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Encyclopedia > Einstein field equations
General relativity
G_{mu nu} = {8pi Gover c^4} T_{mu nu},
Key topics
Introduction to...
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Equivalence principle
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Frame-dragging · Geodetic effect
Event horizon · Singularity
Black hole An illustration of a rotating black hole at the center of a galaxy 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. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... 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. ... 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. ... For the concept in fluid dynamics and meteorology, see Gravity wave. ... 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. ... Simulated view of a black hole in front of the Milky Way. ...

Equations
Linearized Gravity
Post-Newtonian formalism
Einstein field equations
Advanced theories
Kaluza-Klein
Quantum gravity
Solutions
Schwarzschild

Reissner-Nordström
Kerr · Kerr-Newman
Kasner · Milne · Robertson-Walker 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. ... Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism. ... This article does not cite any references or sources. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, is a black hole that carries electric charge , no angular momentum, and mass . ... 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-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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The Einstein field equations (EFE) or Einstein's equations are a set of ten equations in Einstein's theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy.[1] They were first published in 1915.[2]. Einstein redirects here. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... “Gravity” redirects here. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contribution of various energy or force-fields, which are not usually considered to be matter per se (though they may contribute to the mass of objects). ...


The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor). 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. ... 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. ...


The EFE are used to determine the curvature of spacetime resulting from the presence of mass and energy. That is, they determine the metric tensor of spacetime for a given arrangement of stress-energy in the spacetime. Because of the relationship between the metric tensor and the Einstein tensor, the EFE become a set of coupled, non-linear differential equations when used in this way. In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... This article is in need of attention from an expert on the subject. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

Contents

Mathematical form of Einstein's field equation

The Einstein field equations (EFE) may be written in the form:[1]

R_{mu nu} - {textstyle 1 over 2}R,g_{mu nu} = kappa T_{mu nu} = -{8 pi G over c^4} T_{mu nu}.

where Rμν is the Ricci tensor, R the scalar curvature, gμν the metric tensor and Tμν the stress-energy tensor. The constant κ (kappa) is called the Einstein constant (of gravitation), where π (pi) is Archimedes' constant, G the gravitational constant and c the speed of light. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... This article is in need of attention from an expert on the subject. ... For other uses, see Kappa (disambiguation). ... For other uses, see Pi (disambiguation) Pi (upper case Π, lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry. ... 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. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ...


The above form of the EFE is for the −+++ metric sign convention, which is commonly used in general relativity, and which is used by convention here in Wikipedia. Using the +--- metric sign convention leads to an alternate form of the EFE which is In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary. ...

R_{mu nu} - {textstyle 1 over 2}R,g_{mu nu} = -kappa T_{mu nu} = {8 pi G over c^4} T_{mu nu}.

The change of sign on the right hand side occurs because the values of Tμν have signs which are determined by the sign convention. The value of the left hand side are convention independent: Rμν has values which are independent of the convention and the convention dependencies of R and gμν cancel out.


The EFE is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here using the abstract index notation. Each tensor has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number. 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 mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...


Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, the equations can be seen to hold in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations (if the dimension is clear).


Despite the simple appearance of the equation it is, in fact, quite complicated. Given a specified distribution of matter and energy in the form of a stress-energy tensor, the EFE are understood to be equations for the metric tensor gμν, as both the Ricci tensor and Ricci scalar depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic partial 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. ...


One can write the EFE in a more compact form by defining the Einstein tensor Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ...

G_{mu nu} = R_{mu nu} - {1 over 2}R g_{mu nu},

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

G_{mu nu} = {8pi Gover c^4} T_{mu nu},

Using geometrized units where G = c = 1, this can be re-written as 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. ...

G_{mu nu} = 8pi T_{mu nu}.,

The expression on the left represents the curvature of spacetime as determined by the metric and the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how the curvature of spacetime is related to the matter/energy content of the universe.


These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity. In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...


Equivalent formulations

Einstein's field equations can be rewritten in the following equivalent "trace-reversed" form

R_{mu nu} = kappa (T_{munu} - {textstyle 1 over 2}T,g_{munu})

which may be more convenient in some cases (for example, when one's interested in weak-field limit and can replace gμν in the expression on the right with the Minkowski tensor without significant loss of accuracy).


Properties of Einstein's equation

Conservation of energy and momentum

An important consequence of the EFE is the local conservation of energy and momentum; this result arises by using the differential Bianchi identity to obtain 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. ...

nabla_nu G^{mu nu}=G^{mu nu}{}_{;nu}=0

which, by using the EFE, results in

nabla_nu T^{mu nu}= T^{mu nu}{}_{;nu}=0

which expresses the local conservation of stress-energy. This conservation law is a physical requirement. In designing the field equations, Einstein aimed at finding equations which automatically satisfied this conservation condition. Einstein redirects here. ...


Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics which is linear in the wavefunction. 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. ... 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 physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... Fig. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...


The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations. Gravitation is the tendency of masses to move toward each other. ... 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. ...


