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In physics, the Einstein field equation or the Einstein equation is an equation in the theory of gravitation, called general relativity, that describes how matter creates gravity 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 the theory of general relativity, gravity is described by the properties of the local geometry of spacetime. In particular, the gravitational field can be built out of the metric tensor, a quantity describing geometrical properties spacetime such as distance, area, and angle. Matter is described by its stress-energy tensor, a quantity which contains the density and pressure of matter. These tensors are 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.

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 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 holeA rotating black hole Kerr black hole or Kerr-Newman black hole is a black hole that possesses angular momentum. It is one of three possible types of black holes that could exist in the theory of gravitation called General Relativity. Black holes can be c of a given mass and angular momentum , 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 universeAlternate uses: See Universe (disambiguation In the first half of the 20th century, the word universe was used to mean the whole spacetime continuum in which we exist, together with all the energy and matter within it. Attempts to understand the universe and predicts a universe which is not static, but expanding.

1 Mathematical form of the Einstein field equation

The Einstein field equation describes how space-time is curved by matterMatter is anything that has mass and occupies space. One [contemporary] view on matter takes it as all scientifically observable entities whatsoever. Matter can more accurately be defined as the energy that has a low vibratory rate, a compressed energy st, and (the other way round) how matterMatter is anything that has mass and occupies space. One [contemporary] view on matter takes it as all scientifically observable entities whatsoever. Matter can more accurately be defined as the energy that has a low vibratory rate, a compressed energy st is influenced by the curvature of space-time (i.e. how the curvature gives rise to gravity).

The field equation reads as follows

where is the Einstein curvature tensorIn differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds, which, in components, is expressed as :, where is the Ricci tensor, is the metric of the manifold, and is the scalar curvature. The Bianchi identities can be ea, a second order differential equation in terms of the metric tensor , and is the stress-energy tensor. The coupling constant is given in terms of is pi, is the speed of light and is the gravitational constant.

The Einstein curvature tensor can be written as

where in addition is the Ricci curvature tensor,

The field equation therefore also reads as follows:

The metric 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 are the core of the mathematical formulation of general relativity.