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anti de Sitter spacetime in nLab

nLab anti de Sitter spacetime

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Contents

Context

Riemannian geometry

Gravity

gravity, supergravity

Formalism

Definition

Spacetime configurations

Properties

Spacetimes

Quantum theory

Contents

Definition

Up to isometry, the anti de Sitter spacetime of dimension dd, AdS dAdS_d, is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Minkowski spacetime ℝ d−1,2\mathbb{R}^{d-1,2} that solves the equation

∑ i=1 d−1(x i) 2−(x d) 2−(x 0) 2=−R 2, \textstyle{\sum_{i = 1}^{d-1}} (x_i)^2 - (x_d)^ 2 - (x_0)^2 \;=\; - R^2 \,,

for some R≠0R \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,⋯,x d}\{x^0, x^1, x^2, \cdots, x^d\} denote the canonical coordinates. AdS dAdS_d is homeomorphic to ℝ d−1×S 1\mathbb{R}^{d-1} \times S^1, and its isometry group is O(d−1,2)O(d-1, 2).

More generally, one may define the anti de Sitter space of signature (p,q)(p,q) as isometrically embedded in the space ℝ p,q+1\mathbb{R}^{p,q+1} with coordinates (x 1,...,x p,t 1,…,t q+1)(x_1, ..., x_p, t_1, \ldots, t_{q+1}) as the sphere ∑ i=1 px i 2−∑ j=1 q+1t j 2=−R 2\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 ℝ 1+p\mathbb{R}^{1+p} with its canonical coordinate functions

X a:ℝ 1+p→−ℝ,a∈{0,1,⋯,p} 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

ds ℝ 1,p 2≔∑ a=0 pdX a⊗dX a. d s^2_{\mathbb{R}^{1,p}} \;\coloneqq\; \textstyle{\sum_{a = 0}^p} \mathrm{d}X^a \otimes \mathrm{d}X^a \,.

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

r:ℝ 1+p×ℝ >0↠ℝ >0↪ℝ. 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+2AdS_{p+2} of the form

ι:ℝ 1+p×ℝ >0↪−−AdS p+2 \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)ι *ds AdS 2=r 2R 2ds ℝ 1,d 2+R 2r 2dr 2. \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=0r = 0.

In slight variation, in terms of

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

the metric (1) becomes

(2)ι *ds AdS 2=R 2z 2(1R 4ds ℝ 1,d 2+dz 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

ρ≔lnr,hencer=e ρ,dr=rdρ \rho \;\coloneqq\; ln r \,, \;\; \text{hence} \;\; r = e^\rho \,, \;\; \mathrm{d}r = r \, \mathrm{d}\rho

the metric (1) becomes

(3)ι *ds AdS 2=e 2ρR 2ds ℝ 1,d 2+R 2dρ 2. \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

E a ≔ rRdX a a∈{0,1,⋯,d} E p′ ≔ Rrdr \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

ι *ds AdS p+2 2=η abE a⊗E b+E p′⊗E p′. \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′p' – this is meant to be the index value corresponding to the radial direction)

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

(4)dE a = Ω a bE b dE p′ = Ω p′ bE b, \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

Ω ap′=−Ω p′a=−rR 2dX a. \Omega^{a p'} \,=\, - \Omega^{p' a} \;=\; - \tfrac{r}{R^2} \mathrm{d}X^a \,.

The corresponding curvature 2-form

R a 1a 2=−Ω a 1 p′Ω p′a 2 R ad′≔dΩ ap′ \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

R a 1a 2=−r 2R 4dX a 1dX a 2=−1R 2E a 1E a 2 R ap′=−R p′a=−1R 2dX a=−1R 2E aE p′. \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

R a 1a 2 b 1b 2 = +1R 2δ b 1b 2 a 1a 2 R ap′ bp′ = +1R 2δ a b, \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:

Ric a 1a 2 ≔ R a 1 b a 2b+R a 1 p′ a 2p′=pR 2η a 1a 2+1R 2η a 1a 2 = p+1R 2η a 1a 2 Ric p′p′ ≔ R p′ b p′b = p+1R 2. \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 dE a=+Ω a bE b\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 pp-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,ℚ p)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:

See also:

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 Λ\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×S qAdS_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:

See also:

  • Michael Gutperle, Nicholas Klein, A Penrose limit for type IIB AdS 6AdS_6 solutions (arXiv:2105.10824)

Last revised on August 15, 2024 at 17:02:30. See the history of this page for a list of all contributions to it.

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