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Encyclopedia > General relativity
A simulated Black Hole of ten solar masses as seen from a distance of 600 kilometers with the Milky Way in the background (horizontal camera opening angle: 90°).

General relativity predicts a number of novel effects relating to the passage of time, the geometry of space, the motion of bodies in free fall and the propagation of light. Examples are gravitational time dilation, the gravitational redshift of light, and the gravitational time delay; in numerous observations and experiments to date, the theory's predictions for these effects have been confirmed. Although not the only relativistic theory of gravity, general relativity is the simplest such theory that is consistent with the experimental data. Still, a number of open questions remain: the most fundamental is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. For other uses, see Free-fall (disambiguation). ... For other uses, see Light (disambiguation). ... Gravitational time dilation is a consequence of Albert Einsteins theories of relativity and related theories which causes time to pass at different rates in regions of a different gravitational potential; the higher the local distortion of spacetime due to gravity, the slower time passes. ... Graphic representing the gravitational redshift of a neutron star (not exact) In physics, light or other forms of electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational field (and which could be said to have climbed uphill out of a gravity well... This article is in need of attention from an expert on the subject. ... Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ... Alternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einsteins theory of general relativity. ... This box:      Werner Heisenberg and Erwin SchrÃ¶dinger, founders of Quantum Mechanics. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ...

General relativity
$G_{mu nu} = {8pi Gover c^4} T_{mu nu},$
Einstein field equations
Introduction to...
Mathematical formulation of...
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The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... Newtonâ€™s conception and quantification of gravitation held until the beginning of the 20th century, when Albert Einstein extended the special relativity to form the general relativity (GR) theory. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described... This article or section is in need of attention from an expert on the subject. ... This box:      In physics, a gravitational wave is a fluctuation in the curvature of spacetime which propagates as a wave, traveling outward from a moving object or system of objects. ... According to Albert Einsteins theory of general relativity, space and time get pulled out of shape near a rotating body in a phenomenon referred to as frame-dragging. ... The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a spinning, moving body. ... For the science fiction film, see Event Horizon (film). ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... For other uses, see Black hole (disambiguation). ... It has been suggested that Weak-field approximation be merged into this article or section. ... The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... The Friedmann equations relate various cosmological parameters within the context of general relativity. ... In physics, Kaluzaâ€“Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... In physics and astronomy, a Reissner-NordstrÃ¶m black hole, discovered by Gunnar NordstrÃ¶m and Hans Reissner, is a black hole that carries electric charge , no angular momentum, and mass . ... The GÃ¶del solution is an exact solution of the Einstein field equation in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant (see lambdavacuum solution). ... In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... The Kerr-Newman metric is a solution of Einsteins general relativity field equation that describes the spacetime geometry around a charged (), rotating () black hole of mass m. ... The Kasner metric is an exact solution to Einsteins theory of general relativity. ... Milnes model follows the description from special relativity of an observable universes spacetime diagram containing past and future light cones along with elsewhere in spacetime. ... // The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. ... In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einsteins field equation. ... The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation for General Relativity. ... â€œEinsteinâ€ redirects here. ... Hermann Minkowski. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ... Monsignor Georges LemaÃ®tre, priest and scientist. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... Howard Percy Robertson (January 27, 1903 - August 26, 1961) was a scientist known for contributions related to cosmology and the uncertainty principle. ... Roy Patrick Kerr (1934- ) is a New Zealand born mathematician who is best known for discovering the famous Kerr vacuum, an exact solution to the Einstein field equation of general relativity, which models the gravitational field outside an uncharged rotating massive object, or even a rotating black hole. ... Alexander Alexandrovich Friedman or Friedmann (ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ€ ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ€Ð¾Ð²Ð¸Ñ‡ Ð¤Ñ€Ð¸Ð´Ð¼Ð°Ð½) (June 16, 1888 â€“ September 16, 1925) was a Russian cosmologist and mathematician. ... Chandrasekhar redirects here. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... This is a partial list of persons who have made major contributions to the development of standard mainstream general relativity. ...

## From classical mechanics to general relativity

The structure of general relativity and the way the theory is formulated are best understood by examining its similarities with, and departures from, classical physics.[1]

### Geometry of Newtonian gravity

Ball falling to the floor in an accelerated rocket (left), and on Earth (right)

Conversely, it would seem 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. Following from Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors, 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.[4] 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 accelerated rocket.[5] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... This article is about Earth as a planet. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For other uses, see Friction (disambiguation). ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Gravity is a force of attraction that acts between bodies that have mass. ... Gravitation is the tendency of masses to move toward each other. ... Image:Lorand Eotvos. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... For other uses, see Free-fall (disambiguation). ...

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 which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry; however, as can be shown using simple thought experiments following the free-fall trajectories of different test particles, the Newtonian connection is not integrable—space-time is curved. The result is a geometric formulation of Newtonian gravity using only covariant concepts; in other words: this description is valid in any desired coordinate system.[6] In this geometric description, tidal effects—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.[7] In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... Look up connection, connected, connectivity in Wiktionary, the free dictionary. ... In physics, gravitational potential is the measure of potential energy an object possesses due to its position in a gravitational field. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... Integrability is a mathematical concept used in different areas. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In category theory, see covariant functor. ...

### Relativistic generalization

Light cone

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.[8] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant (the defining symmetry of special relativity), not Galilei invariant (the defining symmetry of classical mechanics). The differences between the two become significant when we are dealing with speeds approaching that of light and high-energy phenomena.[9] Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Sphere symmetry group o. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. ...

Lorentz symmetry introduces an additional structure, in mathematical terms: a conformal structure. This is the set of light cones (see the image on the left): For each 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.[10]

Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. As gravity comes into play, assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles; translated into the language of space-time: 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.[11] In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... In the context of special relativity, time-like separated points (or events) in spacetime have a spacetime interval greater than 0 (see sign convention). ...

A priori, it is not clear whether the new local frames in free fall are indeed those in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, and its preferred frames might not be the same as the local free-falling inertial 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; the actual measurements show that free-falling frames in which light propagates as it does in special relativity.[12] 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 Einstein equivalence principle, and is one of the guiding principles when it comes to generalizing special relativistic physics to include gravity.[13] Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ...

With reference to the same experimental data, it becomes clear that time as shown 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 suggestive of a more general geometry: where all reference frames in free fall are equivalent, and approximately Minkowskian, we are dealing with a curved generalization of Minkowski space: instead of Minkowskian, assume the metric tensor to be, more generally, semi-Riemannian. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, a Levi-Civita connection; assume this to be the connection implied by the universality of free fall.[14] In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...

### Einstein's equations

$G_{ab} = kappa, T_{ab},$

where Gab is the Einstein tensor, Tab is the energy-momentum tensor (both written in abstract index notation).[16] Matching the theory's prediction to observational results for planetary orbits, the proportionality constant has the value κ = 8πG/c4, with G the gravitational constant and c the speed of light.[17] The tensors Gab and Tab are both rank-2 symmetric tensors, 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.[18] Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ... This article is about the astronomical term. ... ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ...

While the metric description of gravity follows rather straightforwardly from special relativity and the universality of free fall, it is worth mentioning that 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.[19] Alternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einsteins theory of general relativity. ... This article or section is in need of attention from an expert on the subject. ... Teleparallelism (also called distant parallelism and teleparallel gravity), was an attempt by Einstein to unify electromagnetism and gravity. ... This article is in need of attention from an expert on the subject. ...

