# nLab anti de Sitter spacetime

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Gravity

gravity, supergravity

# Contents

## Definition

Up to isometry, the anti de Sitter spacetime of dimension $d + 1$ is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Cartesian space $\mathbb{R}^{d+2}$ that solves the equation

$\sum_{i = 1}^{d+1} (x^i)^2 - (x^{d+2})^2 = 0$

and equipped with the metric induced from the ambient metric

$g = \sum_{i = 1}^{d+1} d x^i \otimes d x^i - d x^{i+1} \otimes d x^{i+t} \,,$

where $x^i\colon \mathbb{R}^{d+2} \to \mathbb{R}$ denote the canonical coordinates on a Cartesian space.

## Properties

(…)

### Holography

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

## References

• Ingemar Bengtsson, Anti-de Sitter space lecture notes (pdf)

• C. Frances, The conformal boundary of anti-de Sitter space-times, in AdS/CFT correspondence: Einstein metrics and their conformal boundaries , 205–216, IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005 (pdf)

• Gary Gibbons, Anti-de-Sitter spacetime and its uses (arXiv:1110.1206)

• Wikipedia (English): anti de Sitter space

Revised on January 6, 2012 16:35:03 by Urs Schreiber (89.204.137.240)