
| |
| * [http://www.luth.obspm.fr/IHP06/ Series of lectures on General Relativity] given in 2006 at the Institut Henri Poincaré (introductory courses and advanced ones).
| |
| * [http://www.math.ucr.edu/home/baez/gr/ General Relativity Tutorials] by [[John Baez]]
| |
| * {{Cite web|author=Brown, Kevin|title=Reflections on relativity|work=Mathpages.com|url=http://www.mathpages.com/rr/rrtoc.htm|accessdate=May 29, 2005}}
| |
| * {{Cite web|author=Carroll, Sean M.|title=Lecture Notes on General Relativity|url=http://arxiv.org/abs/gr-qc/9712019|accessdate=January 5, 2014}}
| |
| * {{Cite web|author=Moor, Rafi|title=Understanding General Relativity|url=http://www.rafimoor.com/english/GRE.htm|accessdate=July 11, 2006}}
| |
| * {{Cite web|author=Waner, Stefan|title=Introduction to Differential Geometry and General Relativity|url=http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/pdfs/DiffGeom.pdf| accessdate=2006-01-31|format=PDF}}
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