## Definition and basic applications

See also: Mathematics of general relativity and Physical theories modified by general relativity

As a result of the derivation outlined in the previous section, we now have all the information needed to define and characterize general relativity. For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... This article will use the Einstein summation convention. ...

### Definition and basic properties

General relativity is a metric theory of gravitation. At its core are Einstein's equations, which link the geometry of a four-dimensional, semi-Riemannian manifold representing space-time with the energy-momentum contained in that space-time.[20] Phenomena that, in classical mechanics, are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity.[21] The curvature is, in turn, caused by the energy-momentum of matter; paraphrasing the relativist John Archibald Wheeler, space-time tells matter how to move; matter tells space-time how to curve.[22] Gravity redirects here. ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... For other uses, see Geometry (disambiguation). ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... It has been suggested that this article or section be merged with Momentum#Momentum_in_relativistic_mechanics. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Free Fall opens with one of the most stunning first paragraphs I have ever, or am ever likely to, read. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... The Space Shuttle Discovery as seen from the International Space Station. ... In ordinary language, a trajectory is the path followed by a body moving through space, for instance, the path taken by a falling body or the orbit of a planet. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... John Archibald Wheeler (July 9, 1911â€“April 13, 2008) was an eminent American theoretical physicist. ...

While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases: for weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of gravity.[23] In mathematics and physics, a scalar field associates a scalar to every point in space. ... Note: This is a fairly abstract mathematical approach to tensors. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ... The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ... This article or section does not cite its references or sources. ... Gravitation is the tendency of masses to move toward each other. ...

As it is constructed using tensors, general relativity exhibits general covariance, that is, its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.[24] Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.[25] Locally, as expressed in the equivalence principle, space-time is Minkowskian, and the laws of physics have local Lorentz invariance.[26] In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... This article or section is in need of attention from an expert on the subject. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... The general principle of relativity as used in Einsteins general theory of relativity is that the laws of physics must take the same form in all reference frames. ... For a list of set rules, see Laws of science. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... A local Lorentz covariance or local Lorentz symmetry is a local symmetry of space-time which reduces locally (i. ...

### Model-building

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equation 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) on which are defined specific matter fields, in such a way that matter and geometry satisfy Einstein's equations, and that the matter satisfies whatever equations have been 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.[27] This article or section is in need of attention from an expert on the subject. ...

Einstein's equations are non-linear partial differential equations and, as such, very difficult to solve.[28] Nevertheless, a number of exact solutions are known, although only a few of them have direct physical applications.[29] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,[30] and the Friedmann-Lemaître-Robertson-Walker and de Sitter universes, each describing an expanding cosmos.[31] Exact solutions of great theoretical interest include the Gödel universe, the Taub-NUT solution, and Anti-de Sitter space.[32] To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... In physics and astronomy, a Reissner-NordstrÃ¶m black hole, discovered by Gunnar NordstrÃ¶m and Hans Reissner, is a black hole that carries mass , electric charge , and no angular momentum. ... In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... For other uses, see Black hole (disambiguation). ... // The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. ... A de Sitter universe is a solution to Einsteins field equations of General Relativity which is named after Willem de Sitter. ... In mathematics, the constructible universe (or GÃ¶dels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ... This article or section is in need of attention from an expert on the subject. ...

Significant efforts are being made in the field of numerical relativity, where the goal is to find interesting numerical solutions describing, say, two black holes orbiting each other, with the help of powerful computers.[33] Also, there are different methods for finding approximate solutions in the context of perturbation theory. The best-known of these are linearized gravity[34] and its generalization, the Post-Newtonian expansion, which represents a systematic way of describing a space-time containing matter which is not particularly compact and moves slowly compared with the speed of light; the description starts with Newtonian gravity and, in a systematic sequence, takes into account smaller and smaller effects arising from the difference between Newton's theory and general relativity.[35] An extension of this expansion is the Parametrized Post-Newtonian (PPN) formalism, a framework of testing general relativity against alternative theories in a way that allows quantitative comparisons.[36] This article is in need of attention from an expert on the subject. ... For other uses, see Black hole (disambiguation). ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... It has been suggested that Weak-field approximation be merged into this article or section. ... Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein equations for the metric tensor that represents a multi-component, tensor gravitational field potential instead of a single, scalar gravitational potential in the Newtonian gravity. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... Alternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einsteins theory of general relativity. ...

## Consequences of Einstein's theory

General relativity has a number of consequences, some following directly from the theory's axioms, others having become clear only in the course of the ninety years of research that followed Einstein's initial publication.

### Gravitational time dilation and frequency shift

Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body

In general relativity and, in fact, in any theory in which the equivalence principle holds,[37] gravity has an immediate influence on the passage of time. Imagine two observers Alice and Bob, both of whom are at rest in a stationary gravitational field, with Alice closer to the source of gravity ("deeper in the gravity well") and Bob at a greater distance. Then for light sent from Alice to Bob or vice versa, Bob will measure a lower frequency than Alice: light sent down into a gravity well is blue-shifted, light climbing out of a gravity well is redshifted. Also, Alice's clocks tick more slowly than Bob's: whenever the two are compared (either by sending light signals back and forth, or by slowly transporting clocks from one location to the other), the result will be that Bob's clocks are running faster. This effect is not restricted to clocks, but applies to all processes (the rate at which Alice and Bob age, cook five-minute eggs, or play Chopin's Minute Waltz); it is known as gravitational time dilation.[38] Gravitational time dilation is a consequence of Albert Einsteins theories of relativity and related theories which causes time to pass at different rates in regions of a different gravitational potential; the higher the local distortion of spacetime due to gravity, the slower time passes. ... Image File history File links Size of this preview: 450 Ã— 600 pixelsFull resolution (480 Ã— 640 pixel, file size: 211 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Size of this preview: 450 Ã— 600 pixelsFull resolution (480 Ã— 640 pixel, file size: 211 KB, MIME type: image/png) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Stationary can mean: Look up stationary in Wiktionary, the free dictionary. ... A gravity well is the scientific/science fictional term for the distortion in space-time caused by a massive body such as a planet. ... For other uses, see Frequency (disambiguation). ... Blue shift is the opposite of redshift, the latter being much more noted due to its importance to modern astronomy. ... This article is about the light phenomenon. ... The Waltz in D flat major, opus 64, No. ... Gravitational time dilation is a consequence of Albert Einsteins theories of relativity and related theories which causes time to pass at different rates in regions of a different gravitational potential; the higher the local distortion of spacetime due to gravity, the slower time passes. ...

The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound and Rebka[39] and later confirmed by astronomical observations.[40] There are numerous direct measurements of gravitational time dilation using atomic clocks[41] while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).[42] Tests in stronger gravitational fields are provided by the observation of binary pulsars.[43] All results are in agreement with general relativity;[44] however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.[45] The Pound-Rebka experiment is a well known experiment in general relativity. ... An atomic clock is a type of clock that uses an atomic resonance frequency standard as its counter. ... GPS redirects here. ... A binary pulsar is a pulsar with a binary companion, often another pulsar, white dwarf or neutron star. ...

### Light deflection and gravitational time delay

Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray).

