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General relativity

General relativity

Introduction to... Mathematical formulation of...

Fundamental concepts

Special relativity Equivalence principle World line · Riemannian geometry

Phenomena

Black hole · Event horizon · Lenses Waves · Singularity Frame-dragging · Geodetic effect

Equations

Linearized Gravity Post-Newtonian formalism Einstein field equations

Advanced theories

Kaluza-Klein Quantum gravity

Solutions

Schwarzschild · Kasner · Kerr Milne · Reissner-Nordström Robertson-Walker Image File history File links Circle-question-red. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Newton’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. ... Notational point: General relativity articles using tensors will use the abstract index notation . ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... 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. ... Simulated view of a black hole in front of the Milky Way. ... For the science fiction film, see Event Horizon (film). ... This article or section is in need of attention from an expert on the subject. ... For the concept in fluid dynamics and meteorology, see Gravity wave. ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... 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. ... 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. ... This article or section is in need of attention from an expert on the subject. ... Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism. ... This article or section does not adequately cite its references or sources. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... The Kasner metric is an exact solution to Einsteins theory of general relativity. ... 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. ... 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. ... In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, is a black hole that carries mass , electric charge , and no angular momentum. ... // The Friedmann-Lemaître-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. ...

Scientists

Einstein · Minkowski · Eddington Lemaître · Schwarzschild Robertson · Kerr · Friedman Chandrasekhar · Hawking · others “Einstein” redirects here. ... Hermann Minkowski. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ... Father Georges-Henri Lemaître (July 17, 1894 – June 20, 1966) was a Belgian Roman Catholic priest, honorary prelate, professor of physics and astronomer. ... 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 (Александр Александрович Фридман) (June 16, 1888 – 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. ...

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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. These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress-energy tensor T^ab.^[1] (To wit, whenever a field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to the metric should give the stress-energy contribution due to the field.) General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... 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. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... This article is in need of attention from an expert on the subject. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... This article is in need of attention from an expert on the subject. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

Finally, when all the contributions to the stress-energy tensor are added up, the result must satisfy the Einstein field equations (written here in geometrized units) This article or section is in need of attention from an expert on the subject. ... In physics, especially in the general theory of relativity, geometrized units or sometimes geometric units, is a physical unit system in which all physical quantities are expressed in the unit of length: meter. ...

In the above field equations, the tensor field standing on the left hand side, the Einstein tensor, is computed uniquely from the metric tensor which is part of the definition of a Lorentzian manifold. Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence "here and now" of nongravitational energy-momentum causes a proportional amount of Ricci curvature "here and now". Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of nongravitational energy-momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or nongravitational fields. Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. ... In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... 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. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This article or section is in need of attention from an expert on the subject. ...

Known solutions

The well-known solutions are mainly linear and exponential scalar Potential-fields, homogenous and isotropic without infinitesimal tensor properties.

Schwarzschild solutions

Einstein was astonished about two promptly given solutions, by Karl Schwarzschild: Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ...

• The first Schwarzschild solution is well known as the first solution of Einstein’s General relativity and as his “extern solution”, meaning outside of the Schwarzschild radius. It describes in cylinder coordinates a kind of to its end more and more contracting tunnel-shape. The geometric surface define all possible circles. The solution give all valid extern effects of a black hole.
• The second, so called Inner Schwarzschild solution was presented by Schwarzschild some days later only, mainly found in German original and related texts as "innere Schwarzschild Lösung". Schwarzschild had seen that the inner solution of his black hole is equivalently described as an attractor, a super-massive body within the spere of an homogenous isotropic compressed gas, fluid or solid, valid within boundary of R=Schwarzschild radius.

Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Schwarzschild radius (sometimes inappropriately referred to as the gravitational radius[1]) is a characteristic radius associated with every mass. ... A point plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ... Simulated view of a black hole in front of the Milky Way. ...

