Anti de Sitter space

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In mathematics and physics, n-dimensional anti de Sitter space, denoted [AdS_n], is the maximally symmetric, simply-connected, Lorentzian manifold with constant negative curvature. It may be regarded as the Lorentzian analog of n-dimensional hyperbolic space.

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

n-Dimensional anti de Sitter space has SO(n−1, 2) (possibly with reflections) as automorphism group, according to the Erlangen program.

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\left(dt^2-dy^2-\sum_idx_i^2\right).]

In the limit as y = 0, this reduces to a Minkowski metric [dy^2=\left(dt^2-\sum_idx_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 Poincare half-plane metric.

There are two types of AdS space: one where time is periodic, and the universal cover with 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 above 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.

If AdS is periodic in time, 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.

References

Bengtsson, Ingemar: Anti-de Sitter space. Lecture notes. (see

Claessens, L; Detournay, S: Black holes in symmetric spaces : anti-de Sitter spaces. (see )

Frances, Charles: Géométrie et dynamique lorentziennes conformes (thesis). (see , page 57)

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.

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Anti de Sitter space

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References