anti de Sitter space

From Mathematics wiki

1 Embedding
2 Global coordinates
3 Conformal compactification
4 Topology
5 Boundary
6 Symmetry
7 Cover
8 See also
9 References

Embedding

Consider the $n+1\,$ -dimensional Minkowski space $\mathbb{R}^{2,n-1}\,$ with metric

$ds^2 = -(dx^0)^2 - (dx^{n})^2 + (dx^1)^2 + ... + (dx^{n-1})^2\,$

and embed into it the one-sheeted quadric defined by

$-(x^0)^2 - (x^n)^2 + (x^1)^2 + ... (x^{n-1})^2 = -R^2\,$ .

This submanifold has codimension $1\,$ and is known as anti de Sitter space or $\mathrm{AdS}_n\,$ . The induced metric on $\mathrm{AdS}_n\,$ has Lorentzian signature. Since $O(2,n-1)\,$ leave both the ambient metric and the sub-manifold unaltered, the isometry group of $\mathrm{AdS}_n\,$ is $O(2,n-1)\,$ , the indefinite orthogonal group.

Global coordinates

Let

$x^0 = R \cosh\rho\, \cos \tau\,$ ,

$x^n = R \cosh\rho\, \sin \tau\,$ ,

$x^i = R \sinh\rho\, \Omega^i\,$ , where $i = 1...n-1\,$ and $\sum_{i=1}^{n -1}(\Omega^i)^2 = 1\,$ .

Where $0 \leq \rho\,$ and $0 \leq \tau \leq 2\pi\,$ . Then

$ds^2 = R^2(-\cosh^2\rho\,\,d\tau^2 + d\rho^2 +\sinh^2\rho\,\,d\Omega_{n-2}^2)\,$ ,

where $d\Omega_{n-2}^2\,$ is the metric on the n-2 sphere $\mathbb{S}^{n-2}\,$ . Near $\rho \approx 0\,$ , the metric becomes

$ds^2_{\rho\to 0} = R^2(-d\tau^2 + d\rho^2 + d\Omega_{n-2}^2)\,$ ,

This coordinate chart covers the entire hyperboloid and is thus termed global.

Conformal compactification

Global coordinates

We may conformally compactify $\mathrm{AdS}_{n}\,$ by introducing the coordinate $\theta \in \left(0, \frac{\pi}{2}\right)\,$ such that $\tan\theta = \sinh \rho\,$ and

$ds^2 = \frac{R^2}{\cos^2\theta}(-d\tau^2 + d\theta^2 +\sin^2\theta\,\,d\Omega_{n-2}^2)\,$ .

For $\mathrm{AdS}_{2}\,$ one takes $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\,$ .

The above metric is conformally equivalent to

$ds^2 \sim -d\tau^2 + d\theta^2 +\sin^2\theta\,\,d\Omega_{n-2}^2\,$ .

Thus, for $n \neq 2\,$ , surfaces of constant $\tau\,$ are conformally equivalent to $(n-1)\,$ -dimensional hemispheres, while $\mathrm{AdS}_2\,$ is mapped to $\mathbb{R}\times \mathbb{S}^2\,$ .

Stereographic projective coordinates

Let^[1]

$x^0 = \rho \frac{1 + \|y\|^2}{1-\|y\|^2}\,$

$x^\mu = \rho \frac{2 y^{\mu-1} }{1-\|y\|^2}\,$ , $\mu = 1...n\,$ ,

where $\|y\|^2 = (y^0)^2 + ... + (y^{n-2})^2 - (y^{n-1})^2\,$ , i.e. for the purposes of this calculation, we can raise and lower indices with the Minkowski metric $\eta_{\mu\nu} = \operatorname{diag}\,(+1,+1...,+1,-1)\,$ . Then

$dx^0 = d\rho \frac{1 + \|y\|^2}{1-\|y\|^2} + 4\rho \frac{y_\mu dy^\mu} {( 1-\|y\|^2 )^2}\,$ .

