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 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 <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 gives rise to gravity).

The field equation reads as follows

<math>E_{ik} = 8 \pi {G \over c^4} T_{ik}<math>

where <math>E_{ik}<math> is the Einstein curvature tensor, a second order differential equation in terms of the metric tensor <math>g_{ik}<math>, and <math>T_{ik}<math> is the stress-energy tensor. The coupling constant is given in terms of <math>\pi<math> is pi, <math>c<math> is the speed of light and <math>G<math> is the gravitational constant.

The Einstein curvature tensor can be written as

<math>E_{ik} = R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} <math>

where in addition <math>R_{ik}<math> is the Ricci curvature tensor, <math>R<math> is the Ricci curvature scalar and <math>\Lambda<math> is the cosmological constant.

The field equation therefore also reads as follows:

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

The metric <math>g_{ik}<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 are the core of the mathematical formulation of general relativity.

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 Friedman-Lemaître-Robertson-Walker metric.

See also Einstein-Hilbert action

References

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