Friedmann-Lemaître-Robertson-Walker solutions

The first Friedmann equations solution is based on the same mathematical calculus as the second Schwarzschild solution, but explicitly considering molecules of an isotropic homogenous interstellar gas fluid instead of molecules of a super compressed fluid or solid only within its calculated black hole radius. The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... The Friedmann equations relate various cosmological parameters within the context of general relativity. ...

The second solution described for the first time the Big bang as new solution, assuming that the spatial component of the 4D-metric can be time dependent. The main basis of the actual standard solution at its physical Schwarzschild radius has the same problem: While in black holes the time stops for us, the total beginning of the time must be supposed at its Schwarzschild radius, meanwhile declared sufficiently by mainstream but not yet accepted for its critics. According to the Big Bang model, the universe emerged from an extremely dense and hot state. ... The Schwarzschild radius (sometimes inappropriately referred to as the gravitational radius[1]) is a characteristic radius associated with every mass. ...

De Sitter solutions

De Sitter universe is another solution to Albert Einstein's field equations of GR by Willem de Sitter. It is an exponential solution. The exponentially expanding universe of the FLRW form has the scale factor: Willem de Sitter (May 6, 1872, Sneek – November 20, 1934, Leiden [1]) was a Dutch mathematician, physicist and astronomer. ... A de Sitter universe is a solution to Einsteins field equations of General Relativity which is named after Willem de Sitter. ... Willem de Sitter (May 6, 1872 – November 20, 1934) was a mathematician, physicist and astronomer. ... // The Friedmann-Lemaître-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. ...

, (H = Hubble constant)

describing the expansion of physical spatial distances. The metric expansion of space is a key part of sciences current understanding of the universe, whereby space itself is described by a metric which changes over time. ...

• A de Sitter universe is one with no ordinary matter content (supported e.g. by dark energy) but with a positive cosmological constant which sets the expansion rate, H with the effect that a larger cosmological constant leads to a larger expansion rate.

A mathematically equivalent negative exponent would produce implosion. A complex exponent shows an oscillation: A de Sitter universe is a solution to Einsteins field equations of General Relativity which is named after Willem de Sitter. ... 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. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ...

Geoffrey Burbidge’s solution

According to Geoffrey Burbidge and Margaret_Burbidge, the universe is oscillatory and as such expands and contracts periodically over infinite time between the following periodic extremes: Geoffrey Ronald Burbidge (born September 24, 1925) is a British-American physics professor in the University of California, San Diego. ... Geoffrey Ronald Burbidge (born September 24, 1925) is a British-American physics professor in the University of California, San Diego. ... Margaret Burbidge (nee Eleanor Margaret Peachey) (born August 12, 1919) is a British astrophysicist, noted for original research and holding many administrative posts, including director of the Royal Greenwich Observatory. ...

• Einstein had initially preferred a static solution of the GR. Because of its instability he had revised it and named it by himself as his biggest stupor.
• Such an initially maximal huge universe is still affected by a rest of gravity. At thereby initially minimum gas-tension the space must thereby move more and more move to a center inside, finally imploding by gravity.
• At the end of implosion with its minimum space, seen as a maximal compressed interstellar gas will stop to move. Then begins the other extreme, an increasing movement outwards finally exploding.
• The resulting Burbidge’s theory was called the B²FH theory after the participants.

This theory, due to its controversial nature, has brought fame and infamy to Burbidge.

