nLab
anti de Sitter spacetime

Context

Riemannian geometry

Gravity

Contents

Definition

Up to isometry, the anti de Sitter spacetime of dimension dd, AdS dAdS_d, is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Minkowski spacetime d,1\mathbb{R}^{d,1} that solves the equation

i=1 d1(x i) 2(x d) 2(x 0) 2=R 2 \sum_{i = 1}^{d-1} (x_i)^2 - (x_d)^ 2 - (x_0)^2 = -R^2

for some R0R \neq 0 (the “radius” of the spacetime) and equipped with the metric induced from the ambient metric, where {x 0,x 1,x 2,,x d}\{x^0, x^1, x^2, \cdots, x^d\} denote the canonical coordinates. AdS dAdS_d is homeomorphic to d1×S 1\mathbb{R}^{d-1} \times S^1, and its isometry group is O(d1,2)O(d-1, 2).

More generally, one may define the anti de Sitter space of signature (p,q)(p,q) as isometrically embedded in the space p,q+1\mathbb{R}^{p,q+1} with coordinates (x 1,...,x p,t 1,,t q+1)(x_1, ..., x_p, t_1, \ldots, t_{q+1}) as the sphere i=1 px i 2 j=1 q+1t j 2=R 2\sum_{i=1}^p x_i^2 - \sum_{j=1}^{q+1} t_j^2 = -R^2.

Properties

Conformal boundary

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Holography

Asymptotically anti-de Sitter spaces play a central role in the realization of the holographic principle by AdS/CFT correspondence.

References

Revised on December 16, 2016 07:10:11 by David Corfield (129.12.18.176)