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In mathematics and physics, n-dimensional de Sitter space, denoted dSn, is the maximally symmetric, simply-connected, Lorentzian manifold with constant positive curvature. It may be regarded as the Lorentzian analog of an n-sphere (with its canonical Riemannian metric). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ...
Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...
In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ...
For other uses, see sphere (disambiguation). ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
In the language of general relativity, de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equation with a positive cosmological constant Λ. When n = 4, it is also a cosmological model for the physical universe; see de Sitter universe. Jump to: navigation, search General relativity (GR) is the geometrical theory of gravity published by Albert Einstein in 1915. ...
A vacuum solution is a solution of a field equation in which the sources of the field are taken to be identically zero. ...
In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ...
The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ...
A de Sitter universe is a solution to Einsteins field equations of General Relativity which is named after Willem de Sitter. ...
De Sitter space is named for Willem de Sitter. Willem de Sitter (May 6, 1872 – November 20, 1934) was a mathematician, physicist and astronomer. ...
Definition
De Sitter space is most easily defined as a submanifold of Minkowski space in one higher dimension. Take Minkowski space R1,n with the standard metric: This is a glossary of terms specific to differential geometry and differential topology. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
Jump to: navigation, search Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
De sitter space is the submanifold described by the hyperboloid Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...
 where α is some positive constant with dimensions of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. One can check that the induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces α2 with − α2 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.) In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...
Hyperboloid of one sheet Hyperboloid of two sheets In mathematics, a hyperboloid is a quadric, a type of surface in three dimensions, described by the equation: (hyperboloid of one sheet), or (hyperboloid of two sheets) If, and only if, , it is a hyperboloid of revolution. ...
Jump to: navigation, search In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...
Topologically, de Sitter space is R × Sn−1 (we shall assume that n ≥ 3 so that de Sitter space is simply-connected). Given the standard embedding of the (n−1)-sphere in Rn with coordinates yi one can introduce a new coordinate t so that Jump to: navigation, search Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
  The metric in these coordinates (t plus some set of coordinates on Sn−1) is given by where is the standard round metric on the (n−1)-sphere.
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 vectors and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...
The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ...
In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ...
In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...
De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric: An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor: Taking a trace shows that k is equal to s/n, where n is the dimension of M and s is the scalar curvature. ...
In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...
 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 In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ...
 For the case n = 4, we have Λ = 3/α2 and R = 4Λ = 12/α2.
Static coordinates We can introduce static coordinates for de Sitter as follows: In general relativity, a spacetime is said to be static if it admits a global, nowhere zero, timelike hypersurface orthogonal Killing vector field. ...
 where zi 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 r = α. Jump to: navigation, search In cosmology, a cosmological horizon marks a limit to observability, and marks the boundary of a region that an observer cannot see into directly due to cosmological effects. ...
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