In mathematical physics, n-dimensional de Sitter space (often abbreviated to dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere (with its canonical Riemannian metric).
The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations with a positive cosmological constant (corresponding to a positive vacuum energy density and negative pressure). There is cosmological evidence that the universe itself is asymptotically de Sitter, i.e. it will evolve like the de Sitter universe in the far future when dark energy dominates.
de Sitter space and anti-de Sitter space are named after Willem de Sitter (1872–1934),[1][2] professor of astronomy at Leiden University and director of the 萊頓天文台. 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 圖利奧·列維-齊維塔.[3]
定義
de Sitter space can be defined as a submanifold of a generalized 閔考斯基時空 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 its dimension being that 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. (Note that if one replaces with in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space. For a detailed proof, see Minkowski space § Geometry.)
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 勞侖茲群 O(1, n). The metric therefore then has n(n + 1)/2 independent 基靈向量場s and is maximally symmetric. Every maximally symmetric space has constant curvature. The 黎曼曲率張量 of de Sitter is given by[4]
(using the sign convention for the Riemann curvature tensor). 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[4]
For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.
Coordinates
Static coordinates
We can introduce static coordinates for de Sitter as follows:
where gives the standard embedding the (n − 2)-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:
These coordinates, also known as "global coordinates" cover the maximal extension of de Sitter space, and can therefore be used to find its 潘洛斯圖.[5]
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.
See also