Anti de Sitter Space - Definition and Properties

Definition and Properties

Much as elliptical and hyperbolic spaces can be visualized by an isometric embedding in a flat space of one higher dimension (as the sphere and pseudosphere respectively), anti de Sitter space can be visualized as the Lorentzian analogue of a sphere in a space of one additional dimension. To a physicist the extra dimension is timelike, while to a mathematician it is negative; in this article we adopt the convention that timelike dimensions are negative so that these notions coincide.

The anti de Sitter space of signature (p,q) can then be isometrically embedded in the space with coordinates (x1, ..., xp, t1, ..., tq+1) and the pseudometric

as the sphere

where is a nonzero constant with dimensions of length (the radius of curvature). Note that this is a sphere in the sense that it is a collection of points at constant metric distance from the origin, but visually it is a hyperboloid, as in the image shown.

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.

When q = 0, this construction gives ordinary hyperbolic space. The remainder of the discussion applies when q ≥ 1.

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