Einstein field equations

In the context of general relativity is the Einstein's field equations ( Albert Einstein ) Einstein equations, Einstein - Hilbert equations ( according to Albert Einstein and David Hilbert ) or gravitational equations, described the physical phenomenon of gravitation.

The development of the equations based on the basic idea of ​​geometricize gravity, ie to map all properties of gravitation and its effect on physical processes using the properties of a semi- Riemannian manifold.

Basic assumptions and requirements

For the preparation of the field equations physical considerations are initially necessary because the form of the equations must be postulated.

It makes the approach that include the shape of the field equations which generalize the gravity on the right side of the energy -momentum tensor as a source of the field. Therefore, since the tensor is on the right side, this must also apply to the left side. This tensor should the geometrical properties of space-time and represent a combination of the metric tensor and a tensor, which describes the curvature represent. Thus, the field equations take the form

To, the geometric tensor is called Einstein tensor. The constant = 8πG / c4 is called Einstein's gravitational constant or simply Einstein constant and is assumed to be proportional constant. These two quantities are to be determined.

From the previous considerations arise summarized these requirements:

The field equations

For these claims, the field equations follow:

Here, the gravitational constant, the speed of light, the Ricci tensor Krümmungsskalar and the metric tensor.

The field equations can be defined with the opposite sign of the Einstein constant

This sign is purely from the convention used dependent and not physically significant; both conventions are widely used.

In the energy -momentum tensor is taken into account that mass and energy are equivalent; that is, any form of energy induced gravitational mass. The energy -momentum tensor contains in addition to the mass - energy density ( energy per unit mass or volume ) other forms of energy ( eg, the pressure that can exert a radiation field ). A change in the energy-momentum tensor, ie a change in the energy distributions described by him has, thus a change in the structure of spacetime in the vicinity of this energy distribution result. The structure of the curvature of space- time (that is, the space and time) in turn influences the matter located there, namely, energy, space and time are in direct interaction. This influence of matter emanating from the curvature of space and time, is in our experience the world nothing but gravity.

The vacuum field equations

For example, considering the exterior of stars, where resides as an approximation no matter, so is set. They call then

Or equivalent

The vacuum field equations and their solutions, vacuum solutions. For the environment of a non-rotating and electrically neutral sphere of mass M is obtained in spherical coordinates from this example, the outer Schwarzschild solution, the line element has the form


Einstein -Maxwell equations

Used for the electromagnetic energy -momentum tensor

Used in the field equations

This is called the Einstein -Maxwell equations.

The cosmological constant

It turned out that the Einstein's field equations can be further generalized. Thus it is possible to add another additive term in the Einstein tensor, which consists of a constant and the metric tensor. Thus, the requirement of the vanishing divergence is still met, and so the field equations take the form

Of. Here is the cosmological constant, which was incorporated by Einstein in the field equations and chosen so that the universe is static; this was the time most sensible intuition. However, it turned out that the universe as described by the theory is unstable. When Edwin Hubble finally proved that the universe was expanding, Einstein repudiated his constant. He is said to have referred to as the "biggest blunder " of his life; this it was rumored, however, only by George Gamow.

Despite Einstein's error, the cosmological constant today an important and enigmatic size in the range of modern cosmology dar. was long kept its value for 0, but the modern methods of astronomy have shown that the constant must have a positive value to certain things to explain. There have been several models for the development of a universe created, in which the constant plays a major role. The best known and simplest of these models is the de Sitter spacetime.

Originally, the constant was indeed intended as an independent parameter, but it can be identified in the vacuum case with the energy-momentum tensor. It is

Where the constant

Is called the vacuum energy density - it is an energy of empty space underlying, with the empty space, as to be expected, as a "flat" proves (). If a vacuum energy exists, then there exists a non-vanishing cosmological constant, and vice versa.