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.

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 mass are equivalent (as expressed in the equation E = mc²). Space and time values can be converted into time or space units by multiplying or dividing the value by the speed of light (e.g., seconds times meters per second equals meters).

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 ${\displaystyle \textstyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }$ (i.e. the gravitational pull between two objects equals the gravitational constant times the product of their masses divided by the square of the distance between them) is merely an approximation of the gravity effects seen in general relativity. However this approximation becomes inaccurate in extreme physical situations, like relativistic speeds (light, in particular), or very large & dense masses.

In general relativity, gravity is caused by spacetime being curved ("distorted"). It is a common misconception to attribute gravity to curved space; neither space nor time has an absolute meaning in relativity. 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 every-day 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 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 momentarily parallel timelike geodesics^[4] 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 spacetime curvature changed. 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^[4] 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.

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

The analogy used above describes curvature of a two-dimensional space caused by gravity in a flat ambient space of one dimension higher. Similarly, the (curved) de Sitter and anti-de Sitter spaces of four dimensions can be embedded into a (flat) Pseudo-Riemannian space of five dimensions. This allows distances and angles within the embedded space to be directly determined from those in the five-dimensional flat space.

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.

Closed timelike curves and the universal cover

When q ≥ 1, the embedding above has closed timelike curves; for example, the path parameterized by ${\displaystyle t_{1}=\alpha \sin(\tau ),t_{2}=\alpha \cos(\tau ),}$ 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 contains 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 quasi-sphere itself, while others define it as equivalent to the universal cover of the embedding.

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.

Instability

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.^[5] Mathematician Georgios Moschidis proved that given spherical symmetry, the conjecture holds true for the specific cases of the Einstein-null dust system with an internal mirror (2017) and the Einstein-massless Vlasov system (2018).^[6][7]

Global coordinates

${\displaystyle \mathrm {AdS} _{n}}$is parametrized in global coordinates by the parameters ${\displaystyle (\tau ,\rho ,\theta ,\varphi _{1},\cdots ,\varphi _{n-3})}$ as:

${\displaystyle {\begin{cases}X_{1}=\alpha \cosh \rho \cos \tau \\X_{2}=\alpha \cosh \rho \sin \tau \\X_{i}=\alpha \sinh \rho \,{\hat {x}}_{i}\qquad \sum _{i}{\hat {x}}_{i}^{2}=1\end{cases}}}$,

where ${\displaystyle {\hat {x}}_{i}}$ parametrize a ${\displaystyle S^{n-2}}$ sphere, and in terms of the coordinates ${\displaystyle \varphi _{i}}$ they are ${\displaystyle {\hat {x}}_{1}=\sin \theta \sin \varphi _{1}\cdots \sin \varphi _{n-3}}$, ${\displaystyle {\hat {x}}_{2}=\sin \theta \sin \varphi _{1}\cdots \cos \varphi _{n-3}}$, ${\displaystyle {\hat {x}}_{3}=\sin \theta \sin \varphi _{1}\cdots \cos \varphi _{n-2}}$ and so on. The ${\displaystyle \mathrm {AdS} _{n}}$ metric in these coordinates is:

${\displaystyle ds^{2}=\alpha ^{2}\left(-\cosh ^{2}\rho \,d\tau ^{2}+\,d\rho ^{2}+\sinh ^{2}\rho \,d\Omega _{n-2}^{2}\right)}$

where ${\displaystyle \tau \in [0,2\pi ]}$ and ${\displaystyle \rho \in \mathbb {R} ^{+}}$. Considering the periodicity of time ${\displaystyle \tau }$ and in order to avoid closed timelike curves (CTC), one should take the universal cover ${\displaystyle \tau \in \mathbb {R} }$. In the limit ${\displaystyle \rho \to \infty }$ one can approach to the boundary of this spacetime usually called ${\displaystyle \mathrm {AdS} _{n}}$ conformal boundary.

With the transformations ${\displaystyle r\equiv \alpha \sinh \rho }$ and ${\displaystyle t\equiv \alpha \tau }$ we can have the usual ${\displaystyle \mathrm {AdS} _{n}}$ metric in global coordinates:

${\displaystyle ds^{2}=-f(r)\,dt^{2}+{\frac {1}{f(r)}}\,dr^{2}+r^{2}\,d\Omega _{n-2}^{2}}$

where ${\displaystyle f(r)=1+{\frac {r^{2}}{\alpha ^{2}}}}$

Poincaré coordinates

By the following parametrization:

${\displaystyle {\begin{cases}X_{1}={\frac {\alpha ^{2}}{2r}}\left(1+{\frac {r^{2}}{\alpha ^{4}}}\left(\alpha ^{2}+{\vec {x}}^{2}-t^{2}\right)\right)\\X_{2}={\frac {r}{\alpha }}t\\X_{i}={\frac {r}{\alpha }}x_{i}\qquad i\in \{3,\ldots ,n\}\\X_{n+1}={\frac {\alpha ^{2}}{2r}}\left(1-{\frac {r^{2}}{\alpha ^{4}}}\left(\alpha ^{2}-{\vec {x}}^{2}+t^{2}\right)\right)\end{cases}},}$

the ${\displaystyle \mathrm {AdS} _{n}}$ metric in the Poincaré coordinates is:

