nLab
anti de Sitter spacetime

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 ${ℝ}^{d+2}$ that solves the equation

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

and equipped with the metric induced from the ambient metric

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

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

Properties

Conformal boundary

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Holography

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

Related concepts

• anti de Sitter group

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