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Definition

Up to isometry, the anti de Sitter spacetime of dimension $d$, $AdS_d$, is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Minkowski spacetime ${ℝ}^{d<span><del\; class="diffmod">\; ,</del><ins\; class="diffmod">\; -</ins></span>1,2}$ \mathbb{R}^{d,1} \mathbb{R}^{d-1,2} that solves the equation

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

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

Properties

Coordinate charts

(…)

in horospheric coordinates the AdS metric tensor is

In terms of

this becomes

and with

for $n \neq 0$

we get

Conformal boundary

(…)

Holography

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

Related concepts

• anti de Sitter group

• de Sitter spacetime

• super anti de Sitter spacetime

• near-horizon geometry

• singleton representation

References

• Matthias Blau, chapter 38 of Lecture notes on general relativity (web)

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

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 1, chapter I.3.8 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• 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

• Abdelghani Zeghib, On closed anti de Sitter spacetimes, Math. Ann. 310, 695–716 (1998) (pdf)

• Leszek M. Sokolowski, The bizarre anti-de Sitter spacetime, International Journal of Geometric Methods in Modern Physics 13 no.9 (2016) 1630016 (arXiv:1611.01118)