Anti-de Sitter space
In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature.
Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.
Paul Dirac was the first person to rigorously explore anti-de Sitter space, doing so in 1963.
As such, they are exact solutions of the Einstein field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively.
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.
Of course, in general relativity, both the small and large objects mutually influence the curvature of spacetime.
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.
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.
However this approximation becomes inaccurate in extreme physical situations, like relativistic speeds (light, in particular), or very large & dense masses.
Nevertheless, to describe weak gravity, as on the Earth, it is sufficient to consider time distortion in a particular coordinate system.
We find gravity on the Earth very noticeable while relativistic time distortion requires precision instruments to detect.
The reason why we do not become aware of relativistic effects in our everyday life is the huge value of the speed of light (c = 300000 km/s approximately), which makes us perceive space and time as different entities.
De Sitter space involves a variation of general relativity in which spacetime is slightly curved in the absence of matter or energy.
An intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity.
This spacetime geometry results in momentarily parallel timelike geodesics[b] diverging, with spacelike sections having positive curvature.
In anti-de Sitter space, in the absence of matter or energy, the curvature of spacelike sections is negative, corresponding to a hyperbolic geometry, and momentarily parallel timelike geodesics[b] eventually intersect.
This corresponds to a negative cosmological constant, where empty space itself has negative energy density but positive pressure, unlike the standard ΛCDM model of our own universe for which observations of distant supernovae indicate a positive cosmological constant corresponding to (asymptotic) de Sitter space.
[5] Such a relabelling reverses the sign of the curvature, which is conventionally referenced to the directions that are labelled spacelike.
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.
The mathematical description of anti-de Sitter space generalizes the idea of curvature.
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.
This is a (generalized) sphere in the sense that it is a collection of points for which the "distance" (determined by the quadratic form) from the origin is constant, but visually it is a hyperboloid, as in the image shown.
The unproven "AdS instability conjecture" introduced by the physicists Piotr Bizon and Andrzej Rostworowski in 2011 states that arbitrarily small perturbations of certain shapes in AdS lead to the formation of black holes.
The constant time slices of this coordinate patch are hyperbolic spaces in the Poincaré half-space metric.
The adjacent image represents the "half-space" region of anti-de Sitter space and its boundary.
is an n-dimensional vacuum solution for the theory of gravitation with Einstein–Hilbert action with negative cosmological constant
), i.e. the theory described by the following Lagrangian density: where G(n) is the gravitational constant in n-dimensional spacetime.
and in order to avoid closed timelike curves (CTC), one should take the universal cover
Since AdS is maximally symmetric, it is also possible to cast it in a spatially homogeneous and isotropic form like FRW spacetimes (see Friedmann–Lemaître–Robertson–Walker metric).
