General relativity

Overview of GR History Mathematics Resources Tests Black hole Einstein equations Equivalence principle Event horizon Exact solutions FLRW metric Gravitational lens Gravitational radiation Kerr metric Quantum gravity Schwarzschild metric Singularity

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Albert Einstein Astrophysics Cosmology Special relativity Riemannian geometry

edit General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Image File history File links Download high resolution version (1024x768, 7 KB) Description: Gravitational light deflection at a neutron star Source: Gallery of Tempolimit Lichtgeschwindigkeit Date: 09. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... // Development Early investigations The development of general relativity began in 1907 with the publication of an article by Einstein on acceleration under special relativity. ... Notational point: General relativity articles using tensors will use the abstract index notation . ... // Books Popular Geroch, Robert (1981). ... Einsteins general theory of relativity was introduced in 1915. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything from escaping it except through quantum tunneling behavior (known as Hawking Radiation). ... 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. ... In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Event Horizon is a 1997 science fiction and horror film. ... // Introduction 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. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... This article is in need of attention from an expert on the subject. ... This article is in need of attention from an expert on the subject. ... In general relativity, the Kerr metric describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... In Einsteins theory of general relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the gravitational field outside a spherical, non-rotating mass such as a (non-rotating) star, planet, or black hole. ... This article is in need of attention from an expert on the subject. ... Albert Einstein photographed by Oren J. Turner in 1947. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... 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. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...

In physics, the Einstein field equation or Einstein equation is a differential equation in Einstein's theory of general relativity. It is a dynamical equation which describes how matter and energy change the geometry of spacetime, this curved geometry being interpreted as the gravitational field of the matter source. The motion of objects (with a mass much smaller than the matter source) in this gravitational field is described very accurately by the geodesic equation. Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Albert Einstein photographed by Oren J. Turner in 1947. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Matter is commonly referred to as the substance of which physical objects are composed. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... The gravitational field is a field that causes bodies with mass to attract each other. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ...

Mathematical form of Einstein's field equation

The Einstein field equation (EFE) is usually written in the form

Here R_ab is the Ricci tensor, R is the Ricci scalar, g_ab is the metric tensor, T_ab is the stress-energy tensor, and the constant is given in terms of π (pi), c (the speed of light) and G (the gravitational constant). The EFE equation is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here in terms of components. 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 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. ... The stress-energy tensor is a tensor quantity in relativity. ... Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... Cherenkov effect in a swimming pool nuclear reactor. ... 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. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ...

The EFE is understood to be an equation for the metric tensor g_ab (given a specified distribution of matter and energy in the form of a stress-energy tensor). Despite the simple appearance of the equation it is, in fact, quite complicated. This is because both the Ricci tensor and Ricci scalar depend on the metric in a complicated nonlinear manner.

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. ...

which is a symmetric second-rank tensor that is a function of the metric. Working in geometrized units where G = c = 1, the EFE can then be 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. ...

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 mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ...

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. ...

which, by using the EFE, results in

which expresses the local conservation law referred to above.

Nonlinearity of the field equations

The EFE are a set of 10 coupled elliptic-hyperbolic nonlinear partial differential equations for the metric components. This nonlinear feature of the dynamical equations distinguishes general relativity from other physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields (i.e. the sum of two solutions is also a solution); another example is Schrodinger's equation of quantum mechanics where the equation is linear in the wavefunction. Maxwells equations (sometimes called the Maxwell 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. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess a property known as electric charge, and is in turn affected by the presence and motion of such particles. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... Fig. ...

The correspondence principle

Einstein's equation reduces 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's 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. ...

The cosmological constant

One can modify Einstein's field equations by introducing a term proportional to the metric:

The constant Λ is called the cosmological constant. The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ...

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". Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an American astronomer, noted for his discovery of galaxies beyond the Milky Way and the cosmic red shift. ...

Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing wrong (i.e. inconsistent) with the presence of such a term in the equations. Indeed, quite recently, improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations.

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:

The constant

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. ...

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. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... 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. Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) world + λογια (logia) discourse) is the study of the universe in its totality and by extension mans place in it. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything from escaping it except through quantum tunneling behavior (known as Hawking Radiation). ... The deepest visible-light image of the cosmos. ...

Vacuum field equations

If the energy-momentum tensor T_ab is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations, which can be written as:

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 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. ...

The above vacuum equation assumes that the cosmological constant is zero. If it is taken to be nonzero then the vacuum equation becomes:

Mathematicians usually refer to manifolds with a vanishing Ricci tensor as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds. 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. The gravitational field is a field that causes bodies with mass to attract each other. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... 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. ... This article is in need of attention from an expert on the subject. ...

See also

• Einstein-Hilbert action
• Exact solutions of Einstein's field equations
• History of general relativity
• Mathematics of general relativity

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). ... // Development Early investigations The development of general relativity began in 1907 with the publication of an article by Einstein on acceleration under special relativity. ... Notational point: General relativity articles using tensors will use the abstract index notation . ...

References

• Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972) ISBN 0471925675
• Stephani, H., Kramer, D., MacCallum, M., Hoenselaers C. and Herlt, E. Exact Solutions of Einstein's Field Equations (2nd edn.) (2003) CUP ISBN 0521461367