For other topics related to Einstein see Einstein (disambig)

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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 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.

Contents

1 Tensor geometry of gravity 2 Solutions of the field equation 3 Mathematical form of the Einstein field equation 4 Exact solutions of the Einstein field equations 5 References

Tensor geometry of gravity

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.

Matter is described by its stress-energy tensor, a quantity which describes the density and pressure of matter. These tensors are both symmetric second rank tensors, so they have

D(D + 1)/2

independent components in 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 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.

1. 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).
2. 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.
3. 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:

<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):

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

where <math>G_{ab}<math> are the components of the 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).

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:

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

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

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

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

The metric, with components <math>g_{ab}<math>, 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.

es:Ecuaci�n del campo de Einstein ja:アインシュタイン方程式 zh-cn:爱因斯坦场方程

Exact solutions of the Einstein field equations

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.

Another solution, which corresponds to an expanding universe, is known as the Friedmann-Lema�tre-Robertson-Walker metric.

In the study of exact solutions of the field equations, it is sometimes convenient to decompose the Riemann tensor into it's trace and trace-free parts. In four dimensions, this becomes,

<math>R_{abcd}=\frac{R}{6}G_{abcd}+E_{abcd}+C_{abcd}<math>

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 <math>\tilde{R}<math> components are given by:

<math>\tilde{R}_{ab}=R_{ab}-\frac {R}{4}g_{ab}<math>.

Solutions for which the Ricci tensor is identically zero in the region under consideration are termed vacuum solutions. Then the Riemann and Weyl tensors are equal in that region.

See also Einstein-Hilbert action

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-10: 0521461367|ISBN-13: 9780521461368]