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anti de Sitter spacetime

Contents

Context

Riemannian geometry

Gravity

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 d1,2\mathbb{R}^{d-1,2} that solves the equation

i=1 d1(x i) 2(x d) 2(x 0) 2=R 2 \sum_{i = 1}^{d-1} (x_i)^2 - (x_d)^ 2 - (x_0)^2 = -R^2

for some R0R \neq 0 (the “radius” of the spacetime) and equipped with the metric 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 d1×S 1\mathbb{R}^{d-1} \times S^1, and its isometry group is O(d1,2)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 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

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in horospheric coordinates the AdS metric tensor is

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

In terms of

y1/z y \coloneqq 1/z

this becomes

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

and with

y1nr n y \coloneqq \tfrac{1}{n} r^n

for n0n \neq 0

we get

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

Conformal boundary

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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, 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

General

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)

pp-Waves as Penrose limits of AdS p×S qAdS_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:

Review:

Last revised on February 7, 2020 at 03:30:03. See the history of this page for a list of all contributions to it.