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For other topics related to Einstein see Einstein (disambig)

Introduction

In physics, the Einstein field equation or Einstein equation is a tensor equation in the theory of gravitation. It is the dynamical equation of the physical theory called general relativity, describing how matter creates gravity (the curvature of spacetime) and, conversely, how gravity affects matter.

Mathematical form of the Einstein field equation

The Einstein field equation describes how space-time is curved by matter, and (the other way round) how matter is influenced by the curvature of space-time (i.e. how the curvature influences masses).

As it is a tensor equation, the Einstein field equation is usually written out in terms of its components. The resulting set of equations are then called the Einstein field equations (EFE's):

G_{ab}={8\pi G \over c^{4}}T_{ab}

where $G_{ab}$ are the components of the Einstein tensor, which is composed of derivatives of the metric tensor with components $g_{ab}$ , and $T_{ab}$ are the components of the stress-energy tensor and the constant is given in terms of $\pi$ (pi), $c$ (the speed of light) and $G$ (the gravitational constant).

One of the solutions of the EFE's represents an expanding universe. In Einstein's time, nobody actually believed that the universe was expanding (even Einstein). To eliminate such a solution from arising, Einstein changed the equation to:

G_{ab}=R_{ab}-{R \over 2}g_{ab}+\Lambda g_{ab}

where $R_{ab}$ are the components of the Ricci tensor, $R$ is the Ricci scalar and $\Lambda$ is the cosmological constant.

Using the definition of the Einstein tensor, the previous equation now reads:

R_{ab}-{R \over 2}g_{ab}+\Lambda g_{ab}={8\pi G \over c^{4}}T_{ab}

The metric, with components $g_{ab}$ , is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number.

These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity.

(Exact) solutions of the field equation

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). An exact solution is a metric which corresponds to a physically realizable energy-momentum tensor.

Some exact solutions are:

The solution for empty space (vacuum) around a spherically symmetric, static mass distribution, is the Schwarzschild metric and the Kruskal-Szekeres metric. It applies to a star and leads to the prediction of an event horizon beyond which we cannot observe. It predicts the possible existence of a black hole of a given mass $M$ from which no energy can be extracted (in the classical or non-quantummechanical sense).
The solution for empty space (vacuum) around an axial symmetric, rotating mass distribution, is the Kerr metric. It applies to a rotating star and leads to the prediction of the possible existence of a rotating black hole of a given mass $M$ and angular momentum $J$ , from which the rotational energy can be extracted.
The solution for an isotropic and homogeneous universe filled with a constant density and negligible pressure, is the Friedmann-Lemaître-Robertson-Walker. It applies to the universe as a whole and leads to different models of evolution of the universe and predicts a universe which is not static, but expanding.

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.

References

Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972) [ISBN 0471925675]

Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers and Eduard Herlt, "Exact Solutions of Einstein's Field Equations - 2nd Edition (2003) ISBN 0521461367

