De Sitter space


In mathematics and physics, n-dimensional de Sitter space is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere.
In the language of general relativity, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant . There is cosmological evidence that the universe itself is asymptotically de Sitter - see de Sitter universe.
De Sitter space and anti-de Sitter space are named after Willem de Sitter, professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked closely together in Leiden in the 1920s on the spacetime structure of our universe. De Sitter space was also discovered, independently, and about the same time, by Tullio Levi-Civita.

Definition

De Sitter space can be defined as a submanifold of a generalized Minkowski space of one higher dimension. Take Minkowski space R1,n with the standard metric:
De Sitter space is the submanifold described by the hyperboloid of one sheet
where is some nonzero constant with dimensions of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. The induced metric is nondegenerate and has Lorentzian signature.
De Sitter space can also be defined as the quotient of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.
Topologically, de Sitter space is .

Properties

The isometry group of de Sitter space is the Lorentz group. The metric therefore then has independent Killing vector fields and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by
De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:
This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by
The scalar curvature of de Sitter space is given by
For the case, we have and.

Static coordinates

We can introduce static coordinates for de Sitter as follows:
where gives the standard embedding the -sphere in Rn−1. In these coordinates the de Sitter metric takes the form:
Note that there is a cosmological horizon at.

Flat slicing

Let
where. Then in the coordinates metric reads:
where is the flat metric on 's.
Setting, we obtain the conformally flat metric:

Open slicing

Let
where forming a with the standard metric. Then the metric of the de Sitter space reads
where
is the standard hyperbolic metric.

Closed slicing

Let
where s describe a. Then the metric reads:
Changing the time variable to the conformal time via we obtain a metric conformally equivalent to Einstein static universe:
This serves to find the Penrose diagram of de Sitter space.

dS slicing

Let
where s describe a. Then the metric reads:
where
is the metric of an dimensional de Sitter space with radius of curvature in open slicing coordinates. The hyperbolic metric is given by:
This is the analytic continuation of the open slicing coordinates under and also switching and because they change their timelike/spacelike nature.