In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dSn, is the Lorentzian manifold analog of an n-sphere (with its canonical Riemannian metric); it is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. De Sitter space and anti-de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked in the 1920s in Leiden closely together on the spacetime structure of our universe.
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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
De Sitter space was also discovered, independently, and about the same time, by Tullio Levi-Civita.
More recently it has been considered as the setting for special relativity rather than using Minkowski space, since a group contraction reduces the isometry group of de Sitter space to the Poincaré group, allowing a unification of the spacetime translation subgroup and Lorentz transformation subgroup of the Poincaré group into a simple group rather than a semi-simple group. This alternate formulation of special relativity is called de Sitter relativity.
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
De Sitter space can also be defined as the quotient O(1, n) / O(1, n − 1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.
Topologically, de Sitter space is R × Sn−1 (so that if n ≥ 3 then de Sitter space is simply connected).
Properties
The isometry group of de Sitter space is the Lorentz group O(1, n). The metric therefore then has n(n + 1)/2 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 n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.
Static coordinates
We can introduce static coordinates
where
Note that there is a cosmological horizon at
Flat slicing
Let
where
where
Open slicing
Let
where
where
is the metric of a Euclidean hyperbolic space.
Closed slicing
Let
where
Changing the time variable to the conformal time via
This serves to find the Penrose diagram of de Sitter space.
dS slicing
Let
where
where
is the metric of an
This is the analytic continuation of the open slicing coordinates under