de Sitter space

In mathematics and physics, an n-dimensional de Sitter space (after Willem de Sitter ), noted that Lorentz manifold analogous to an n- sphere (with its canonical Riemannian manifold); he is maximal symmetric, has a constant positive curvature and is simply connected for.

In the four-dimensional Minkowski space (3 spatial dimensions plus time) or in space-time is the de Sitter space the analog of a sphere in ordinary Euclidean space.

In the language of general relativity theory of de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive ( repulsive ) cosmological constant (corresponding to a positive vacuum energy density and negative pressure ) and thus a cosmological model of the physical universe; see de Sitter model.

The de Sitter space was discovered by Willem de Sitter and at the same time - regardless of de Sitter - by Tullio Levi -Civita.

Lately, the de Sitter space is considered more as a setting for the special theory of relativity rather than Minkowski space. Such a formulation is called de Sitter relativity.

Definition

The de Sitter space can be defined as a submanifold of Minkowski space of a higher dimension.

So we consider the Minkowski space with the usual metric tensor

Then the de Sitter space is the submanifold defined by the hyperboloid

Will be described, wherein a positive constant with the dimension of a length. The metric tensor of the de Sitter space is the one that is generated by the metric tensor of the Minkowski space. One can verify that the generated metric is non- degenerate, and a signature of the mold (1, k, 0). ( When in the above definition is replaced by, one obtains a two -shelled hyperboloid. In this case, the metric generated is positive definite, and each of the two shells is a copy of a hyperbolic geometry - n. )

The de Sitter space can also be defined as the quotient of two Lorentz groups, which shows that he is a non- Riemannian symmetric space.

Topology, is the de Sitter space on the form.

Properties

The isometry group of de Sitter space is the Lorentz group. Therefore, the metric has independent Killing vectors and is maximal symmetric. Every maximal symmetric space has constant curvature. The Riemann curvature tensor of de Sitter space

The de Sitter space is an Einstein manifold, since the Ricci tensor is proportional to the metric:

That is, the de Sitter space is a vacuum solution of Einstein's field equations with cosmological constant

This is Krümmungsskalar this space

For n = 4, Λ = 3/α2 results and R = 4Λ = 12/α2.

Static coordinates

For the de Sitter space is static coordinates (time, radius, ...) as follows can be introduced:

The standard embedding of the sphere in Rn- 1 is.

In these coordinates the de Sitter metric takes the following form:

There is at a cosmological horizon: Please note.

Slicing coordinates

Flat

Approach:

In which

Then is the metric of the de Sitter space in coordinates:

Closed

Approach:

The one - sphere describe.

Then is the metric of the de Sitter space:

If the time - variable change in the conformal time:

One obtains a metric that is conformally equivalent to the static Einstein universe:

Thus one finds the Penrose diagram of de Sitter space.

Open

Approach:

Wherein a sphere formed with the standard metric

Then is the metric of the de Sitter space

With the metric of a hyperbolic Euclidean space.

DS

Approach:

The one - sphere describe.

Then is the metric of the de Sitter space:

In which

The metric one -dimensional de Sitter space in the open slicing coordinates, with radius of curvature.

The hyperbolic metric is:

This is the analytic continuation of the open slicing coordinates

And also the exchange of and because they change their time-or space-like properties.