In mathematics and physics, n-dimensional anti de Sitter space, denoted AdS_n, is the maximally symmetric, simply-connected, Lorentzian manifold with constant negative curvature. It may be regarded as the Lorentzian analog of n-dimensional hyperbolic space. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... 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. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...

In the language of general relativity, anti de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equation with a negative cosmological constant Λ. General relativity (GR) is the geometrical theory of gravitation 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. ...

n-Dimensional anti de Sitter space has SO(n-1,2) (possibly with reflections) as automorphism group, according to the Erlangen program. In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). ... Look up reflection in Wiktionary, the free dictionary. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...

A coordinate patch covering part of the space gives the half-space coordinatization of anti de Sitter space. The metric for this patch is In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...

In the limit as y = 0, this reduces to a Minkowksi metric ; thus, the anti-de Sitter space contains a conformal Minkowski space at infinity ("infinity" having y-coordinate zero in this patch).

There are two types of AdS space: one where time is periodic, and the universal cover with non-periodic time. The coordinate patch above covers half of a single period of the spacetime. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

There is another coordinate system in which the constant time slices are hyperbolic geometries. A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ...

Its conformal boundary at infinity contains conformal Minkowski space.

Because the conformal infinity of AdS is timelike, specifying the initial data on a spacelike hypersurface would not determine the future evolution uniquely (i.e. deterministically) unless there are boundary conditions associated with the conformal infinity. To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

Image File history File links the half-space region of anti deSitter space and its boundary. ...

The image above represents the "half-space" region of anti deSitter space and its boundary. The interior of the cylinder corresponds to anti deSitter spacetime, while its cylindrical boundary corresponds to its conformal boundary. The green shaded region in the interior corresponds to the region of AdS covered by the half-space coordinates and it is bounded by two null aka lightlike geodesic hyperplanes; the green shaded area on the surface corresponds to the region of conformal space covered by Minkowski space.

If AdS is periodic in time, the green shaded regions covers half of the AdS space and half of the conformal spacetime; the left ends of the green discs will touch in the same fashion as the right ends.

See also

• De Sitter space
• AdS/CFT