de Sitter space

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 - see de Sitter universe.

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 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.[3]


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. (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 geometry of Minkowski space.)

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).


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 coordinatesEdit

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 slicingEdit



where  . Then in the   coordinates metric reads:


where   is the flat metric on  's.

Setting  , we obtain the conformally flat metric:


Open slicingEdit



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




is the standard hyperbolic metric.

Closed slicingEdit



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.[clarification needed]

dS slicingEdit



where  s describe a  . Then the metric reads:




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 alsoEdit


  1. ^ de Sitter, W. (1917), "On the relativity of inertia: Remarks concerning Einstein's latest hypothesis", Proc. Kon. Ned. Acad. Wet., 19: 1217–1225
  2. ^ de Sitter, W. (1917), "On the curvature of space", Proc. Kon. Ned. Acad. Wet., 20: 229–243
  3. ^ Levi-Civita, Tullio (1917), "Realtà fisica di alcuni spazî normali del Bianchi", Rendiconti, Reale Accademia Dei Lincei, 26: 519–31

Further readingEdit

External linksEdit