The cosmological constant

One can modify the EFE by introducing a term proportional to the metric: In mathematics a metric or distance function is a function which defines a distance between elements of a set. ...

R_{mu nu} - {1 over 2}R g_{mu nu} = -{8 pi} T_{mu nu}+ Lambda g_{mu nu}.

The constant Λ is called the cosmological constant. Since Λ is constant, the energy conservation law is unaffected. In physical cosmology, 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. ...


The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our universe is, in fact, not static but expanding. So Λ was abandoned, with Einstein calling it the "biggest blunder [he] ever made".[3] For many years the cosmological constant was almost universally considered to be 0. Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an American astronomer. ... Accelerating universe is a term for the idea that our universe is undergoing divergent rapid expansion. ...


Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations.[4] [5] Einstein redirects here. ... A giant Hubble mosaic of the Crab Nebula, a supernova remnant Astronomy (also frequently referred to as astrophysics) is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earths atmosphere (such as the cosmic background radiation). ...


Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor:

T_{mu nu}^{mathrm{(vac)}} = frac{Lambda}{8pi}g_{mu nu}.

The constant

rho_{mathrm{vac}} = frac{Lambda}{8pi}

is called the vacuum energy. The existence of a cosmological constant is equivalent to the existence of a non-zero vacuum energy. The terms are now used interchangeably in general relativity. Vacuum energy is an underlying background energy that exists in space even when devoid of matter (known as free space). ...


Solutions of the field equations

The solutions of the Einstein field equations are metrics of spacetime. The solutions are hence often called 'metrics'. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions.[6] This article or section is in need of attention from an expert on the subject. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... 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. ... 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). ...


The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe. Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ... Simulated view of a black hole in front of the Milky Way. ... The Universe is defined as the summation of all particles and energy that exist and the space-time in which all events occur. ...


Vacuum field equations

If the energy-momentum tensor Tμν is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting Tμν = 0 in the full field equations, the vacuum equations can be written as A field equation is an equation in a physical theory that describes how a fundamental force (or a combination of such forces) interacts with matter. ...

R_{mu nu} = {1 over 2}R g_{mu nu}.

By reversing the trace of this equation, we get the precisely equivalent form

R_{mu nu} = 0

In the case of nonzero cosmological constant, the equations are

R_{mu nu} = {1 over 2}R g_{mu nu} - Lambda g_{mu nu},

for which the trace-reversed form is

R_{mu nu} = Lambda g_{mu nu}.

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution. In General relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... 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. ...


Manifolds with a vanishing Ricci tensor, Rμν = 0, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci tensor vanishes. ... An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor: Taking a trace shows that k is equal to s/n, where n is the dimension of M and s is the scalar curvature. ...


The linearised EFE

Main articles: Linearised Einstein field equations, Linearized gravity This article is in need of attention from an expert on the subject. ... It has been suggested that Weak-field approximation be merged into this article or section. ...


The nonlinearity of the EFE makes finding exact solutions quite difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric. This linearisation procedure can be used to discuss the phenomena of gravitational radiation. A gravitational field is a model used within physics to explain how gravity exists in the universe. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... 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 physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


See also

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... 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). ... // Geroch, Robert (1981). ... // 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. ... For a less technical introduction to this topic, please see Introduction to mathematics of general relativity. ... This article or section is in need of attention from an expert on the subject. ...

External links

Mapúa Institute of Technology (MIT, MapúaTech or simply Mapúa) is a private, non-sectarian, Filipino tertiary institute located in Intramuros, Manila. ...

References

See General relativity resources. // Geroch, Robert (1981). ...

  1. ^ a b Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. 
  2. ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungun der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844-847. Retrieved on 2006-09-12. 
  3. ^ Gamow, George. My World Line : An Informal Autobiography. Viking Adult date = April 28, 1970. ISBN-10: 0670503762. Retrieved on 3-14-2007. 
  4. ^ Wahl, Nicolle. "Was Einstein's 'biggest blunder' a stellar success?", November 22, 2005. Retrieved on 2007-03-14. 
  5. ^ Turner, Michael S. (May, 2001). "A Spacetime Odyssey". Int.J.Mod.Phys. A17S1: 180-196. Retrieved on 2007-03-14. 
  6. ^ Stephani, Hans; D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt (2003). Exact Solutions of Einstein's Field Equations. Cambridge University Press. ISBN 0-521-46136-7. 

  Results from FactBites:
 
Einstein field equations - Wikipedia, the free encyclopedia (1319 words)
The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor).
Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations.
The study of exact solutions of Einstein's field equations is one of the activities of cosmology.
General relativity - Wikipedia, the free encyclopedia (5428 words)
The solutions of the Einstein field equations that call for this behavior for the current universe, which may require the reintroduction of the cosmological constant, are for a stress-energy which is at least 70% dark energy.
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.
Einstein did not believe in an expanding universe, and so he added a cosmological constant to the field equations to permit the creation of static universe solutions.
  More results at FactBites »

 
 

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