Halton Arp's solution

Intrinsic redshift meant by principle only that some stellar objects anyhow not obey standard laws (e.g. Hubble's law) and that something like this has to be declared. Arp et. al. discovered highly discontinuous redshifts within filamentary superclusters, voids and related quasars, until now meaning that a too rare statistic probability cannot completely overrule intrinsic effects by generally different considerations of the standard model. It seems that redshift of several "child galaxies" are significantly higher than its "parent galaxies". For Quasars it could be based on gravitational redshift (Einstein’s clocks run differently at different gravity-centres by Einstein effects: a same kind of emitted photon starts at different places at different frequencies affected by different gravities). A problem arised because Intrinsic redshift became then a hypothesis from various non-standard cosmologies that a significant portion of the observed redshift of extragalactic objects (e.g. quasars and galaxies) may be caused by a phenomenon other than known redshift mechanisms (cosmological redshift, Doppler redshift, gravitational redshift). A proposed Redshift quantization could not be confirmed until now. Therefore also this controversial theory is widely not accepted. Halton Arp in London, Oct 2000 Halton Christian Arp is an American astronomer. ... Seeing Red. ... This article or section is in need of attention from an expert on the subject. ... The Einstein effect may mean: Gravitational redshift (Einstein shift) Gravitational lensing There are also: The Bose-Einstein effect The Einstein-de Haas effect Category: ... This article is under development Redshift quantization or redshift periodicity is the hypothesis that the redshifts of cosmologically distant objects (in particular galaxies) tend to cluster around multiples of some particular value. ...

Fritz Zwicky’s solution

Einstein himself wrote - until today confusing physics - that photons have a "zero rest mass (remark: considered in standard cosmology) but non-zero relativistic mass". Zwicky related to the relativistic mass and the Einstein effects because Einstein confirmed Plancks view that photons are particles with a relativistic mass. Gravitational redshifts was utilized by him simply by another view about photons, meaning now: Fritz Zwicky (February 14, 1898 – February 8, 1974) was an American-based Swiss astronomer. ... The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... The term mass in special relativity can be used in different ways, occasionally leading to confusion. ... The Einstein effect may mean: Gravitational redshift (Einstein shift) Gravitational lensing There are also: The Bose-Einstein effect The Einstein-de Haas effect Category: ... This article is about Planck, the German physicist. ... This article or section is in need of attention from an expert on the subject. ...

• Gravity influences photons in direction of its related radius or simply always centripetally in direction of its centre, meanwhile partly supported by some newer graviton’s theories.
• By this non-conform "relativistic mass view" about physics an increasing gravitational potential “slows” the energy of a photon by redshift by loosing its momentum.

Not accccepted by the mainstream, increasing gravitational potential was recently used to declare the Pioneer anomaly. In physics, the graviton is mainly still considered to be a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ... In physics, gravitational potential is the measure of potential energy an object possesses due to its position in a gravitational field. ... 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. ...

Difficulties with the definition

Take any Lorentzian manifold, compute its Einstein tensor G^ab, which is a purely mathematical operation, divide by 8π, and declare the resulting symmetric second rank tensor field to be the stress-energy tensor T^ab. Thus any Lorentzian manifold is a solution of the Einstein field equation with some right hand side. Which of course doesn't make general relativity useless, but only shows that there are two complementary ways to use it. One can fix the form of the stress-energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress-energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model). Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what most of cosmologists do for the last 5-10 years: they assume that the Universe is homogenous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... This article is in need of attention from an expert on the subject. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure. ... 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. ...

Within the first approach the alleged stress-energy tensor must arise in the standard way from a "reasonable" matter distribution or nongravitational field. In practice, this notion is pretty clear, especially if you restrict the admissible nongravitational fields to the only one known in 1916, the electromagnetic field. But ideally we would like to have some mathematical characterization that states some purely mathematical test which we can apply to any putative "stress-energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. Unfortunately, no such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator. But these conditions, it seems, can satisfy no-one. On the one hand, they are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect. The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... 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. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In physics, the Casimir effect is a physical force exerted between separate objects, which is due to neither charge, gravity, nor the exchange of particles, but instead is due to resonance of all-pervasive energy fields in the intervening space between the objects. ...

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...

In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as closed timelike curves. Some of the best known exact solutions, in fact, have this character. This article or section does not cite its references or sources. ... This article or section does not cite its references or sources. ... In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is closed. ...

