Anti-de Sitter space

Anti de Sitter space

In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analog of n-dimensional hyperbolic space, just as Minkowski space and de Sitter space are the analogs of Euclidean and elliptical spaces respectively. It is best known for its role in the AdS/CFT correspondence.

In the language of general relativity, anti de Sitter space is a maximally symmetric, vacuum solution of Einstein's field equation with an attractive cosmological constant Lambda (corresponding to a negative vacuum energy density and positive pressure).

In mathematics, anti de Sitter space is sometimes defined more generally as a space of arbitrary signature (p,q). Generally in physics only the case of one timelike dimension is relevant. Because of differing sign conventions, this may correspond to a signature of either (n−1, 1) or (1, n−1).

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 mathbb{R}^{p,q+1} with coordinates (x1, ..., xp, t1, ..., tq+1) and the pseudometric

ds^2 = sum_{i=1}^p dx_i^2 - sum_{j=1}^{q+1} dt_j^2
as the sphere
sum_{i=1}^p x_i^2 - sum_{j=1}^{q+1} t_j^2 = -alpha^2
where alpha 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.

Closed timelike curves and the universal cover

When q ≥ 1, the embedding above has closed timelike curves; for example, the path parameterized by t_1 = alpha sin(tau), t_2 = alpha cos(tau), and all other coordinates zero is such a curve. When q ≥ 2 these curves are inherent to the geometry (unsurprisingly, as any space with more than one temporal dimension will contain closed timelike curves), but when q = 1, they can be eliminated by passing to the universal covering space, effectively "unrolling" the embedding. A similar situation occurs with the pseudosphere, which curls around on itself although the hyperbolic plane does not; as a result it contains self-intersecting straight lines (geodesics) while the hyperbolic plane does not. Some authors define anti de Sitter space as equivalent to the embedded sphere itself, while others define it as equivalent to the universal cover of the embedding. Generally the latter definition is the one of interest in physics.

Symmetries

If the universal cover is not taken, (p,q) anti de Sitter space has O(p,q+1) as its isometry group. If the universal cover is taken the isometry group is a cover of O(p,q+1).

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_idx_i^2right).

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 dy^2=left(dt^2-sum_idx_i^2right); 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, r geqslant 0 and the hyperpolar coordinates α, θ and φ.

ds^2 = - left(k^2r^2 + 1right)dt^2 + frac{1}{k^2r^2+1}dr^2 + r^2 dOmega^2

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.

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 with parity aka reflectional symmetry and time reversal symmetry can be seen as a quotient of two groups AdS_n=frac{ O(2,n-1) }{ O(1,n-1) } whereas AdS without P or C can be seen as frac{ Spin^+(2,n-1) }{ Spin^+(1,n-1) }.

This quotient formulation gives to AdS_n a 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 0cdots leftarrow v^trightarrow end{pmatrix} begin{pmatrix}
      vdots & uparrow
        0    & v 
      vdots & downarrow
end{pmatrix} & B end{pmatrix} , where B is a skew-symmetric 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^trightarrow cdots 0cdots end{pmatrix} begin{pmatrix}
     uparrow   & vdots
         w      &  0
     downarrow & vdots
end{pmatrix} & 0 end{pmatrix}. These two fulfil mathcal{G}=mathcal{H}oplusmathcal{Q} . Then explicit matrix computation shows that [mathcal{H},mathcal{Q}]subseteqmathcal{Q}, quad [mathcal{Q},mathcal{Q}]subseteqmathcal{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.
  • Ellis, G. F. R.; Hawking, S. W. The large scale structure of space-time. Cambridge university press (1973). (see pages 131-134).
  • Frances, C: The conformal boundary of anti-de Sitter space-times AdS/CFT correspondence: Einstein metrics and their conformal boundaries, 205--216, IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005.
  • 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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