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

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

for some $R \neq 0$ (the “radius” of the spacetime) and equipped with the metric tensor 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

A comprehesive account of the AdS metric tensor in various coordinate charts is given in Blau §39.3.

Poincaré and horospheric coordinates (e.g. Blau §39.3.7). Consider the Cartesian space $\mathbb{R}^{1+p}$ with its canonical coordinate functions

$$X^a \;\colon\; \mathbb{R}^{1+p} \xrightarrow{\phantom{-}} \mathbb{R} \,, \;\;\; a \in \{0,1,\cdots, p\}$$

and denote its standard Minkowski metric tensor by

$$d s^2_{\mathbb{R}^{1,p}} \;\coloneqq\; \textstyle{\sum_{a = 0}^p} \mathrm{d}X^a \otimes \mathrm{d}X^a \,.$$

Moreover consider $\mathbb{R}^{1+p} \times \mathbb{R}_{\gt 0}$ equipped with the pullback of the above coordinate function as well as with

$$r \;\colon\; \mathbb{R}^{1+p} \times \mathbb{R}_{\gt 0} \twoheadrightarrow \mathbb{R}_{\gt 0} \hookrightarrow \mathbb{R} \,.$$

Then there is a chart of $AdS_{p+2}$ of the form

$$\iota \;\colon\; \mathbb{R}^{1+p} \times \mathbb{R}_{\gt 0} \xhookrightarrow{\phantom{--}} AdS_{p+2}$$

such that the pullback of the AdS metric tensor is

(1)$$\iota^\ast \mathrm{d}s^2_{AdS} \;=\; \tfrac{r^2}{R^2} \mathrm{d}s^2_{\mathbb{R}^{1,d}} \,+\, \tfrac{R^2}{r^2} \mathrm{d}r^2 \,.$$

This is the form of the AdS-metric which arises naturally as the near horizon geometry of black p-branes in supergravity (e.g. AFFHS98 (5)). The black brane singularity itself would be at $r = 0$.

In slight variation, in terms of

$$z \,\coloneqq\, 1/r \,, \;\;\; \text{hence} \;\;\; r = z^{-1} \,, \;\; \mathrm{d}r \;=\; -\tfrac{1}{z^2} \mathrm{dz} \;\;$$

the metric (1) becomes

(2)$$\iota^\ast \mathrm{d}s^2_{AdS} \;=\; \frac{R^2}{z^2} \big( \tfrac{1}{R^4} \mathrm{d}s^2_{\mathbb{R}^{1,d}} \,+\, \mathrm{d}z^2 \big) \,.$$

This is called horospheric coordinates by Gibbons 2000 (12).

On the other hand, in terms of

$$\rho \;\coloneqq\; ln r \,, \;\; \text{hence} \;\; r = e^\rho \,, \;\; \mathrm{d}r = r \, \mathrm{d}\rho$$

the metric (1) becomes

(3)$$\iota^\ast \mathrm{d}s^2_{AdS} \;=\; \tfrac{e^{2\rho}}{R^2} \mathrm{d}s^2_{\mathbb{R}^{1,d}} \,+\, R^2 \mathrm{d}\rho^2 \,.$$

This is called horospheric coordinates in arXiv:1412.2054 (37).

Cartan geometry

We spell out the curvature tensors of anti de Sitter spacetime, using a Cartan connection (i.e. first order formulation).

An evident choice of an orthonormal coframe field for the AdS metric in Poincaré coordinates (1) is

$$\begin{array}{ccll} E^a &\coloneqq& \tfrac{r}{R} \mathrm{d}X^a & a \in \{0,1, \cdots, d\} \\ E^{p'} &\coloneqq& \tfrac{R}{r} \mathrm{d}r \end{array}$$

in that

$$\iota^\ast \mathrm{d}s^2_{AdS_{p+2}} \;=\; \eta_{a b} E^a \otimes E^b + E^{p'} \otimes E^{p'} \,.$$

(no sum over $p'$ – this is meant to be the index value corresponding to the radial direction)

The torsion-free spin connection $\Omega$ for this coframe field, characterized by