In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics—a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity.[46] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),[47] several effects of gravity on light propagation emerge. In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described... This article is in need of attention from an expert on the subject. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... Cherenkov effect in a swimming pool nuclear reactor. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein equations for the metric tensor that represents a multi-component, tensor gravitational field potential instead of a single, scalar gravitational potential in the Newtonian gravity. ...

The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also be derived by extending the universality of free fall to light,[48] the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity; from the standpoint of Einstein's theory they take into account the effect of gravity on time, but not its consequences for the warping of space.[49] An important example of this is starlight being deflected as it passes the Sun. In this case, the positions of stars observed in the Sun's vicinity during a solar eclipse appear shifted by an angular deflection of up to 4GM / c2R, or 1.75 arc seconds.[50] This effect was first measured by a British expedition directed by Arthur Eddington, and confirmed with significantly higher accuracy by subsequent measurements.[51] In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Sol redirects here. ... Photo taken during the 1999 eclipse. ... A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ...

Closely related to the bending of light is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of radar signals sent from Earth to planets such as Venus or Mercury and thence reflected back;[52] later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.[53] In both cases, what was measured was the propagation of signals in the Sun's gravitational field. More recent measurements have detected the Shapiro effect in signals sent by a pulsar that is part of a binary system; in that case, the gravitational field causing the time delay is that of the other pulsar.[54] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called γ that reflects the influence of gravity on the geometry of space.[55] In General relativity, the Shapiro effect, or gravitational time delay, is one of the four classic solar system tests of general relativity. ... For other uses, see Venus (disambiguation). ... This article is about the planet. ... It has been suggested that Radio pulsar be merged into this article or section. ... The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ...

### Gravitational waves

Main article: Gravitational waves
Ring of test particles floating in space
Ring of test particles influenced by gravitational wave

There are several analogies between weak-field gravity and electromagnetism. One is that, for electromagnetic waves, there are corresponding gravitational waves: ripples in spacetime that propagate at the speed of light.[56] In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ... In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right).[57] Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by 10 − 21 or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.[58] The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there is no linear superposition for gravitational waves. Describing such more general waves is not an easy task. There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty space[59] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,[60] while, when it comes to describing the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.[61] The principle of Linear superposition describes the sum of two or more disturbances at a point resulting from the simultaneous presence of these disturbances attempting to occupy the same place. ... Category: Mathematics stubs ... Gowdy universes or, alternatively, Gowdy solutions of Einsteins equations are simple model spacetimes in general relativity which represent an expanding universe filled with a regular pattern of gravitational waves. ... This article is in need of attention from an expert on the subject. ...

### Orbital effects and the relativity of direction

General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The most striking one concerns the relativistic apside shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction. In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described... A diagram of Keplerian orbital elements. ...

#### Precession of apsides

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star

In general relativity, the apsides of orbits (the points of an orbiting body 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 rosette-like shape (see image). Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle;[62] the result can also be obtained by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)[63] or the much more general post-Newtonian formalism.[64] The effect is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body's gravity (in other words, the special kind of nonlinearity exhibited by Einstein's theory).[65] Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... A diagram of Keplerian orbital elements. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... Precession (also called gyroscopic precession) is the phenomenon by which the axis of a spinning object (e. ... For other uses, see Ellipse (disambiguation). ... Rosettes can refer for: A small, circular, device that can be awarded with medals (see: Rosette (decoration)). A type of plant with their leaves at an upset stem in a typical form. ... In metric theories of gravitation, particularly general relativity, a test particle is an idealized model of a small object whose mass is so small that it does not appreciably disturb the ambient graviational field. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein equations for the metric tensor that represents a multi-component, tensor gravitational field potential instead of a single, scalar gravitational potential in the Newtonian gravity. ... In theoretical physics, a particles self-energy represents the contribution to the particles energy or effective mass due to interactions between the particle and the system it is apart of. ... In mathematics, a nonlinear system is one whose behavior cant be expressed as a sum of the behaviors of its parts (or of their multiples. ...

#### Orbital decay

Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.[69]

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor using the binary pulsar PSR1913+16 that they had discovered in 1974; it amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993.[71] Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars.[72] Russell Alan Hulse (born November 28, 1950) is an American physicist and winner of the Nobel Prize in Physics, shared with his thesis advisor Joseph Hooton Taylor Jr. ... Joseph H. Taylor, Jr. ... Graph PSR B1913+16 (also known as J1915+1606) is a pulsar in a binary star system, in orbit with another star around a common center of mass. ... Artists impression. ...

#### Geodetic precession and frame-dragging

Main articles: Geodetic precession and Frame dragging

Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass get "dragged around"; this is most extreme for rotating black holes where, for an object entering a zone known as the ergosphere, rotation is inevitable.[77] Such effects can again be tested through their influence on the orientation of a gyroscope in free fall:[78] somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction;[79] a precision measurement is the main aim of the Gravity Probe B mission, whose final results are expected in May 2008.[80] According to Albert Einsteins theory of general relativity, space and time get pulled out of shape near a rotating body in a phenomenon referred to as frame-dragging. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... A rotating black hole (Kerr black hole or Kerr-Newman black hole) is a black hole that possesses angular momentum. ... The LAGEOS-1 satellite. ... Gravity Probe B with solar panels folded Gravity Probe B (GP-B) is a satellite-based mission which launched in 2004. ...

## Astrophysical applications

### Gravitational lensing

Main article: Gravitational lensing
Einstein cross: four images of the same astronomical object, produced by a gravitational lens

Gravitational lensing has developed into a tool of observational astronomy. Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data are also used to understand the structural evolution of galaxies.[86] Mayall telescope at Kitt Peak National Observatory Observational astronomy is a division of the astronomical science that is concerned with getting data, in contrast with theoretical astrophysics which is mainly concerned with finding out the measureable implications of physical models. ... For other uses, see Dark matter (disambiguation). ... Hubbles law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. ... For other uses, see Galaxy (disambiguation). ...

### Gravitational wave astronomy

Main article: Gravitational waves
Artist's impression of the space-borne gravitational wave detector LISA

From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational waves (see the section on Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, this being one of the major goals of current relativity-related research.[87] To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.[88] A joint US-European mission to launch a space-based detector, LISA, is currently under development,[89] with a precursor mission (LISA Pathfinder) due for launch in late 2009.[90] In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The LISA is the Laser Interferometer Space Antenna experiment. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... A gravitational wave detector is any experiment designed to measure gravitational waves, minute distortions of spacetime that are predicted by Einsteins theory of general relativity. ... Geo 600 is a gravitational wave detector located in Hannover, Germany. ... LIGO stands for Lesser Inner Greater Outer. ... TAMA 300 is a gravitational wave detector located in Japan. ... Virgo (Latin for virgin, symbol , Unicode â™) is a constellation of the zodiac. ... The LISA is the Laser Interferometer Space Antenna experiment. ... LISA Pathfinder is the revised name for SMART-2, an ESA space probe to be launched in 2008. ...