@@ Line 1: / Line 1: @@
 ''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]]. It is the central statement of the physical theory called [[general relativity]], describing how matter creates [[gravity]] (the heaviness of massive bodies) and, conversely, how gravity affects matter.
-The Einstein field equation reduces to [[gravity|Newton's law of gravity]] in the non-relativistic limit (that is, at low velocities and weak gravitational fields). In other words, [[Albert Einstein]]'s work on gravity builds on [[Isaac Newton]]'s, and corrects it, while not essentially contradicting it for everyday physics.
-==Tensor geometry of gravity==
+== Introduction ==
-In the theory of general relativity, gravity is described by the properties of the local geometry of [[spacetime]].  In particular, the gravitational field can be thought of as arising from the [[metric tensor]], a quantity describing the geometrical properties of spacetime such as distance, area, and angle.
+In [[physics]], the '''Einstein field equation''' or '''Einstein equation''' is a [[tensor]] equation in the [[theory of gravitation]]. It is the   of the physical theory called [[general relativity]], describing how matter creates [[gravity]] (the  of ) and, conversely, how gravity affects matter.
-Matter is described by its [[stress-energy tensor]], a quantity which describes the density and pressure of matter. These tensors are both [[symmetric tensor|symmetric]] [[rank of a tensor|second rank]] [[tensor]]s, so they have
-:''D''(''D'' + 1)/2
-independent components in [[dimension|D-dimensional]] spacetime.
-In 4-dimensional spacetime, then, these tensors have 10 independent components.  Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6.
-The strength of coupling between matter and gravity is determined by the [[Physical constants|gravitational constant]].
-==Solutions of the field equation==
-A solution of the  '''Einstein field equation''' is a certain metric appropriate for the given mass and pressure distribution of the matter.
-Some solutions for a given physical situation are as follows.
-# The solution for empty space (vacuum) around a spherically symmetric, static mass distribution, is the [[Schwarzschild metric]] and the [[Kruskal-Szekeres  metric]].  It applies to a star and leads to the prediction of an [[event horizon]] beyond which we cannot observe. It predicts the possible existence of a [[black hole]] of a given mass <math>M</math> from which no energy can be extracted (in the classical or non-quantummechanical sense).
-# The solution for empty space (vacuum) around an axial symmetric, rotating mass distribution, is the [[Kerr metric]].  It applies to a rotating star and leads to the prediction of the possible existence of a [[rotating black hole]] of a given mass <math>M</math> and angular momentum <math>J</math>, from which the rotational energy can be extracted.
-# The solution for an isotropic and homogeneous universe filled with a constant density and negligible pressure, is the [[Robertson-Walker metric]]. It applies to the universe as a whole and leads to different models of evolution of the [[universe]] and predicts a universe which is not static, but expanding.
 ==Mathematical form of the Einstein field equation==
-The '''Einstein field equation''' describes how [[space-time]] is curved by [[matter]], and (the other way round) how [[matter]] is influenced by the curvature of space-time (i.e. how the curvature influences masses). In tensor notation, it is expressed simply as:
+The Einstein field equation describes how [[space-time]] is curved by [[matter]], and (the other way round) how [[matter]] is influenced by the curvature of space-time (i.e. how the curvature influences masses).
-:<math> \mathbf{G} = \kappa \mathbf{T} </math>
-where <math>\mathbf{G}</math> is the Einstein tensor, <math>\mathbf{T}</math> is the stress-energy tensor and
-:<math>\kappa = {8 \pi G \over c^4} </math>
-is the coupling constant.
 As it is a tensor equation, the Einstein field equation is usually written out in terms of its components. The resulting set of equations are then called the '''Einstein field equations''' ('''EFE's'''):
@@ Line 45: / Line 16: @@
 where <math>G_{ab}</math> are the components of the [[Einstein tensor|Einstein tensor]], which is composed of derivatives of the [[metric tensor]] with components
-<math>g_{ab}</math>, and <math>T_{ab}</math> are the components of the [[stress-energy tensor]]. The coupling constant is given in terms of <math>\pi</math> ([[pi]]), <math>c</math> (the [[speed of light]]) and <math>G</math> (the [[gravitational constant]]).
+<math>g_{ab}</math>, and <math>T_{ab}</math> are the components of the [[stress-energy tensor]]   constant is given in terms of <math>\pi</math> ([[pi]]), <math>c</math> (the [[speed of light]]) and <math>G</math> (the [[gravitational constant]]).
 One of the solutions of the EFE's represents an [[expanding universe]]. In Einstein's time, nobody actually believed that the universe was expanding (even Einstein). To eliminate such a solution from arising, Einstein changed the equation to:
@@ Line 66: / Line 37: @@
 [[zh-cn:&#29233;&#22240;&#26031;&#22374;&#22330;&#26041;&#31243;]]
-==Exact solutions of the Einstein field equations==
+== of the field equation==
-One of the earliest solutions was found by [[Karl Schwarzschild]], and the metric found by him which solves the Einstein equations is called the [[Schwarzschild metric]].
+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). An [[exact solution]] is a metric which corresponds to a physically realizable energy-momentum tensor.
-Another solution, which corresponds to an expanding universe, is known as the [[Friedmann-Lemaître-Robertson-Walker]] metric.
+Some exact solutions are:
-In the study of [[exact solutions]] of the field equations, it is sometimes convenient to decompose the Riemann tensor into its trace and trace-free parts. In four dimensions, this becomes,
+# The solution for empty space (vacuum) around a spherically symmetric, static mass distribution, is the [[Schwarzschild metric]] and the [[Kruskal-Szekeres  metric]].  It applies to a star and leads to the prediction of an [[event horizon]] beyond which we cannot observe. It predicts the possible existence of a [[black hole]] of a given mass <math>M</math> from which no energy can be extracted (in the classical or non-quantummechanical sense).
-<math>R_{abcd}=\frac{R}{6}G_{abcd}+E_{abcd}+C_{abcd}</math>
+# The solution for empty space (vacuum) around an axial symmetric, rotating mass distribution, is the [[Kerr metric]].  It applies to a rotating star and leads to the prediction of the possible existence of a [[rotating black hole]] of a given mass <math>M</math> and angular momentum <math>J</math>, from which the rotational energy can be extracted.
+# The solution for an isotropic and homogeneous universe filled with a constant density and negligible pressure, is the [[Robertson-Walker]]. It applies to the universe as a whole and leads to different models of evolution of the [[universe]] and predicts a universe which is not static, but expanding.
-where the [[Weyl tensor]] is the trace-free part (as it satisfies <math>C^a{}_{bad}=0)</math> and the tensors <math>G</math> and <math>E</math> have the following components:
-:<math>G_{abcd}=g_{a[c}g_{d]b}</math>
-:<math>E_{abcd}=\tilde{R}_{a[c}g_{d]b}+\tilde{R}_{b[d}g_{c]a}</math>
-where the [[trace-free Ricci tensor]] components are given by:
-<math>\tilde{R}_{ab}=R_{ab}-\frac {R}{4}g_{ab}</math>.
-Solutions for which the energy-momentum tensor is identically zero in the region under consideration are termed [[vacuum solutions]].
+== 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.