De Sitter space
In mathematical physics, "n"-dimensional de Sitter space (often abbreviated to dS) 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 formula_1 (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), 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 R with the standard metric: de Sitter space is the submanifold described by the hyperboloid of one sheet where formula_4 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 formula_5 with formula_6 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 "".) 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 (so that if then de Sitter space is simply connected). 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 formula_11 for de Sitter as follows: where formula_13 gives the standard embedding the -sphere in R. In these coordinates the de Sitter metric takes the form: Note that there is a cosmological horizon at formula_15. Flat slicing. Let where formula_17. Then in the formula_18 coordinates metric reads : where formula_20 is the flat metric on formula_21's. Setting formula_22, we obtain the conformally flat metric: Open slicing. Let where formula_25 forming a formula_26 with the standard metric formula_27. Then the metric of the de Sitter space reads where is the standard hyperbolic metric. Closed slicing. Let where formula_13s describe a formula_32. Then the metric reads: Changing the time variable to the conformal time via formula_34 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  Penrose diagram. dS slicing. Let where formula_13s describe a formula_38. Then the metric reads: where is the metric of an formula_41 dimensional de Sitter space with radius of curvature formula_4 in open slicing coordinates. The hyperbolic metric is given by: This is the analytic continuation of the open slicing coordinates under formula_44 and also switching formula_45 and formula_46 because they change their timelike/spacelike nature.