### Black holes and other compact objects

Main article: Black holes
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves

Astronomically, the most important property of compact objects is that they provide a superbly efficient mechanism for converting gravitational into radiation energy.[97] Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as Microquasars.[98] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.[99] Interestingly, to a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity.[100] General relativity plays a central role in modelling all these phenomena,[101] relativistic lensing effects being thought to play a role for the signals received from X-ray pulsars.[102] Accretion, means any growth or increase in size by a gradual external addition or inclusion. ... Look up dust in Wiktionary, the free dictionary. ... A stellar black hole is a black hole formed by the gravitational collapse of a massive star (3 or more solar masses) at the end of its lifetime. ... Top: artists conception of a supermassive black hole tearing apart a star. ... An active galaxy is a galaxy where a significant fraction of the energy output is not emitted by the normal components of a galaxy: stars, dust and interstellar gas. ... Microquasars are smaller cousins of quasars. ... Relativistic Jet. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... In astronomy, superluminal motion is the apparently faster-than-light motion seen in some radio galaxies, quasars and recently also in some galactic sources called microquasars. ... An X-ray pulsar is a neutron star with a powerful magnetic field that gives rise to regular X-ray pulses. ...

Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"),[103] observations of stellar dynamics in the center of our own Milky Way galaxy,[104] and indications that at least some of the compact objects in question appear to have no solid surface[105] provide strong indirect evidence for the existence of black holes. Direct evidence, such as observing the "shadow" of the Milky Way galaxy's central black hole horizon, is eagerly sought for.[106] Eddington luminosity (sometimes also called the Eddington limit) is the largest luminosity that can pass through a layer of gas in hydrostatic equilibrium, supposing spherical symmetry. ... For other uses, see Milky Way (disambiguation). ...

Black holes are also sought-after targets in the search for gravitational waves (see the section Gravitational waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity;[107] the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances;[108] the gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole's geometry.[109] For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... This article is in need of attention from an expert on the subject. ... A standard candle is an astronomical object that has a known luminosity. ...

### Cosmology

Main article: Physical cosmology

Each solution of Einstein's equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. The current models are based on an extension of the original form of Einstein's equations which include the cosmological constant Λ, an additional term that has an important influence on the large-scale dynamics of the cosmos, This article is about the physics subject. ... In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Î›) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ...

$G_{ab} + Lambda g_{ab} = kappa, T_{ab}$

where gab is the spacetime metric.[110] In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...

Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope WMAP

The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,[119] and they have proven a sound basis to explaining the evolution of the universe's large-scale structure.[120] On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observations[121]) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly); there is currently no generally accepted description of this new kind of matter within the framework of particle physics[122] or otherwise.[123] A similar open question is that of dark energy. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation 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, namely dark energy;[124] the nature of this new form of energy remains unclear.[125] For other uses, see Dark matter (disambiguation). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe. ... For other uses, see Supernova (disambiguation). ... In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Î›) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... In physics and thermodynamics, an equation of state is a relation between state variables. ...

A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")[126] have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around 10 − 33 seconds, known as an inflationary phase.[127] While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,[128] problems remain. There is a bewildering variety of possible inflationary scenarios not restricted by current observations.[129] Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity; an authoritative answer would require a complete theory of quantum gravity, which does not exist at the moment[130] (cf. the section on quantum gravity, below). In physical cosmology, cosmic inflation is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density. ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...

### Causal structure and global geometry

Main article: Causal structure
Penrose diagram of an infinite Minkowski universe

In general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose-Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.[131] Image File history File links Penrose. ... Image File history File links Penrose. ... Hermann Minkowski. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ... In theoretical physics, a Penrose diagram (named after Roger Penrose who invented them) is usually a two-dimensional diagram that captures the causal relations between different points in spacetime. ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... The Minkowski diagram is a graphical tool used in special relativity to visualize spacetime with regard to an inertial reference frame. ...

Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are now termed global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) to derive general results.[132] Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Global spacetime structure is the structure of spacetime on a global level, i. ... This article or section is in need of attention from an expert on the subject. ... In general relativity, Raychaudhuris equation is a fundamental result describing the motion of nearby bits of matter. ... This article is about matter in physics and chemistry. ... The energy conditions refer to various constraints which can be imposed on a spacetime that any physically reasonable matter distributions in physics are expected to satisfy. ...

### Cosmic partitions: horizons

Main articles: Horizon (general relativity), No hair theorem, and Black hole mechanics

$r_s=frac{2GM}{c^2},$

where G is the gravitational constant and c the speed of light. Imagine a circular hoop with the circumference rs. A mass M small enough to fit through that hoop, regardless of their relative orientation, is compact enough to form a black hole.[134] According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

The ergosphere of a rotating black hole, which plays a key role when it comes to extracting energy from such a black hole

Initial black hole studies relied on simplified models obtained from explicit solutions of Einstein's equation, especially the spherically-symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Subsequent studies using global geometry 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 the result of what are called the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like 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.[135] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A rotating black hole (Kerr black hole or Kerr-Newman black hole) is a black hole that possesses angular momentum. ... Category: Mathematics stubs ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... In general relativity, a spacetime is said to be static if it admits a global, nowhere zero, timelike hypersurface orthogonal Killing vector field. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... In general relativity, a spacetime is said to be stationary if it admits a global, nowhere zero timelike Killing vector field. ... A rotating black hole (Kerr black hole or Kerr-Newman black hole) is a black hole that possesses angular momentum. ... In physics, momentum is a physical quantity related to the velocity and mass of an object. ... This gyroscope remains upright while spinning due to its angular momentum. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In astrophysics, the no-hair theorem states that black holes are completely characterized only by three externally observable parameters: mass, electrical charge, and angular momentum. ... In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any...

Even more remarkably, there is a general set of laws known as black hole mechanics, analogous to the laws of thermodynamics. For example, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, just as the entropy of a thermodynamic system. This law sets a limit to the energy that can be extracted from a rotating black hole (e.g. by the Penrose process).[136] In fact, there is strong evidence that the laws of black hole mechanics are indeed a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy:[137] semi-classical calculations indicate that black holes do emit thermal radiation, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation, and we will come back to it in the section on general relativity and quantum theory, below.[138] In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. ... The laws of thermodynamics, in principle, describe the specifics for the transport of heat and work in thermodynamic processes. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Radiant heat redirects here. ... Black body spectrum For a general introduction, see black body. ... In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation thought to be emitted by black holes due to quantum effects. ...

Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("particle horizon"), and some regions of the future cannot be influenced (event horizon); in both cases, the location of the horizon in spacetime depends on the event in question.[139] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons[140] (associated with a semi-classical radiation known as Unruh radiation).[141] It has been suggested that this article or section be merged into Observable universe. ... In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. ... The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none, that is, the accelerating observer will find themselves in a warm background. ...

### Singularities

Main article: Spacetime singularity

Another general—and quite disturbing—feature of general relativity is the appearance of spacetime boundaries known as singularities. Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all timelike and lightlike geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges"—regions where the paths of light and falling particles come to an abrupt end and geometry becomes ill-defined. By definition, these are spacetime singularities. In more interesting cases, the geometrical quantities characterizing spacetime curvature (e.g. the Ricci scalar) take on infinite values at such "curvature singularities".[142] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,[143] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.[144] The Friedmann-Lemaître-Robertson-Walker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely big bang singularities.[145] A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... // The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. ... For other uses, see Big Bang (disambiguation). ...