Types of exact solution

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress-energy tensor:

• vacuum solutions: T^ab = 0; these describe regions in which no matter or nongravitational fields are present,

• electrovacuum solutions: T^ab must arise entirely from an electromagnetic field which solves the source-free Maxwell equations on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,

• null dust solutions: T^ab must correspond to a stress-energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,

• fluid solutions: T^ab must arise entirely from the stress-energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the mass, momentum, and stress of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields: In General relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. ... In general relativity, an electrovacuum solution is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) source-free Maxwell equations appropriate to the given geometry. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... A null dust solution is an exact solution of Einsteins field equation in which the Einstein tensor is null. ... In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. ... In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p. ...

• scalar field solutions: T^ab must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,

• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): T^ab arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known. In general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a massless scalar field (often taken to have minimal curvature coupling). ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... Mesons of spin 1 form a nonet In particle physics, a meson is a strongly interacting boson, that is, it is a hadron with integral spin. ... Look up Quintessence in Wiktionary, the free dictionary. ... In general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ...

Below we have sketched a classification by physical interpretation. This is probably more useful for most readers than the Segre classification of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts: The Segre classification is an algebraic classification of rank two symmetric tensors. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

• nonnull electrovacuums have Segre type and isotropy group SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type and isotropy group E(2),
• perfect fluids have Segre type and isotropy group SO(3),
• Lambdavacuums have Segre type and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress-energy tensor. This article is about the mathematical concept. ...

Constructing solutions

The Einstein field equation, when fully written out as a system of partial differential equations, takes the form of a rather complicated system of coupled, nonlinear partial differential equations. As such, in general, it is very hard to solve. 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. ...

Nonetheless, several effective techniques for obtaining exact solutions are available.

The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum). In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In general relativity, a spacetime is said to be stationary if it admits a global, nowhere zero timelike Killing vector field. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... 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. ... It has been suggested that this article or section be merged into Schwarzschild solution. ...

This naive approach usually works best if one uses a frame field rather than a coordinate basis. In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold which is physically interpreted as a model of spacetime. ...

A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form. In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ... The Segre classification is an algebraic classification of rank two symmetric tensors. ...

This second kind of symmetry approach has often been used with the Newman-Penrose formalism, which uses spinorial quantities for more efficient bookkeeping. The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose[1] for General Relativity. ...

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS). In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödingers equation in two dimensions. ...

But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie's notion of the point symmetry of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature. In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... The symmetry group of an object (e. ...

Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible.^[2] Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... In mathematics and physics, an integrable system refers to a system of partial differential equations that may be integrated to obtain the solutions to the equations. ... In mathematics, the inverse scattering transform is a procedure for integrating certain nonlinear partial differential equations (PDEs) by first converting them into a system of linear ordinary differential equations (ODEs). ... The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ... A soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ...

There are also various transformations which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of soliton equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood, or even globally objectionable. In mathematics, Bäcklund transforms relate solutions of differential equations. ... 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. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a delicate balance between nonlinear and dispersive effects in the medium. ...

Existence of solutions

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. One of the most basic questions one can ask is: do solutions exist, and if so, how many?

To get started, we should adopt a suitable initial value formulation of the field equation, which gives two new systems of equations, one giving a constraint on the initial data, and the other giving a procedure for evolving this initial data into a solution. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.

To get some idea of "how many" solutions we might optimistically expect, we can appeal to Einstein's constraint counting method. A typical conclusion from this style of argument is that a generic vacuum solution to the Einstein field equation can be specified by giving four arbitrary functions of three variables and six arbitrary functions of two variables. These functions specify initial data, from which a unique vacuum solution can be evolved. (In contrast, the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions, are specified by giving just two functions of two variables, which are not even arbitrary, but must satisfy a system of two coupled nonlinear partial differential equations. This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.) In mathematics, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation. ...

However, this crude analysis falls far short of the much more difficult question of global existence of solutions. The global existence results which are known so far turn out to involve another idea.

