Anti-de Sitter space

In mathematics and physics, n-dimensional anti de Sitter space, denoted $AdS_{n}$, is the Lorentzian analog of n-dimensional hyperbolic space. It is a maximally symmetric, Lorentzian manifold with constant negative curvature.

In the language of general relativity, anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with a negative cosmological constant $\Lambda$.

Anti de Sitter space is the negative curvature analogue of De Sitter space, which is named for Willem de Sitter. It is used in the AdS/CFT correspondence.

Definition and properties

Anti de Sitter space can be defined as a submanifold of R^{2, n−1} of codimension 1. Take the space R^2,n−1 with the standard metric:

$ds^{2}=-dx_{0}^{2}-dx_{1}^{2}+\sum _{i=2}^{n}dx_{i}^{2}.$

Anti de Sitter space is the submanifold described by the hyperboloid

$-x_{0}^{2}-x_{1}^{2}+\sum _{i=2}^{n}x_{i}^{2}=-\alpha ^{2}$

where $\alpha$ is some non-zero constant with dimensions of length. The metric on anti de Sitter space is the metric induced from the ambient metric. One can check that the induced metric is nondegenerate and has Lorentzian signature.

Anti de Sitter space can also defined as the quotient O(2,n−1)/O(1,n−1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.

n-Dimensional anti de Sitter space has O(n−1, 2) as its isometry group. It is not simply-connected; it is homeomorphic to the product S¹×Rⁿ⁻¹, so its fundamental group is the integers, and it has a contractible universal cover. Anti de Sitter spacetime has closed time-like loops, though its universal cover does not. Some authors use Anti de Sitter space to refer to the simply connected universal cover.

Coordinate patches

A coordinate patch covering part of the space gives the half-space coordinatization of anti de Sitter space. The metric for this patch is

$ds^{2}={\frac {1}{y^{2}}}\left(dt^{2}-dy^{2}-\sum _{i}dx_{i}^{2}\right).$

In the limit as y = 0, this reduces to a Minkowski metric $dy^{2}=\left(dt^{2}-\sum _{i}dx_{i}^{2}\right)$; thus, the anti-de Sitter space contains a conformal Minkowski space at infinity ("infinity" having y-coordinate zero in this patch). The constant time slices of this coordinate patch are hyperbolic spaces in the Poincaré half-plane metric.

In AdS space time is periodic, and the universal cover has non-periodic time. The coordinate patch above covers half of a single period of the spacetime.

Because the conformal infinity of AdS is timelike, specifying the initial data on a spacelike hypersurface would not determine the future evolution uniquely (i.e. deterministically) unless there are boundary conditions associated with the conformal infinity.

The image on the right represents the "half-space" region of anti deSitter space and its boundary. The interior of the cylinder corresponds to anti deSitter spacetime, while its cylindrical boundary corresponds to its conformal boundary. The green shaded region in the interior corresponds to the region of AdS covered by the half-space coordinates and it is bounded by two null aka lightlike geodesic hyperplanes; the green shaded area on the surface corresponds to the region of conformal space covered by Minkowski space.

The green shaded regions covers half of the AdS space and half of the conformal spacetime; the left ends of the green discs will touch in the same fashion as the right ends.

Anti de Sitter as homogeneous and symmetric space

In the same way that the sphere $S^{2}={\frac {O(3)}{O(2)}}$, anti de Sitter can be seen as a quotient of two groups $AdS_{n}={\frac {O(2,n-1)}{O(1,n-1)}}$. This quotient formulation gives to $AdS_{n}$ an homogeneous space structure. The Lie algebra of $O(1,n)$ is given by matrices

${\mathcal {H}}={\begin{pmatrix}{\begin{matrix}0&0\\0&0\end{matrix}}&{\begin{pmatrix}\cdots 0\cdots \\\leftarrow v^{t}\rightarrow \end{pmatrix}}\\{\begin{pmatrix}\vdots &\uparrow \\0&v\\\vdots &\downarrow \end{pmatrix}}&B\end{pmatrix}}$,

where $B$ is a skewsymmetric matrix. A complementary in the Lie algebra of ${\mathcal {G}}=O(2,n)$ is

${\mathcal {Q}}={\begin{pmatrix}{\begin{matrix}0&a\\-a&0\end{matrix}}&{\begin{pmatrix}\leftarrow w^{t}\rightarrow \\\cdots 0\cdots \\\end{pmatrix}}\\{\begin{pmatrix}\uparrow &\vdots \\w&0\\\downarrow &\vdots \end{pmatrix}}&0\end{pmatrix}}.$

These two fulfil ${\mathcal {G}}={\mathcal {H}}\oplus {\mathcal {Q}}$. Then explicit matrix computation shows that $[{\mathcal {H}},{\mathcal {Q}}]\subseteq {\mathcal {Q}},\quad [{\mathcal {Q}},{\mathcal {Q}}]\subseteq {\mathcal {H}}$. So anti de Sitter is a reductive homogeneous space, and a non-Riemannian symmetric space.

References

• Bengtsson, Ingemar: Anti-de Sitter space. Lecture notes.
• Qingming Cheng (2001) [1994], "Anti de Sitter space", Encyclopedia of Mathematics, EMS Press
• 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.