Given just these examples, which are all highly symmetric and thus simplified, one might think the occurrence of singularities to be an idealization. The famous singularity theorems proved using the methods of global geometry suggest 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[146] and also at the beginning of a wide class of expanding universes.[147] However, these theorems say very 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 conjecture).[148] As problematic as singularities are, there are indications that all realistic future singularities (where no symmetry is perfect, and matter has realistic properties) are safely hidden away behind a horizon, and thus invisible for all distant observers. This is postulated by the cosmic censorship hypothesis (Penrose 1969); while no formal proof of this conjecture exists, numerical simulations offer supporting evidence of its validity.[149] The Penrose-Hawking singularity theorems are a set of results in general relativity which attempt to answer the question of whether gravity is necessarily singular. ... A BKL singularity is a non-symmetric, chaotic, vacuum solution to Einsteins field equations conjectured to represent the actual interior geometry of a physical black hole formed by gravitational collapse. ... It has been suggested that Naked singularity be merged into this article or section. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ...

### Evolution equations

Main article: Initial value formulation (general relativity)

Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields, for example, they are self-interacting (that is, non-linear even in the absence of other fields, and they have no fixed background structure—the stage itself evolves as the cosmic drama is played out).[150] The initial value formulation is a way of expressing the formalism of Einsteins theory of general relativity in a way that describes a universe evolving over time. ... This article or section is in need of attention from an expert on the subject. ... For other uses of this term, see Spacetime (disambiguation). ... For a system with internal state (also called stateful system), time evolution means the change of state brought about by the passage of time. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...

Nevertheless, in order to understand Einstein's equations as partial differential equations, it is crucial to re-formulate them in a way that describes the evolution of the universe over time. This is achieved by so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension, such as the ADM formalism.[151] These decompositions show that the spacetime evolution equations of general relativity are indeed well-behaved, meaning that solutions always exist and are uniquely defined (once suitable initial conditions are specified).[152] Formulations like this are also the basis of numerical relativity: attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers.[153] In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation for General Relativity. ... In mathematics, an existence theorem is a theorem with a statement beginning there exist(s) .., or more generally for all x, y, ... there exist(s) .... That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. ... The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. ... This article is in need of attention from an expert on the subject. ...

### Global and quasi-local quantities

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 for this is that the gravitational field—like any physical field—must be ascribed a certain energy. However, it is fundamentally impossible to localize that energy.[154] The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity. ... For other uses, see Mass (disambiguation). ...

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)[155] or suitable symmetries (Komar mass).[156] 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 Bondi mass at null infinity.[157] Just as in classical physics, it can be shown that these masses are positive.[158] Analogous global definitions exist for momentum and angular momentum.[159] In addition, there have 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 systems, such as a more precise formulation of the hoop conjecture.[160] In theoretical physics, the ADM energy (short for Richard Arnowitt, Stanley Deser and Charles Misner) is a special way to define the energy in general relativity which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity - for example a spacetime... // The Komar mass of a system is one of several formal concepts of mass that used in general relativity. ... The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity. ... Classical Physics refers to the ideas and laws developed before Relativity and Quantum Theory. ... This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ... In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... The hoop conjecture was proposed by Kip Thorne in 1972. ...

## Relationship with quantum theory

Along with general relativity, quantum theory, the basis of our understanding of matter from elementary particles to solid state physics is considered one of the two pillars of modern physics.[161] However, it is still an open question of how the concepts of quantum theory can be reconciled with those of general relativity. Look up quantum in Wiktionary, the free dictionary. ... For the novel, see The Elementary Particles. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ...

### Quantum field theory in curved spacetime

The unification of quantum theory and special relativity has led to the highly successful quantum field theories which form the basis of modern elementary particle physics. These theories 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.[162] Quantum field theory in curved spacetimes is an extension of the standard quantum field theory to curved spacetimes. ... Quantum field theory (QFT) is the quantum theory of fields. ... Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

Short of constructing a theory of quantum gravity, in which all interactions, including general relativity's description of gravity, are formulated within the framework of quantum theory, there is a way to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself: use classical general relativity to describe a curved background space-time, and define a generalized quantum field theory to describe the behavior of quantum matter within that space-time.[163] The corresponding models have led to highly interesting results. Most notably, they indicate that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that black holes evaporate over time.[164] As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.[165] Look up Classical in Wiktionary, the free dictionary. ... In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation thought to be emitted by black holes due to quantum effects. ... In physics, Hawking radiation is thermal radiation emitted by black holes due to quantum effects. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...

### Quantum gravity

Main article: Quantum gravity

As interesting as the results of quantum field theory in curved space-time are, there are strong indications that it is not the last word when it comes to the relationship between general relativity and quantum theory. There is evidence that there is no convincing way to combine classical general relativity and standard quantum theory in a consistent way—put simply, matter is the source of space-time curvature, and once matter has quantum properties, we can expect space-time to have them as well.[166] Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...

Then, there is the troubling nature of singularities (cf. above): "ragged edges" of space-time. They indicate the limits of general relativity, and, given the small length scales involved, they are commonly seen as an indication that what is needed for an adequate description of the interior of black holes and time evolution close to the big bang is a full theory of quantum gravity: a theory in which gravity and the associated geometry of space-time are described in the language of quantum theory.[167] A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...

At present, no complete and consistent theory of quantum gravity is known, even though efforts to find such a theory have been made for more than 70 years. Still, there are a number of promising candidates for a theory of quantum gravity.[168]

Projection of a Calabi-Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory

One suggestive starting point are ordinary quantum field theories which, after all, are successful in describing the other three basic fundamental forces in the context of the standard model of elementary particle physics. However, while this leads to an acceptable effective (quantum) field theory of gravity at low energies,[169] gravity turns out to be much more problematic at higher energies. Where, for ordinary field theories such as quantum electrodynamics, a technique known as renormalization is an integral part of deriving predictions which take into account higher-energy contributions,[170] gravity turns out to be nonrenormalizable: at high energies, applying the recipes of ordinary quantum field theory yields models that are devoid of all predictive power.[171] Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (840 Ã— 840 pixel, file size: 327 KB, MIME type: image/png) I created this image myself to replace Image:Calabi-Yau. ... Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (840 Ã— 840 pixel, file size: 327 KB, MIME type: image/png) I created this image myself to replace Image:Calabi-Yau. ... Calabi-Yau manifold (an artists impression) In mathematics, a Calabi-Yau manifold is a compact KÃ¤hler manifold with a vanishing first Chern class. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. ... In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Figure 1. ... In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ...

Simple spin network of the type used in loop quantum gravity

Another approach to quantum gravity starts with the canonical quantization procedures of quantum theory. Starting with the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is an analogue of the Schrödinger equation: the Wheeler-deWitt equation which, regrettably, turns out to be ill-defined.[177] A major break-through came with the introduction of what are now known as Ashtekar variables, which represent geometric gravity using mathematical analogues ofelectric and magnetic fields.[178] The resulting candidate for a theory of quantum gravity is Loop quantum gravity, in which space is represented by a network structure called a spin network, evolving over time in discrete steps.[179] A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... In theoretical physics, the Wheeler-deWitt equation is an equation that a wave function of the Universe should satisfy in a theory of quantum gravity. ... In theoretical physics, Ashtekar (new) variables (named after Abhay Ashtekar who invented them) represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... For the indie-pop band, see The Magnetic Fields. ... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ... A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. ...

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,[180] there are numerous other attempts to arrive at a viable theory of quantum gravity, some example being dynamical triangulations,[181] causal sets,[182] twistor models[183] or the path-integral based models of quantum cosmology.[184] The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the geometric objects of the four dimensional space-time (Minkowski space) into the geometric objects in the 4-dimensional complex space with the metric signature (2,2). ... In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... In theoretical physics, quantum cosmology is a young field attempting to study the effect of quantum mechanics on the earliest moments of the universe after the Big Bang. ...