$dx^\mu = d\rho \frac{2 y^\mu }{1-\|y\|^2} + \rho \frac{2 }{(1-\|y\|^2)^2} \left[ (1 -\|y\|^2) \delta_\nu^\mu + 2 y^\mu y_\nu \right] dy^\nu\,$ ,

from which the metric tensor on the embedding space is

$ds^2 = -d\rho^2 + \frac{4\rho^2}{(1-\|y\|^2)^2}(dy)^2\,$ .

Furthermore,

$-(x^0)^2 - (x^n)^2 + (x^1)^2 + ... (x^{n-1})^2 = \frac{\rho^2}{(1-\|y\|^2)^2} \left[ -(1+\|y\|^2)^2 + 4\|y\|^2 \right] = -\rho^2\,$ ,

so that $\mathrm{AdS}_n\,$ is described by the surface $\rho^2 = R^2\,$ . The induced metric on $\mathrm{AdS}_n\,$ is then

$ds^2 = \frac{4R^2}{(1-\|y\|^2)^2} (dy)^2\,$ ,

i.e.,

$g_{\mu\nu} = \frac{4R^2}{(1-\|y\|^2)^2}\eta_{\mu\nu}\,$ .

Poincaré coordinates

AdSn hyperboloid intersected by the hyperplane . Coordinates other than those shown are held fixed.

AdS n

hyperboloid intersected by the hyperplane $x^0 = x^{n-1}\,$ . Coordinates other than those shown are held fixed.

We first define the lightcone coordinates $u\,$ and $v\,$ :

$u = \frac{1}{R^2}(x^0 - x^{n-1})\,$ ,

$v = \frac{1}{R^2}(x^0 + x^{n-1})\,$ .

along with

$\tilde{x}^i = \frac{x^i}{R u}\,$ (spacelike), where $i = 1...n-2\,$ ,

$t = \frac{x^{n}}{R v}\,$ (timelike),

whence

$R^4 uv + R^2 u^2 (t^2 - \|\tilde{x}\|^2) = R^2\,$ ,

where $\|\tilde{x}\|=\textstyle\sum_i (\tilde{x}^i)^2\,$ is the Euclidean norm. This allows us to solve for $v\,$ . Thus

$x^0 = \frac{1}{2u} \left(1 + u^2 (R^2 + \|\tilde{x}\|^2 - t^2 )\right)\,$ ,

$x^n = R u t\,$ ,

$x^{n-1} = \frac{1}{2u} \left(1 + u^2 (-R^2 + \|\tilde{x}\|^2 - t^2 ) \right)\,$ ,

$x^i = R u \tilde{x}^i\,$ , where $i = 1...n-2\,$ ,

so that

$ds^2 = R^2\left( \frac{du^2}{u^2} + u^2( -dt^2 + (d\tilde{x})^2 )\right)\,$ .

Next, we define the coordinate $z = \frac{1}{u}\,$ , so that

$x^0 = \frac{1}{2z} \left(z^2 + R^2 +\|\tilde{x}\|^2 - t^2 \right)\,$ ,

$x^{n} = \frac{R t}{z}\,$ ,

$x^{n-1} = \frac{1}{2z} \left(z^2 -R^2 +\|\tilde{x}\|^2 - t^2) \right)\,$ ,

$x^i = \frac{R \tilde{x}^i}{z}\,$ , where $i = 1...n-2\,$ ,

and

$ds^2 = \frac{R^2}{z^2}\left( - dt^2 + dz^2 + (d\tilde{x})^2 \right)\,$ .

