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# Anti de Sitter space

In mathematics, n-dimensional anti de Sitter space is a geometry with Lorentzian signature. It has as automorphism group SO(n,2) (possibly with reflections), 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 is now

$ds^2=\frac{1}{y^2}\left(dt^2-dy^2-\sum_idx_i^2\right).$

y = 0 corresponds to the conformal Minkowski space at infinity.

There are two types of AdS space: one where time is periodic, and its universal cover. The coordinates above cover half of the periodic version.

In another coordinate system, the constant time slices are hyperbolic geometries.

Its conformal boundary at infinity contains conformal Minkowski space.

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 and 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.

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).