Currently, there is still no complete and consistent quantum theory of gravity, and the candidate models still need to overcome major formal and conceptual problems. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests, although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[185]

## History and current status

First page from Einstein's manuscript explaining general relativity

Soon after publishing his theory of special relativity in 1905, Einstein began to think about how to incorporate gravity into his new relativistic framework. His considerations led him from a simple thought experiment involving an observer in free fall to the equivalence principle and thence to a fully geometric theory of gravity:[186] from explorations of some consequences of the equivalence principle such as the influence of gravity and acceleration on the propagation of light published in 1907[187] to the main work in the years 1911 to 1915 with the realization of the role of differential geometry (with help from Marcel Grossmann on the intricacies of that field of mathematics) and a long search, including detours and false starts, for the field equations relating geometry and the mass-energy content of spacetime. In November of 1915, these efforts culminated in Einstein's presentation to the Prussian Academy of Science of the Einstein field equations.[188] // Creation of General Relativity Early investigations The development of general relativity began in 1907 with the publication of an article by Albert Einstein on acceleration under special relativity. ... This article or section is in need of attention from an expert on the subject. ... In theoretical physics, the current Gold Standard Theory of Gravitation is the general theory of relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Gravity is a force of attraction that acts between bodies that have mass. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Marcel Grossmann (born in Budapest on April 9, 1878 - died in Zurich on September 7, 1936) was a mathematician, a friend, and a classmate of Albert Einstein. ... The Prussian Academy of Sciences (German: ) was an academy established in Berlin on July 11, 1700. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ...

Already in 1916, Schwarzschild found the eponymous solution to the Einstein field equations, laying the groundwork for the description of gravitational collapse and, eventually, black holes. The same year saw the first steps of generalization to electrically charged objects that would result in the Reissner-Nordström solution.[189] In 1917, Einstein initiated the field of relativistic cosmology. However, in line with contemporary thinking, he tried to describe a static universe, adding the cosmological constant to his original field equations for that purpose.[190] When it became clear in 1929 with the work of Hubble and others that our universe is indeed expanding (and thus better described by expanding cosmological solutions found by Friedmann in 1922), Lemaître formulated the earliest version of the big bang models.[191] Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... In physics and astronomy, a Reissner-NordstrÃ¸m black hole is a black hole that carries electric charge , no angular momentum, and mass . ... In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Î›) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... Edwin Powell Hubble (November 20, 1889 â€“ September 28, 1953) was an American astronomer. ... Alexander Alexandrovich Friedman (June 16, 1888 – September 16, Russian cosmologist and mathematician. ... Monsignor Georges LemaÃ®tre, priest and scientist. ... For other uses, see Big Bang (disambiguation). ...

From all these developments, general relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed every unambiguous observational and experimental test to which it has been subjected. Still, there are strong indications the theory is incomplete.[198]

The problem of quantum gravity, and the associated question of the reality of space-time singularities, remain open.[199] Observational data like that for dark energy and dark matter could indicate the need for new physics,[200] and while the so-called Pioneer anomaly might yet admit of a conventional explanation, it, too, could be a harbinger of new physics.[201] Even while staying within the frame of Einstein's theory, general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties Einstein's equations,[202] ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run,[203] and the race for the first direct detection of gravitational waves continues apace,[204] with opportunities to test the theory beyond the limited approximations it has been tested so far even in the binary pulsar measurements.[205] More than ninety years after the theory was first published, general relativity remains a highly active area of research.[206] Unsolved problems in physics: What causes the apparent residual sunward acceleration of the Pioneer spacecraft? The Pioneer anomaly or Pioneer effect is the observed deviation from expectations of the trajectories of various unmanned spacecraft visiting the outer solar system, notably Pioneer 10 and Pioneer 11. ... A binary pulsar is a pulsar with a binary companion, often another pulsar, white dwarf or neutron star. ...

Look up general relativity in
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Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Image File history File links Wikisource-logo. ... The original Wikisource logo. ... In theoretical physics, the current Gold Standard Theory of Gravitation is the general theory of relativity. ... This is a partial list of persons who have made major contributions to the development of standard mainstream general relativity. ... | name = David Hilbert | image = Hilbert1912. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... // Geroch, Robert (1981). ... This article or section is in need of attention from an expert on the subject. ... // Creation of General Relativity Early investigations The development of general relativity began in 1907 with the publication of an article by Albert Einstein on acceleration under special relativity. ... An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ...

