Anti de Sitter Space - Coordinate Patches

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

We easily see that this metric is conformally equivalent to a flat half-space Minkowski spacetime.

The constant time slices of this coordinate patch are hyperbolic spaces in the Poincaré half-plane metric. In the limit as y = 0, this half-space metric reduces to a Minkowski metric ; thus, the anti-de Sitter space contains a conformal Minkowski space at infinity ("infinity" having y-coordinate zero in this patch).

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.

Another commonly used coordinate system which covers the entire space is given by the coordinates t, and the hyperpolar coordinates α, θ and φ.

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-de Sitter 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 region 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.