nLab anti de Sitter spacetime

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 $\mathbb{R}^{d-1,2}$ that solves the equation

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

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 $\sum_{i=1}^p x_i^2 - \sum_{j=1}^{q+1} t_j^2 = -R^2$.

graphics grabbed from Yan 19

Properties

Coordinate charts

(…)

in horospheric coordinates the AdS metric tensor is

$$g_{AdS} \;=\; \frac{1}{z^2} \left( g_{(\mathbb{R}^{p,1})} + (d z)^2 \right)$$

In terms of

$$y \coloneqq 1/z$$

this becomes

$$g_{AdS} \;=\; y^2 \, g_{(\mathbb{R}^{p,1})} + \frac{1}{y^2}(d y)^2$$

and with

$$y \coloneqq \tfrac{1}{n} r^n$$

for $n \neq 0$

we get

$$g_{AdS} \;=\; \tfrac{1}{n^2}r^{2n} \, g_{(\mathbb{R}^{p,1})} + \frac{1}{r^2}(d r)^2$$

Conformal boundary

(…)

Holography

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

In $p$-adic geometry

A 2-adic arithmetic geometry-version of AdS spacetime is identified with the Bruhat-Tits tree for the projective general linear group $PGL(2,\mathbb{Q}_p)$:

graphics from Casselman 14

In the p-adic AdS/CFT correspondence this may be regarded (at some finite depth truncation) as a tensor network state:

graphics from Sati-Schreiber 19c

and as such validates the Ryu-Takayanagi formula for holographic entanglement entropy.

Related concepts

• thermal AdS3

• anti de Sitter group

• de Sitter spacetime

• super anti de Sitter spacetime

• near-horizon geometry

• singleton representation

• AdS/CFT correspondence, AdS/QCD correspondence

References

General

Reviews:

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

• Ingemar Bengtsson, Anti-de Sitter space, lecture notes 1998 (pdf)

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

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

• Makoto Natsuume, section 6 of: AdS/CFT Duality User Guide, Lecture Notes in Physics 903, Springer 2015 (arXiv:1409.3575)

• 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)

See also:

• Wikipedia, anti de Sitter space

Further discussion:

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

• Jiri Podolsky, Ondrej Hruska, Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes, Phys. Rev. D 95, 124052 (2017) (arXiv:1703.01367)

Discussion of thermal Wick rotation on global anti-de Sitter spacetime (which is already periodic in real time) to Euclidean field theory with periodic imaginary time is in

• B. Allen, A. Folacci, Gary Gibbons, Anti-de Sitter space at finite temperature, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (doi:10.1016/0370-2693(87)91437-7)

Discussion of black holes in anti de Sitter spacetime:

• Hawking, Stephen W., and Don N. Page. “Thermodynamics of black holes in anti-de Sitter space.” Communications in Mathematical Physics 87.4 (1983): 577-588.

• M. Socolovsky, Schwarzschild Black Hole in Anti-De Sitter Space (arXiv:1711.02744)

• Peng Zhao, Black Holes in Anti-de Sitter Spacetime (pdf)

• Jakob Gath, The role of black holes in the AdS/CFT correspondence (pdf)

Relation to Teichmüller theory:

• Francesco Bonsante, Andrea Seppi, Anti-de Sitter geometry and Teichmüller theory (arXiv:2004.14414)

Phenomenology

• Anjan A. Sen, Shahnawaz A. Adil, Somasri Sen, Do cosmological observations allow a negative $\Lambda$? (arXiv:2112.10641)

As string vacua

On (in-)stability of non-supersymmetric AdS vacua in string theory:

• Iosif Bena, Krzysztof Pilch, Nicholas Warner, Brane-Jet Instabilities, J. High Energ. Phys. 2020, 91 (2020) (arXiv:2003.02851)

pp-Waves as Penrose limits of $AdS_p \times S^q$ spacetimes

Discussion of pp-wave spacetimes as Penrose limits (Inönü-Wigner contractions) of AdSp x S^q spacetimes and of the corresponding limit of AdS-CFT duality:

• David Berenstein, Juan Maldacena, Horatiu Nastase, Section 2 of: Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills, JHEP 0204 (2002) 013 (arXiv:hep-th/0202021)

• N. Itzhaki, Igor Klebanov, Sunil Mukhi, PP Wave Limit and Enhanced Supersymmetry in Gauge Theories, JHEP 0203 (2002) 048 (arXiv:hep-th/0202153)

• Nakwoo Kim, Ari Pankiewicz, Soo-Jong Rey, Stefan Theisen, Superstring on PP-Wave Orbifold from Large-N Quiver Gauge Theory, Eur. Phys. J. C25:327-332, 2002 (arXiv:hep-th/0203080)

• E. Floratos, Alex Kehagias, Penrose Limits of Orbifolds and Orientifolds, JHEP 0207 (2002) 031 (arXiv:hep-th/0203134)

• E. M. Sahraoui, E. H. Saidi, Metric Building of pp Wave Orbifold Geometries, Phys.Lett. B558 (2003) 221-228 (arXiv:hep-th/0210168)

Review:

• Darius Sadri, Mohammad Sheikh-Jabbari, The Plane-Wave/Super Yang-Mills Duality, Rev. Mod. Phys. 76:853, 2004 (arXiv:hep-th/0310119)

• Badis Ydri, Section 3.1.10 of: Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory (arXiv:1708.00734), published as: Matrix Models of String Theory, IOP 2018 (ISBN:978-0-7503-1726-9)

See also:

• Michael Gutperle, Nicholas Klein, A Penrose limit for type IIB $AdS_6$ solutions (arXiv:2105.10824)