# nLab anti de Sitter spacetime

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

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

cf. e.g. Bayona & Braga 2007 (11). (These are 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.

## References

### Geometry

Review:

With attention to the conformal geometry:

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)

### Quantum field theory

Discussion of (scalar) quantum field theory on AdS backgrounds:

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

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:

### 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:

Review:

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