${\displaystyle ds^{2}=-{\frac {r^{2}}{\alpha ^{2}}}\,dt^{2}+{\frac {\alpha ^{2}}{r^{2}}}\,dr^{2}+{\frac {r^{2}}{\alpha ^{2}}}\,d{\vec {x}}^{2}}$

in which ${\displaystyle 0\leq r}$. The codimension 2 surface ${\displaystyle r=0}$ is the Poincaré Killing horizon and ${\displaystyle r\to \infty }$ approaches to the boundary of ${\displaystyle \mathrm {AdS} _{n}}$ spacetime. So unlike the global coordinates, the Poincaré coordinates do not cover all ${\displaystyle \mathrm {AdS} _{n}}$ manifold. Using ${\displaystyle u\equiv {\frac {r}{\alpha ^{2}}}}$ this metric can be written in the following way:

${\displaystyle ds^{2}=\alpha ^{2}\left({\frac {\,du^{2}}{u^{2}}}+u^{2}\,dx_{\mu }\,dx^{\mu }\right)}$

where ${\displaystyle x^{\mu }=\left(t,{\vec {x}}\right)}$. By the transformation ${\displaystyle z\equiv {\frac {1}{u}}}$ also it can be written as:

${\displaystyle ds^{2}={\frac {\alpha ^{2}}{z^{2}}}\left(\,dz^{2}+\,dx_{\mu }\,dx^{\mu }\right).}$

This latter coordinates are the coordinates which are usually used in AdS/CFT correspondence, with the boundary of AdS at ${\displaystyle z\to 0}$.

FRW open slicing coordinates

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). The spatial geometry must be negatively curved (open) and the metric is

${\displaystyle ds^{2}=-dt^{2}+\alpha ^{2}\sin ^{2}(t/\alpha )dH_{n-1}^{2},}$

where ${\displaystyle dH_{n-1}^{2}=d\rho ^{2}+\sinh ^{2}\rho d\Omega _{n-2}^{2}}$ is the standard metric on the ${\displaystyle (n-1)}$-dimensional hyperbolic plane. Of course, this does not cover all of AdS. These coordinates are related to the global embedding coordinates by

${\displaystyle {\begin{cases}X_{1}=\alpha \cos(t/\alpha )\\X_{2}=\alpha \sin(t/\alpha )\cosh \rho \\X_{i}=\alpha \sin(t/\alpha )\sinh \rho \,{\hat {x}}_{i}\qquad 3\leq i\leq n+1\end{cases}}}$

where ${\displaystyle \sum _{i}{\hat {x}}_{i}^{2}=1}$ parameterize the ${\displaystyle S^{n-3}}$.

de Sitter slicing

Let

${\displaystyle {\begin{aligned}X_{1}&=\alpha \sinh \left({\frac {\rho }{\alpha }}\right)\sinh \left({\frac {t}{\alpha }}\right)\cosh \xi ,\\X_{2}&=\alpha \cosh \left({\frac {\rho }{\alpha }}\right),\\X_{3}&=\alpha \sinh \left({\frac {\rho }{\alpha }}\right)\cosh \left({\frac {t}{\alpha }}\right),\\X_{i}&=\alpha \sinh \left({\frac {\rho }{\alpha }}\right)\sinh \left({\frac {t}{\alpha }}\right)\sinh \xi \,{\hat {x}}_{i},\qquad 4\leq i\leq n+1\end{aligned}}}$

where ${\displaystyle \sum _{i}{\hat {x}}_{i}^{2}=1}$ parameterize the ${\displaystyle S^{n-3}}$. Then the metric reads:

${\displaystyle ds^{2}=d\rho ^{2}+\sinh ^{2}\left({\frac {\rho }{\alpha }}\right)ds_{dS,\alpha ,n-1}^{2},}$

where

${\displaystyle ds_{dS,\alpha ,n-1}^{2}=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {t}{\alpha }}\right)dH_{n-2}^{2}}$

is the metric of an ${\displaystyle n-1}$ dimensional de Sitter space with radius of curvature ${\displaystyle \alpha }$ in open slicing coordinates. The hyperbolic metric is given by:

${\displaystyle dH_{n-2}^{2}=d\xi ^{2}+\sinh ^{2}(\xi )d\Omega _{n-3}^{2}.}$

Geometric properties

${\displaystyle \mathrm {AdS} _{n}}$ metric with radius ${\displaystyle \alpha }$ is one of the maximal symmetric n-dimensional spacetimes. It has the following geometric properties:

Riemann curvature tensor

${\displaystyle R_{\mu \nu \alpha \beta }={\frac {-1}{\alpha ^{2}}}(g_{\mu \alpha }g_{\nu \beta }-g_{\mu \beta }g_{\nu \alpha })}$

Ricci curvature

${\displaystyle R_{\mu \nu }={\frac {-(n-1)}{\alpha ^{2}}}g_{\mu \nu }}$

Scalar curvature

${\displaystyle R={\frac {-n(n-1)}{\alpha ^{2}}}}$