The (inverse of the) Poincaré coordinate chart is singular at $u \to 0\,$ or equivalently $z \to \pm \infty\,$ . Thus the hyperboloid is divided in two by the hyperplane $x^0 - x^{n-1} = 0\,$ , and one chart ( $z > 0\,$ ) covers one half of the hyperboloid while the other ( $z < 0\,$ ) covers the other. Defining

$z^0 = z\,$ ,

$z^{n-1} = t\,$ ,

$z^i = \tilde{x}^i\,$ , where $i = 1...n-2\,$ ,

we can write

$ds^2 = \frac{R^2}{(z^0)^2} (dz)^2\,$ ,

$g_{\mu\nu} = \frac{R^2}{(z^0)^2}\eta_{\mu\nu}\,$ .

Topology

In the $\tau,\rho,\Omega^i\,$ coordinates we may alternatively fix $\tau\,$ or $\rho\,$ to notice that the topology of $\mathrm{AdS}_{n}\,$ is $\mathbb{S}^1\times \mathbb{R}^{n-1}\,$ .

Boundary

In the $(z,t, \tilde{x}^i)\,$ coordinate chart the boundary of $\mathrm{AdS}_n\,$ corresponds to $z = 0\,$ , (where the remaining coordinates describe the structure of $\mathbb{R}^{1,n-2}\,$ or $\mathbb{M}^{n-1}\,$ , i.e., $(n-1)\,$ -dimensional Minkowski space) along with a single "point at infinity" $z = \infty\,$ . The boundary of $\mathrm{AdS}_n\,$ is therefore a conformal compactification of $\mathbb{M}^{n-1}\,$ obtained by adding a point at infinity^[2].

Symmetry

The isometry group of $\mathrm{AdS}_n\,$ , $SO(2,n-1)\,$ has maximal compact subgroup $SO(2)\times SO(n-1)\,$ where $SO(2)\,$ is to be identified with the isometry group of $\mathbb{S}^1\,$ while $SO(n-1)\,$ is is to be identified with the isometry group of $\mathbb{S}^{n-2}\,$ , or rotations in $\mathbb{R}^{n-1}\,$ .

Cover

The universal cover of $\mathrm{AdS}_n\,$ , denoted $\mathrm{CAdS}_n\,$ , is obtained by allowing $\tau \in (-\infty, \infty)\,$ .

References

Further reading:^[3] ^[4] ^[5] ^[6] ^[7] ^[8] ^[9] ^[10]

↑ Jens L. Petersen, (1999). "Introduction to the Maldacena Conjecture on AdS/CFT". Int.J.Mod.Phys. A 14: 3597-3672. arXiv:hep-th/9902131v2.
↑ E. Witten (1998). "Anti De Sitter Space And Holography". Adv.Theor.Math.Phys. 2: 253-291. arXiv:hep-th/9802150.
↑ Bengtsson, Ingemar: Anti-de Sitter space. Lecture notes.
↑ Qingming Cheng, "Anti de Sitter space" SpringerLink Encyclopaedia of Mathematics (2001)
↑ Ellis, G. F. R.; Hawking, S. W. The large scale structure of space-time. Cambridge university press (1973). (see pages 131-134).
↑ Matsuda, H. A note on an isometric imbedding of upper half-space into the anti de Sitter space. Hokkaido Mathematical Journal Vol.13 (1984) p. 123-132.
↑ Wolf, Joseph A. Spaces of constant curvature. (1967) p. 334.
↑ S. J. Avis, C. J. Isham, D. Storey, Quantum field theory in anti-de Sitter space-time Phys. Rev. D 18, 3565 - 3576 (1978)
↑ C. Bayona, N. Braga, Anti-de Sitter boundary in Poincaré coordinates (2006); arXiv:hep-th/0512182
↑ U. Moschella, The de Sitter and anti-de Sitter Sightseeing Tour, Séminaire Poincaré 1 (2005) 1 - 12

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Category: Manifolds

anti de Sitter space

From Mathematics wiki

Contents

Embedding

Global coordinates

Conformal compactification

Global coordinates

Stereographic projective coordinates

Poincaré coordinates

Topology

Boundary

Symmetry

Cover

See also

References

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