Global stability theorems

We can imagine "disturbing" the gravitational field outside some isolated massive object by "sending in some radiation from infinity". We can ask: what happens as the incoming radiation interacts with the ambient field? In the approach of classical perturbation theory, we can start with Minkowksi vacuum (or another very simple solution, such as the de Sitter lambdavacuum), introduce very small metric perturbations, and retain only terms up to some order in a suitable perturbation expansion-- somewhat like evaluating a kind of Taylor series for the geometry of our spacetime. This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar. However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations. 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. ... A binary pulsar is a pulsar with a binary companion, often another pulsar, white dwarf or neutron star. ...

The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation. This requires the introduction of many new ideas. The desired result, sometimes expresed by the slogan that the Minkowski vacuum is nonlinearly stable, was finally proven by Demetrios Christodoulou and Sergiu Klainerman only in 1993. Analogous results are known for lambdavac perturbations of the de Sitter lambdavacuum (Helmut Friedrich) and for electrovacuum perturbations of the Minkowski vacuum (Nina Zipser). Demetrios Christodoulou (born October 19, 1951) is a mathematical physicist, well known in the field of general relativity for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski vacuum. ...

The positive energy theorem

Another issue we might worry about is whether the net mass-energy of an isolated concentration of positive mass-energy density (and momentum) always yields a well-defined (and non-negative) net mass. This result was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stress-energy tensor. Richard Melvin Schoen is an American mathematician. ... Shing-Tung Yau (Chinese: ; pinyin: ; born April 4, 1949) is a prominent mathematician working in differential geometry, and involved in the theory of Calabi-Yau manifolds. ...

The original proof is very difficult; Edward Witten soon presented a much shorter "physicist's proof", which has been justified by mathematicians--- using further very difficult arguments! Roger Penrose and others have also offered alternative arguments for variants of the original positive energy theorem. Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ... 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. ...

Examples

Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy-momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions, including: The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ...

• Kerr-Newman-NUT-de Sitter solution contains contributions from an electromagnetic field and a positive vacuum energy, as well as a kind of vacuum perturbation of the Kerr vacuum which is specified by the so-called NUT parameter,
• Gödel dust contains contributions from a pressureless perfect fluid (dust) and from a positive vacuum energy.

Some hypothetical possibilities which don't fit into our rough classification are:

• certain wormhole metrics (which can serve as a speculative toy model of a stargate held open by a hypothetical kind of exotic matter, as in 2001: A Space Odyssey; also a toy model of hypothetical time machine, see below),
• Alcubierre metric (which has been used as a speculative toy model of effectively superluminal space travel, as in the warp drive from Star Trek).
• "Time machines", i.e. initially nice spacetimes in which at some stage of evolution closed causal curves appear.

Some doubt has been cast upon whether sufficient quantity of exotic matter needed for wormholes and Alcubierre bubbles can exist.^[3] Later, however, these doubts were shown^[4] to be mostly groundless. The third of these examples, in particular, is an instructive example of the procedure mentioned above for turning any Lorentzian manifold into a "solution". It is along this way that Hawking succeeded in proving^[5] that time machines of a certain type (those with a "compactly generated Cauchy horizon") cannot appear withouth exotic matter. Such spacetimes are also a good illustration of the fact that unless a spacetime is especially nice ("globally hyperbolic") the Einstein equations do not determine its evolution uniquely. Any spacetime may evolve into a time machine, but it never has to do so.^[6] The class of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. ... In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory. ... An activated Stargate, the central object of the fictional Stargate universe, here depicted in the SG-1 television series. ... Exotic matter is a hypothetical concept of particle physics. ... Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ... This article is in need of attention from an expert on the subject. ... Faster-than-light (also superluminal or FTL) communications and travel are staples of the science fiction genre. ... This does not cite any references or sources. ... The current Star Trek franchise logo Star Trek is an American science fiction entertainment series and media franchise. ...