## Notes

1. ^ The following exposition re-traces that of Ehlers 1973, section 1.
2. ^ See, for instance, Arnold 1989, chapter 1.
3. ^ See Ehlers 1973, pp. 5f..
4. ^ See Will 1993, section 2.4 or Will 2006, section 2.
5. ^ Cf. Wheeler 1992, chapter 2; similar accounts can be found in most other popular-science books on general relativity.
6. ^ See Ehlers 1973, section 1.2, Havas 1964, and Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959.
7. ^ See Ehlers 1973, pp. 10f..
8. ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006.
9. ^ An in-depth comparison between the two symmetry groups can be found in Giulini 2006a.
10. ^ For instance Rindler 1991, section 22; a thorough treatment can be found in Synge 1972, ch. 1 and 2.
11. ^ Cf. Ehlers 1973, sec. 1.4. and Schutz 1985, sec. 5.1.
12. ^ See Ehlers 1973, p. 17ff.; a derivation can be found e.g. in Mermin & 2005 ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below.
13. ^ Cf. Rindler 2001, sec. 1.13; for an elementary account, see chapter 2 of Wheeler 1990; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985.
14. ^ Ehlers 1973, sec. 1.4. for the experimental evidence, see once more section Gravitational time dilation and frequency shift.
15. ^ Cf. Ehlers 1973, p. 16; Kenyon 1990, sec. 7.2; Weinberg 1972, sec. 2.8.
16. ^ See Ehlers & 1973 pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. It can be shown that 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 space-time of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972.
17. ^ E.g. Kenyon 1990, sec. 7.4.
18. ^ E.g. Schutz 1985, sec. 8.3.
19. ^ Cf. Brans & Dicke 1961 and section 3 in ch. 7 of Weinberg 1972, Goenner 2004, sec. 7.2, and Trautman 2006, respectively.
20. ^ E.g. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other text-book on general relativity.
21. ^ At least approximately, cf. Poisson 2004.
22. ^ E.g. p. xi in Wheeler 1990.
23. ^ E.g. Wald 1984, sec. 4.4.
24. ^ E.g. in Wald 1984, sec. 4.1.
25. ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b.
26. ^ E.g. section 5 in ch. 12 of Weinberg 1972.
27. ^ Cf. the introductory chapters of Stephani et al. 2003.
28. ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996.
29. ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006.
30. ^ E.g. chapters 3, 5, and 6 of Chandrasekhar 1983.
31. ^ E.g. ch. 4 and sec. 3.3. in Narlikar 1993.
32. ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5.
33. ^ See Lehner 2002 for an overview.
34. ^ For instance Wald 1984, sec. 4.4.
35. ^ E.g. Will 1993, sec. 4.1 and 4.2.
36. ^ Cf. section 3.2 of Will 2006 as well as Will 1993, ch. 4.
37. ^ Cf. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198.
38. ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5.
39. ^ See Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186.
40. ^ E.g. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow et al. 2005.
41. ^ Starting with the Hafele-Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186.
42. ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003.
43. ^ Reviews are given in Stairs 2003 and Kramer 2004.
44. ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, section 4.2.
45. ^ Cf. Ohanian & Ruffini 1994, pp. 164–172.
46. ^ The fact that light follows null geodesics is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, section 5.
47. ^ A brief descriptions and pointers to the literature can be found in Blanchet 2006, section 1.3.
48. ^ See Rindler 2001, section 1.16; for the historical examples, Israel 1987, p. 202–204.; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997.
49. ^ E.g. Rindler 2001, sec. 11.11.
50. ^ See this calculation for the numerical derivation of 1.75 arc seconds.
51. ^ Cf. Kennefick 2005; for an overview of more recent measurements, see Ohanian & Ruffini 1994, chapter 4.3. The most precise direct modern observations measure the deflection of the light of distant quasars by the Sun, cf. Shapiro et al. 2004.
52. ^ Shapiro 1965; a pedagogical introduction can be found in Weinberg 1972, ch. 8, sec. 7.
53. ^ The most recent measurements are Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200.
54. ^ Cf. Stairs 2003, section 4.4.
55. ^ Will 1993, sec. 7.1 and 7.2.
56. ^ For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Note, however that for gravitational waves, the dominant contribution is not the dipole, but the quadrupole cf. Schutz 2001.
57. ^ Any textbook on general relativity will contain a description of these properties, e.g. Schutz 1981, ch. 9.
58. ^ For example Jaranowski & Królak 2005.
59. ^ Rindler 2001, ch. 13.
60. ^ See Gowdy 1971, Gowdy 1974.
61. ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy.
62. ^ Pais 1982, pp. 253–254
63. ^ See Rindler 2001, section241.
64. ^ See Will 1993, pp. 177–181.
65. ^ In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3.
66. ^ See Schutz 2003, pp. 48–49 and Pais 1982, pp. 253–254.
67. ^ The most precise measurements are VLBI measurements of planetary positions; see Will 1993, chapter 5, Will 2006, section 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407.
68. ^ See Kramer, Stairs & Manchester 2006.
69. ^ A figure that includes error bars is figure 7, in section 5.1, of Will 2006.
70. ^ See Stairs 2003 and Schutz 2003, pp. 317–321; an accessible account can be found in Bartusiak 2000, pp. 70–86.
71. ^ An overview can be found in Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994.
72. ^ Cf. Kramer 2004.
73. ^ See e.g. Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, sec. §11.4.
74. ^ See Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1982, sec. 7.8.
75. ^ See Bertotti, Ciufolini & Bender 1987 and, for a more recent review, Nordtvedt 2003.
76. ^ See Kahn 2007.
77. ^ E.g. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471.
78. ^ E.g. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004.
79. ^ E.g. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006; see the entry frame-dragging for an account of the debate.
80. ^ A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt et al. 2007; further updates will be available on the mission website Kahn 1996–2007.
81. ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998.
82. ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3.
83. ^ See Walsh, Carswell & Weymann 1979.
84. ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007.
85. ^ For an overview, see Roulet & Mollerach 1997.
86. ^ See Narayan & Bartelmann 1997, sec. 3.7.
87. ^ For an overview, Barish 2005; accessible accounts can be found in Bartusiak 2000 and Blair & McNamara 1997.
88. ^ An overview is given in Hough & Rowan 2000.
89. ^ See Danzmann & Rüdiger 2003.
90. ^ See Landgraf, Hechler & Kemble 2005.
91. ^ Cf. Thorne 1995.
92. ^ See Cutler & Thorne 2001, sec. 2.
93. ^ See Cutler & Thorne 2001, sec. 3.
94. ^ See Miller 2002, lectures 19 and 21.
95. ^ E.g. Celotti, Miller & Sciama 1999, sec. 3.
96. ^ Cf. Springel & al. 2005 and the accompanying summary Gnedin 2005.
97. ^ Cf. Blandford 1987, section 8.2.4,
98. ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996.
99. ^ For a review, see Begelman, Blandford & Rees 1984.
100. ^ See Rees 1966.
101. ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2.
102. ^ Cf. Kraus 1998.
103. ^ See Celotti, Miller & Sciama 1999.
104. ^ Cf. Schödel et al. 2003.
105. ^ Examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 and, for an overview, Narayan 2006, sec. 5.
106. ^ Cf. Falcke, Melia & Agol 2000.
107. ^ Cf. Seidel 1998.
108. ^ Cf. Dalal et al. 2006.
109. ^ E.g. Barack & Cutler 2004.
110. ^ Originally Einstein 1917; cf. the description in Pais 1982, pp. 285–288.
111. ^ See Carroll 2001, ch. 2.
112. ^ See Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991.
113. ^ E.g. with WMAP data, see Spergel et al. 2003.
114. ^ See Peebles 1966; for a recent account of predictions, see Coc et al. 2004; an accessible account can be found in Weiss 2006.
115. ^ See Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005.
116. ^ A review can be found in Lahav & Suto 2004.
117. ^ Cf. Alpher & Herman 1948 and, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965, andfor precision measurements by satellite observatories see Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP).
118. ^ This additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997.
119. ^ See, e.g., fig. 2 in Bridle et al. 2003.
120. ^ For a review, see Bertschinger 1998; more recent results can be found in Springel et al. 2005.
121. ^ These additional observations involve the dynamics of galaxies and galaxy clusters cf. chapter 18 of Peebles 1993, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005.
122. ^ See Peacock 1999, ch. 12, and Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12.
123. ^ 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 Mannheim 2006, sec. 9.
124. ^ See Carroll 2001; an accessible overview is given in Caldwell 2004.
125. ^ 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. Mannheim 2006, 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. Buchert 2007.
126. ^ More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1.
127. ^ A good introduction is Linde 1990; for a more recent review, see Linde 2005.
128. ^ See Spergel et al. 2007, sec. 5 & 6.
129. ^ 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.
130. ^ See Brandenberger 2007, sec. 2.
131. ^ See Frauendiener 2004, Wald 1984, section 11.1, and Hawking & Ellis 1973, section 6.8 & 6.9
132. ^ E.g. Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6.
133. ^ For an account of the evolution of this concept, see Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004.
134. ^ See Thorne 1972; for an account of more recent numerical studies, see Berger 2002, sec. 2.1.
135. ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chrusciel 2006 as overviews of more recent results.
136. ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see chapter 2 of Wald 2001. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969.
137. ^ See Bekenstein 1973, Bekenstein 1974.
138. ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in chapter 3 of Wald 2001.
139. ^ Cf. Narlikar 1993, sec. 4.4.4 and 4.4.5.
140. ^ Cf. Rindler 2001, sec. 12.4
141. ^ Unruh 1976, cf. Wald 2001, chapter 3.
142. ^ See Hawking & Ellis 1973, section 8.1, Wald 1984, section 9.1.
143. ^ See Townsend 1997, chapter 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, chapter 3.
144. ^ See Townsend 1997, chapter 4; for a more extensive treatment, cf. Chandrasekhar 1983, chapter 6.
145. ^ See Ellis & van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
146. ^ Namely when there are trapped null surfaces, cf. Penrose 1965.
147. ^ See Hawking 1966.
148. ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007.
149. ^ 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 Penrose 1969; a text-book level account is given in Wald 1984, pp. 302-305. For numerical results, see the review Berger 2002, sec. 2.1.
150. ^ Cf. Hawking & Ellis 1973, sec. 7.1.
151. ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7.
152. ^ Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998.
153. ^ See Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001.
154. ^ Cf. Misner, Thorne & Wheeler 1973, §20.4.
155. ^ Arnowitt, Deser & Misner 1962.
156. ^ Cf. Komar 1959; for a pedagogical introduction, see Wald 1984, 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. Ashtekar & Magnon-Ashtekar 1979.
157. ^ For a pedagogical introduction, see Wald 1984, sec. 11.2.
158. ^ See the various references given on p. 295 of Wald 1984; 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.
159. ^ E.g. Townsend 1997, ch. 5.
160. ^ Such quasi-local mass-energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy-momentum based on twistor methods; cf. the review article Szabados 2004.
161. ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003.
162. ^ Cf. textbooks such as Ramond 1990, Weinberg 1995, or Peskin & Schroeder 1995; a more accessible overview can be found in Auyang 1995.
163. ^ Cf. Wald 1994 and Birrell & Davies 1984.
164. ^ For Hawking radiation Hawking 1974, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000.
165. ^ Cf. chapter 3 in Wald 2001.
166. ^ See section 2 in Carlip 2001.
167. ^ E.g. p. 407ff. in Schutz 2003.
168. ^ A timeline and overview can be found in Rovelli 2000.
169. ^ See Donoghue 1995.
170. ^ Cf. chapters 17 and 18 of Weinberg 1996.
171. ^ Cf. Goroff & Sagnotti 1985.
172. ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b.
173. ^ E.g. Ibanez 2000.
174. ^ For the graviton as part of the string spectrum, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3; for the extra dimensions, ibid sec. 4.2.
175. ^ E.g. Weinberg 2000, ch. 31.
176. ^ E.g. Townsend 1996, Duff 1996.
177. ^ Cf. section 3 in Kuchař 1973.
178. ^ See Ashtekar 1986, Ashtekar 1987.
179. ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.
180. ^ See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.
181. ^ See Loll 1998.
182. ^ See Sorkin 2005.
183. ^ See ch. 33 in Penrose 2004 and references therein.
184. ^ Cf. Hawking 1987.
185. ^ E.g. Ashtekar 2007, Schwarz 2007.
186. ^ This development is traced in chapters 9 through 15 of Pais 1982 and in Janssen 2005; an accessible overview can be found in Renn 2005, p. 110ff..
187. ^ Einstein 1907, cf. Pais 1982, ch. 9.
188. ^ Published in the Academy's proceedings as Einstein 1915; cf. Pais 1982, ch. 11–15.
189. ^ See Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918).
190. ^ Einstein 1917, cf. Pais 1982, ch. 15e.
191. ^ Hubble's original article is Hubble 1929; a readable overview is given in Singh 2004, ch. 2-4.
192. ^ Cf. Pais 1982, p. 253-254.
193. ^ Cf. Kennefick 2005 and Kennefick 2007.
194. ^ Cf. Pais 1982, ch. 16.
195. ^ Cf. Israel 1987, ch. 7.8-7.10 and Thorne 1993, ch. 3-9.
196. ^ Cf. the sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein.
197. ^ Cf. the section Cosmology and references therein; the historical development is traced in Overbye 1999.
198. ^ Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
199. ^ Cf. the section The search for a theory of quantum gravity, above.
200. ^ Cf. the section Cosmology, above.
201. ^ See Nieto 2006.
202. ^ See Friedrich 2005.
203. ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.
204. ^ See 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 GEO 600 and LIGO.
205. ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2007; for a general review of experimental tests of general relativity, see Will 2006.
206. ^ A good starting point for a snapshot of present-day research in relativity is the electronic review journal Living Reviews in Relativity.