(4)$$\begin{array}{ccl} \mathrm{d}E^a &=& \Omega^a{}_b \, E^b \\ \mathrm{d}E^{p'} &=& \Omega^{p'}{}_b \, E^b \mathrlap{\,,} \end{array}$$

has non-vanishing components

$$\Omega^{a p'} \,=\, - \Omega^{p' a} \;=\; - \tfrac{r}{R^2} \mathrm{d}X^a \,.$$

The corresponding curvature 2-form

$$\begin{array}{l} R^{a_1 a_2} \;=\; - \Omega^{a_1}{}_{p'} \Omega^{p' a_2} \\ R^{a d'} \;\coloneqq\; \mathrm{d}\Omega^{a p'} \end{array}$$

has non-vanishing components

$$\begin{array}{l} R^{a_1 a_2} \;=\; - \tfrac{r^2}{R^4} \mathrm{d}X^{a_1}\, \mathrm{d}X^{a_2} \;=\; - \tfrac{1}{R^2} E^{a_1} \, E^{a_2} \\ R^{a p'} \,=\, - R^{p' a} \;=\; - \tfrac{1}{R^2} \mathrm{d}X^a \;=\; - \tfrac{1}{R^2} E^a \, E^{p'} \,. \end{array}$$

Hence the Riemann tensor has non-vanishing components

$$\begin{array}{ccl} R^{a_1 a_2}{}_{b_1 b_2} &=& + \tfrac{1}{R^2} \delta^{a_1 a_2}_{b_1 b_2} \\ R^{a p'}{}_{b p'} &=& + \tfrac{1}{R^2} \delta^a{}_b \mathrlap{\,,} \end{array}$$

so that the Ricci tensor is proportional to the metric, as befits an Einstein manifold:

$$\begin{array}{ccl} Ric_{a_1 a_2} &\coloneqq& R_{a_1}{}^{b}{}_{a_2 b} \,+\, R_{a_1}{}^{p'}{}_{a_2 p'} \;=\; \tfrac{p}{R^2} \, \eta_{a_1 a_2} + \tfrac{1}{R^2} \, \eta_{a_1 a_2} \\ &=& \tfrac{p+1}{R^2} \, \eta_{a_1 a_2} \\ Ric_{p' p'} &\coloneqq& R_{p'}{}^b{}_{p' b} \\ &=& \tfrac{p+1}{R^2} \,. \end{array}$$

The above convention $\mathrm{d}E^a = + \Omega^a{}_b \, E^b$ (4) makes this come out positive, following the old convention by Freund & Rubin 1980, see there.

Conformal boundary

(…) conformal boundary (…) [e.g. Frances 2011]

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

Review:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 1, chapter I.3.8 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.3: pdf,]

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

• Matthias Blau, chapter 37 of: Lecture notes on general relativity [web]

• Gary Gibbons, Anti-de-Sitter spacetime and its uses, in: Mathematical and Quantum Aspects of Relativity and Cosmology, Lecture Notes in Physics 537, Springer (2000) [arXiv:1110.1206, doi:10.1007/3-540-46671-1_5]

• Charles Frances, The conformal boundary of anti-de Sitter space-times, in: AdS/CFTCorrespondence: Einstein Metrics and Their Conformal Boundaries, IRMA Lectures in Mathematical and Theoretical Physics, EMS (2011) 205-216 [ems:irma/9/206, hal:03195056, pdf]

• Makoto Natsuume, section 6 of: AdS/CFT Duality User Guide, Lecture Notes in Physics 903 Springer (2015) [arXiv:1409.3575, doi:10.1007/978-4-431-55441-7]

• Leszek M. Sokolowski, The bizarre anti-de Sitter spacetime, International Journal of Geometric Methods in Modern Physics 13 9 (2016) 1630016 [arXiv:1611.01118]

• Ricard M. Calvo, Geometry of (Anti-)De Sitter space-time (2018) [pdf, pdf]

See also:

• Wikipedia, anti de Sitter space

Further discussion:

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

Dedicated 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)