See also

• Electrovacuum solution
• Fluid solution
• Lambdavacuum solution
• Null dust solution
• Petrov classification, for algebraic symmetries of the Weyl tensor
• Scalar field solution
• Solutions of the Einstein field equations
• Vacuum solution

In general relativity, an electrovacuum solution is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) source-free Maxwell equations appropriate to the given geometry. ... In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. ... In general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term. ... A null dust solution is an exact solution of Einsteins field equation in which the Einstein tensor is null. ... The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ... In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. ... In general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a massless scalar field (often taken to have minimal curvature coupling). ... This article or section is in need of attention from an expert on the subject. ... In General relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. ...

References

1. ^ Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. (2003). Exact Solutions of Einstein's Field Equations (2nd edn.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7.  The definitive resource for exact solutions in general.
2. ^ Belinski, V.; & Verdaguer, E. (2001). Gravitational solitons. Cambridge: Cambridge University Press. ISBN 0-521-80586-4.  A monograph on the use of soliton methods to produce stationary axisymmetric vacuum solutions, colliding gravitational plane waves, and so forth.
3. ^ L. H. Ford and T. A. Roman (1996) "Quantum field theory constrains traversable wormhole geometries" Phys. Rev. D 53 5496, See also the eprint version. arXiv.
4. ^ S. Krasnikov (2003) "The quantum inequalities do not forbid spacetime shortcuts" Phys. Rev. D 67 104013, See also the eprint version. arXiv.
5. ^ S. W. Hawking (1992) "Chronology protection conjecture" Phys. Rev. D 46 603
6. ^ S. Krasnikov (2002) "No time machines in classical general relativity" Class. and Quantum Grav. 19 4109, See also the eprint version. arXiv.

• Krasiński, A. (1997). Inhomogeneous Cosmological Models. Cambridge: Cambridge University Press. ISBN 0-521-48180-5. 

• MacCallum, M. A. H.. Finding and using exact solutions of the Einstein Equations. arXiv eprint server. Retrieved on February 5, 2006. An up-to-date review article, but too brief, compared to the review articles by Bičák or Bonnor et al. (see below).

• Rendall, Alan M.. Local and Global Existence Theorems for the Einstein Equations. Living Reviews in Relativity. Retrieved on August 11, 2005. A thorough and up-to-date review article.

• Friedrich, Helmut. Is general relativity `essentially understood' ?. arXiv eprint server. Retrieved on August 11, 2005. An excellent and more concise review.

• Bičák, Jiří (2000). "Selected exact solutions of Einstein's field equations: their role in general relativity and astrophysics". Lect. Notes Phys. 540: 1-126.  See also the eprint version. arXiv. Retrieved on June 23, 2005. An excellent modern survey.

• Bonnor, W. B.; Griffiths, J. B.; & MacCallum, M. A. H. (1994). "Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions". Gen. Rel. Grav. 26: 637-729. 

• Bonnor, W. B. (1992). "Physical interpretation of vacuum solutions of Einstein's equations. Part I. Time-independent solutions". Gen. Rel. Grav. 24: 551-573.  A wise review, first of two parts.

• Griffiths, J. B. (1991). Colliding Plane Waves in General Relativity. Oxford: Clarendon Press. ISBN 0-19-853209-1.  The definitive resource on colliding plane waves, but also useful to anyone interested in other exact solutions.available online by the author
• Hoenselaers, C.; & Dietz, W. (1985). Solutions of Einstein's Equations: Techniques and Results. New York: Springer. ISBN 3-540-13366-6. 

• Ehlers, Jürgen; & Kundt, Wolfgang (1962). "Exact solutions of the gravitational field equations". Witten, L. Gravitation: An Introduction to Current Research: 49–101, New York: Wiley.  A classic survey, including important original work such as the symmetry classification of vacuum pp-wave spacetimes.