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arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Jacob David Bekenstein (born May 1, 1947) is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation. ... Jacob David Bekenstein (born May 1, 1947) is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Carl Henry Brans (1935â€“) is an American mathematical physicist. ... Robert Henry Dicke (May 6, 1916 â€“ March 4, 1997) was an American experimental physicist, who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Chandrasekhar redirects here. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Wolfgang Rindler is a leading physicist working in the field of General Relativity where he is well known for introducing the term event horizon, rindler coordinates, and (in collaboration with Roger Penrose) for popularizing the use of spinors in general relativity. ... â€œEinsteinâ€ redirects here. ... â€œEinsteinâ€ redirects here. ... â€œEinsteinâ€ redirects here. ... â€œEinsteinâ€ redirects here. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Edwin Powell Hubble (November 20, 1889 â€“ September 28, 1953) was an American astronomer. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Andrei Linde is an American physicist and professor of Physics at Californias Stanford University. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Andrei Linde is an American physicist and professor of Physics at Californias Stanford University. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... In solid-state physics, N. David Mermin is a polymathic physicist at Cornell University best known for the eponymous Mermin-Wagner theorem and his application of the term Boojum to superfluidity. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Gunnar NordstrÃ¶m (1881-1923) was a Finnish theoretical physicist who is best remembered for his theory of gravitation, which was an early competitor of general relativity. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... J. Robert Oppenheimer[1] (April 22, 1904 â€“ February 18, 1967) was an American theoretical physicist, best known for his role as the director of the Manhattan Project, the World War II effort to develop the first nuclear weapons, at the secret Los Alamos laboratory in New Mexico. ... Abraham (Bram) Pais (May 19, 1918, Amsterdam, The Netherlands â€” July 28, 2000, Copenhagen, Denmark) was a Dutch-born American physicist and science historian. ... Philip James Edwin Peebles (born April 25, 1935) is an Canadian-American theoretical cosmologist. ... Philip James Edwin Peebles (born April 25, 1935) is an Canadian-American theoretical cosmologist. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Arno Allan Penzias (born April 26, American physicist. ... Robert Woodrow Wilson (born January 10, 1936) is an American astronomer. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Pierre Ramond (b. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Irwin I. Shapiro is an American astrophysicist. ... Irwin I. Shapiro is an American astrophysicist. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Bill Unruh teaching in UBC William G. Unruh (born August 21, 1945) is a Canadian physicist at the University of British Columbia, Vancouver, who discovered the Unruh effect. ... Robert Wald (b. ... Robert Wald (b. ... In physics and especially relativity, General Relativity is a popular textbook on Einsteins theory of general relativity written by Robert Wald. ... Robert Wald (b. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... John Archibald Wheeler (July 9, 1911â€“April 13, 2008) was an eminent American theoretical physicist. ... Clifford Martin Will (b. ... Clifford Martin Will (b. ...

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 General relativity - Wikipedia, the free encyclopedia (5386 words) In general relativity, phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion in a curved spacetime. General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi-Walker transport along world lines in spacetime. General relativity was developed by Einstein in a process that began in 1907 with the publication of an article on the influence of gravity and acceleration on the behavior of light in special relativity.
 General Relativity (1940 words) General relativity is a theory of gravitation and to understand the background to the theory we have to look at how theories of gravitation developed. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame. The final steps to the theory of general relativity were taken by Einstein and Hilbert at almost the same time.
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