Einstein's field equation

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 equation, gravity is given in terms of a metric tensor, a quantity describing geometrical properties of four-dimensional spacetime. Matter is decribed by its stress-energy tensor, a quantity which contains the density and pressure of matter. These tensors are symmetric 4 x 4 tensorss, so they 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 predics 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 hole 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 universe and predicts a universe which is not static, but expanding.

Mathematical form of the Einstein field equation

The field equation reads as follows

where is the Einstein curvature tensor, 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,

is the Ricci curvature scalar and  is the cosmological constant.

For further explanation, see general relativity