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In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. The anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in the 1920s in Leiden on the spacetime structure of the universe.
Contents
- Non technical explanation
- Technical terms translated
- Spacetime in general relativity
- de Sitter space in general relativity
- Anti de Sitter space distinguished from de Sitter space
- de Sitter space and anti de Sitter space viewed as embedded in five dimensions
- Caveats
- Definition and properties
- Closed timelike curves and the universal cover
- Symmetries
- Coordinate patches
- As a homogeneous symmetric space
- A simple definition for anti de Sitter space and its properties
- Global coordinates
- Poincar coordinates
- Geometric properties
- References
Manifolds of constant curvature are most familiar in the case of two dimensions, where the surface of a sphere is a surface of constant positive curvature, a flat (Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature.
Einstein's theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of Einstein's field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively.
Anti-de Sitter space generalises to any number of space dimensions. In higher dimensions, it is best known for its role in the AdS/CFT correspondence, which suggests that it is possible to describe a force in quantum mechanics (like electromagnetism, the weak force or the strong force) in a certain number of dimensions (for example four) with a string theory where the strings exist in an anti-de Sitter space, with one additional dimension.
Non-technical explanation
This non-technical explanation first defines the terms used in the introductory material of this entry. Then, it briefly sets forth the underlying idea of a general relativity-like spacetime. Then it discusses how de Sitter space describes a distinct variant of the ordinary spacetime of general relativity (called Minkowski space) related to the cosmological constant, and how anti-de Sitter space differs from de Sitter space. It also explains that Minkowski space, de Sitter space and anti-de Sitter space, as applied to general relativity, can all be thought of as being embedded in a flat five-dimensional spacetime. Finally, it offers some caveats that describe in general terms how this non-technical explanation fails to capture the full detail of the mathematical concept.
Technical terms translated
A maximally symmetric Lorentzian manifold is a spacetime in which no point in space and time can be distinguished in any way from another, and (being Lorentzian) the only way in which a direction (or tangent to a path at a spacetime point) can be distinguished is whether it is spacelike, lightlike or timelike. The space of special relativity (Minkowski space) is an example.
A constant scalar curvature means a general relativity gravity-like bending of spacetime that has a curvature described by a single number that is the same everywhere in spacetime in the absence of matter or energy.
Negative curvature means curved hyperbolically, like a saddle surface or the Gabriel's Horn surface, similar to that of a trumpet bell. It might be described as being the "opposite" of the surface of a sphere, which has a positive curvature.
Spacetime in general relativity
General relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from the presence of matter or energy. Energy and matter are equivalent (as expressed in the equation E = mc2), and space and time can be translated into equivalent units based on the speed of light (c in the E = mc2 equation).
A common analogy involves the way that a dip in a flat sheet of rubber, caused by a heavy object sitting on it, influences the path taken by small objects rolling nearby, causing them to deviate inward from the path they would have followed had the heavy object been absent. Of course, in general relativity, both the small and large objects mutually influence the curvature of spacetime.
The attractive force of gravity created by matter is due to a negative curvature of spacetime, represented in the rubber sheet analogy by the negatively curved (trumpet-bell-like) dip in the sheet.
A key feature of general relativity is that it describes gravity not as a conventional force like electromagnetism, but as a change in the geometry of spacetime that results from the presence of matter or energy.
The analogy used above describes the curvature of a two-dimensional space caused by gravity in general relativity in a three-dimensional superspace in which the third dimension corresponds to the effect of gravity. A geometrical way of thinking about general relativity describes the effects of the gravity in the real world four-dimensional space geometrically by projecting that space into a five-dimensional superspace with the fifth dimension corresponding to the curvature in spacetime that is produced by gravity and gravity-like effects in general relativity.
As a result, in general relativity, the familiar Newtonian equation of gravity
Some of the differences between the familiar Newtonian equation of gravity and the predictions of general relativity flow from the fact that gravity in general relativity bends both time and space, not just space. In normal circumstances, gravity bends time so slightly that the differences between Newtonian gravity and general relativity are detectable only with precise instruments.
de Sitter space in general relativity
de Sitter space involves a variation of general relativity in which spacetime is slightly curved in the absence of matter or energy. This is analogous to the relationship between Euclidean geometry and non-Euclidean geometry.
An intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity. This corresponds to the vacuum having an energy density and pressure. This spacetime geometry results in initially parallel timelike geodesics diverging, with spacelike sections having positive curvature.
Anti-de Sitter space distinguished from de Sitter space
An anti-de Sitter space in general relativity is similar to a de Sitter space, except with the sign of the curvature changed. In this case, in the absence of matter or energy, the curvature of spacelike sections is negative, corresponding to a hyperbolic geometry, and initially parallel timelike geodesics eventually intersect. This corresponds to a negative cosmological constant (which does not match cosmological observations). Here, empty space itself has negative energy density but positive pressure.
In an anti-de Sitter space, as in a de Sitter space, the inherent spacetime curvature corresponds to the cosmological constant.
de Sitter space and anti-de Sitter space viewed as embedded in five dimensions
As noted above, the analogy used above describes curvature of a two-dimensional space caused by gravity in general relativity in a three-dimensional embedding space that is flat, like the Minkowski space of special relativity. Embedding de Sitter and anti-de Sitter spaces of five flat dimensions allows the properties of the embedded spaces to be determined. Distances and angles within the embedded space may be directly determined from the simpler properties of the five-dimensional flat space.
While anti-de Sitter space does not correspond to gravity in general relativity with the observed cosmological constant, an anti-de Sitter space is believed to correspond to other forces in quantum mechanics (like electromagnetism, the weak nuclear force and the strong nuclear force). This is called the AdS/CFT correspondence.
Caveats
The remainder of this article explains the details of these concepts with a much more rigorous and precise mathematical and physical description. People are ill suited to visualizing things in five or more dimensions, but mathematical equations are not similarly challenged and can represent five-dimensional concepts in a way just as appropriate as the methods that mathematical equations use to describe easier to visualize three and four-dimensional concepts.
There is a particularly important implication of the more precise mathematical description that differs from the analogy-based heuristic description of de Sitter space and anti-de Sitter space above. The mathematical description of anti-de Sitter space generalizes the idea of curvature. In the mathematical description, curvature is a property of a particular point and can be divorced from some invisible surface to which curved points in spacetime meld themselves. So, for example, concepts like singularities (the most widely known of which in general relativity is the black hole) which cannot be expressed completely in a real world geometry, can correspond to particular states of a mathematical equation.
The full mathematical description also captures some subtle distinctions made in general relativity between space-like dimensions and time-like dimensions.
Definition and properties
Much as spherical 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. In this article we adopt the convention that the metric in a timelike direction is negative.
The anti-de Sitter space of signature (p, q) can then be isometrically embedded in the space
as the quasi-sphere
where
The metric on anti-de Sitter space is that induced from the ambient metric. It is nondegenerate and has Lorentzian signature.
When q = 0, this construction gives a standard 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
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). This is most easily understood by defining anti-de Sitter space as a symmetric space, using the quotient space construction, given below.
Coordinate patches
A coordinate patch covering part of the space gives the half-space coordinatization of anti-de Sitter space. The metric tensor for this patch is
with
The constant time slices of this coordinate patch are hyperbolic spaces in the Poincaré half-space metric. In the limit as
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,
The image on the right represents the "half-space" region of anti-de Sitter 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.
As a homogeneous, symmetric space
In the same way that the 2-sphere
is a quotient of two orthogonal groups, anti-de Sitter with parity (reflectional symmetry) and time reversal symmetry can be seen as a quotient of two generalized orthogonal groups
whereas AdS without P or C can be seen as the quotient
of spin groups.
This quotient formulation gives
where
These two fulfill
A simple definition for anti-de Sitter space and its properties
where G(n) is the gravitational constant in n-dimensional spacetime. Therefore, it is a solution of the Einstein field equations:
where
Global coordinates
where
where
With the transformations
where
Poincaré coordinates
By the following parametrization:
the
in which
where
Geometric properties