Anti-de Sitter space

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 embedded in a space of one additional dimension, in which the additional dimension is a second timelike dimension. In this article we adopt the convention that the metric tensor in a timelike direction is negative.

The anti-de Sitter space of signature (p, 1) can then be isometrically embedded in the space ${\displaystyle {\mathbb{R}}^{p,2}}$ with coordinates (x₁, ..., x_p, t₁, t₂) and the metric

${\displaystyle d{s}^2=\sum _{i=1}^{p}d{x}_i^2-d{t}_1^2-d{t}_2^2}$

as the quasi-sphere

${\displaystyle \sum _{i=1}^p{x}_i^2-t_1^2-t_2^2=-{\alpha }^2,}$

where α is a nonzero constant with dimensions of length (the radius of curvature). Every point in the embedding has a fixed "distance" (as determined by the quadratic form) from the origin, but may be depicted as a hyperboloid, as in the image shown.

The metric on anti-de Sitter space is that induced from the ambient metric. It is nondegenerate and has Lorentzian signature.

Closed timelike curves and the universal cover

The embedding above has closed timelike curves; for example, the path parameterized by t₁ = α sin(τ), t₂ = α cos(τ), and all other coordinates zero, is such a curve. Such curves 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, 1) anti-de Sitter space has O(p, 2) as its isometry group. If the universal cover is taken, the isometry group is a cover of O(p, 2). 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. 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).

Coordinate patches

A coordinate patch covering part of the space gives the half-space coordinatization of anti-de Sitter space, and is similar to the Poincaré half-plane model of hyperbolic space but with one of the terms in the metric negated, corresponding to one of the tangent directions of the boundary of the half-space. The metric for this patch is

${\displaystyle d{s}^2=\frac{1}{y^2}\left(-d{t}^2+d{y}^2+\sum _{i}d{x}_i^2\right),}$

with ${\displaystyle y>0}$ giving the half-space. 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-space metric. In the limit as ${\displaystyle y\to 0}$, this half-space metric is conformally equivalent to the Minkowski metric $d{s}^2=-d{t}^2+\sum _{i}d{x}_i^2$. 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.

The adjacent image 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

${\displaystyle S^2=\mathrm{O}(3)/\mathrm{O}(2)}$

is a quotient of two orthogonal groups, anti-de Sitter space with parity (reflectional symmetry) and time-reversal symmetry can be seen as a quotient of two generalized orthogonal groups

${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n=\mathrm{O}(2,n-1)/\mathrm{O}(1,n-1)}$

whereas AdS without P or C can be seen as the quotient

${\displaystyle {\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(2,n-1)/{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(1,n-1)}$

of spin groups.

This quotient formulation gives ${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ the structure of a homogeneous space. The Lie algebra of the generalized orthogonal group ${\displaystyle \mathcal{o}(1,n)}$ is given by matrices

${\displaystyle \mathcal{H}=\left(\begin{array}{cc}\begin{array}{cc}0& 0\\ 0& 0\end{array}& \left(\begin{array}{c}\cdots 0\cdots \\ \leftarrow v^{\text{t}}\to \end{array}\right)\\ \left(\begin{array}{cc}⋮& \uparrow \\ 0& v\\ ⋮& \downarrow \end{array}\right)& B\end{array}\right)}$,

where ${\displaystyle B}$ is a skew-symmetric matrix. A complementary generator in the Lie algebra of ${\displaystyle \mathcal{G}=\mathcal{o}(2,n)}$ is

${\displaystyle \mathcal{Q}=\left(\begin{array}{cc}\begin{array}{cc}0& a\\ -a& 0\end{array}& \left(\begin{array}{c}\leftarrow w^{\text{t}}\to \\ \cdots 0\cdots \end{array}\right)\\ \left(\begin{array}{cc}\uparrow & ⋮\\ w& 0\\ \downarrow & ⋮\end{array}\right)& 0\end{array}\right).}$

These two fulfill ${\displaystyle \mathcal{G}=\mathcal{H}\oplus \mathcal{Q}}$. Explicit matrix computation shows that ${\displaystyle [\mathcal{H},\mathcal{Q}]\subseteq \mathcal{Q}}$ and ${\displaystyle [\mathcal{Q},\mathcal{Q}]\subseteq \mathcal{H}}$. Thus, anti-de Sitter space is a reductive homogeneous space, and a non-Riemannian symmetric space.

An overview of AdS spacetime in physics and its properties

${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ is an n-dimensional vacuum solution for the theory of gravitation with Einstein–Hilbert action with negative cosmological constant ${\displaystyle \mathrm{\Lambda }}$, (${\displaystyle \mathrm{\Lambda }<0}$), i.e. the theory described by the following Lagrangian density:

${\displaystyle \mathcal{L}=\frac{1}{16\pi G_{(n)}}(R-2\mathrm{\Lambda })}$,

where G_(n) is the gravitational constant in n-dimensional spacetime. Therefore, it is a solution of the Einstein field equations:

${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=0,}$

where ${\displaystyle G_{\mu \nu }}$ is the Einstein tensor and ${\displaystyle g_{\mu \nu }}$ is the metric of the spacetime. Introducing the radius ${\displaystyle \alpha }$ as $\mathrm{\Lambda }=\frac{-1}{{\alpha }^2}\frac{(n-1)(n-2)}{2}$, this solution can be immersed in an ${\displaystyle (n+1)}$-dimensional flat spacetime with the metric ${\displaystyle \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,-1,+1,\dots ,+1)}$ in coordinates ${\displaystyle (X_1,X_2,X_3,\dots ,X_{n+1})}$ by the following constraint:

${\displaystyle -X_1^2-X_2^2+\sum _{i=3}^{n+1}{X}_i^2=-{\alpha }^2.}$

Global coordinates

${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ is parametrized in global coordinates by the parameters ${\displaystyle (\tau ,\rho ,\theta ,{\phi }_1,\dots ,{\phi }_{n-3})}$ as:

${\displaystyle \{\begin{array}{l}{X}_1=\alpha \cosh \rho \cos \tau \\ X_2=\alpha \cosh \rho \sin \tau \\ X_i=\alpha \sinh \rho {\hat{x}}_i\sum _i{\hat{x}}_i^2=1\end{array}}$,

where ${\displaystyle {\hat{x}}_i}$ parametrize a ${\displaystyle S^{n-2}}$ sphere, and in terms of the coordinates ${\displaystyle {\phi }_i}$ they are ${\displaystyle {\hat{x}}_1=\sin \theta \sin {\phi }_1\dots \sin {\phi }_{n-3}}$, ${\displaystyle {\hat{x}}_2=\sin \theta \sin {\phi }_1\dots \cos {\phi }_{n-3}}$, ${\displaystyle {\hat{x}}_3=\sin \theta \sin {\phi }_1\dots \cos {\phi }_{n-2}}$ and so on. The ${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ metric in these coordinates is:

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

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 ${\displaystyle \rho \to \mathrm{\infty }}$ one can approach to the boundary of this spacetime usually called ${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ conformal boundary.

With the transformations ${\displaystyle r\equiv \alpha \sinh \rho }$ and ${\displaystyle t\equiv \alpha \tau }$ we can have the usual ${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ metric in global coordinates:

${\displaystyle d{s}^2=-f(r)d{t}^2+\frac{1}{f(r)}d{r}^2+r^{2}d{\mathrm{\Omega }}_{n-2}^2}$

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

Hyperboloid model + time coordinate

If we take the formulation of AdS space from the Definition and properties section and convert the coordinates ${\displaystyle t_1}$ and ${\displaystyle t_2}$ to polar coordinates where the radial coordinate is ${\displaystyle x_0}$ and the angular coordinate is ${\displaystyle \phi }$, such that ${\displaystyle t_1=x_0\cos \phi }$ and ${\displaystyle t_2=x_0\sin \phi }$, where the space is periodic in ${\displaystyle \phi }$ with the period ${\displaystyle 2\pi }$, the quasi-sphere becomes

${\displaystyle \sum _{i=1}^p{x}_i^2-x_0^2=-{\alpha }^2,}$

and the metric becomes

${\displaystyle d{s}^2=\sum _{i=1}^{p}d{x}_i^2-d{x}_0^2-(x_{0}d\phi {)}^2.}$

We see that the quasi-sphere takes on the same equation as the manifold in the hyperboloid model of p-dimensional hyperbolic space, while the metric also takes on the equation of the metric in the hyperboloid model, except for the extra term ${\displaystyle -(x_{0}d\phi {)}^2}$. This tells us that the quasi-sphere is translationally symmetric in the ${\displaystyle \phi }$ direction, and that curves in the quasi-sphere with a fixed ${\displaystyle \phi }$-value are spacelike, while ${\displaystyle \phi }$ itself is timelike, where the proper time ${\displaystyle \tau }$ evolves faster the greater ${\displaystyle x_0}$ is as ${\displaystyle \phi }$ increases if all other coordinates are fixed, i.e.

${\displaystyle {\frac{d\tau }{d\phi }|}_{x_0,x_1,\dots ,x_p}=x_0.}$

If we write ${\displaystyle \phi }$ as a function of ${\displaystyle t_1}$ and ${\displaystyle t_2}$, we see that the branch point is at ${\displaystyle t_1=t_2=0}$, but since the constraint posed by the quasi-sphere requires that ${\displaystyle t_1^2+t_2^2\ge 1}$, the branch point exists outside of the quasi-sphere, and we can therefore—just like in global coordinates—unwrap the space into its universal cover by removing the requirement that the space is periodic in ${\displaystyle \phi }$, and in that way avoid closed timelike curves (CTC).

Poincaré coordinates

By the following parametrization:

${\displaystyle \{\begin{array}{l}{X}_1=\frac{{\alpha }^2}{2r}\left(1+\frac{r^2}{{\alpha }^4}\left({\alpha }^2+{\overrightarrow{x}}^2-t^2\right)\right)\\ X_2=\frac{r}{\alpha }t\\ X_i=\frac{r}{\alpha }{x}_{i}i\in \{3,\dots ,n\}\\ X_{n+1}=\frac{{\alpha }^2}{2r}\left(1-\frac{r^2}{{\alpha }^4}\left({\alpha }^2-{\overrightarrow{x}}^2+t^2\right)\right)\end{array},}$

the ${\displaystyle {\mathrm{A}\mathrm{d}\mathrm{S}}_n}$ metric in the Poincaré coordinates is:

${\displaystyle d{s}^2=-\frac{r^2}{{\alpha }^2}d{t}^2+\frac{{\alpha }^2}{r^2}d{r}^2+\frac{r^2}{{\alpha }^2}d{\overrightarrow{x}}^2}$

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

${\displaystyle d{s}^2={\alpha }^2\left(\frac{d{u}^2}{u^2}+u^{2}d{x}_{\mu }d{x}^{\mu }\right)}$

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

${\displaystyle d{s}^2=\frac{{\alpha }^2}{z^2}\left(d{z}^2+d{x}_{\mu }d{x}^{\mu }\right).}$

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

${\displaystyle d{s}^2=-d{t}^2+{\alpha }^2{\sin }^2(t/\alpha )d{H}_{n-1}^2,}$

where ${\displaystyle d{H}_{n-1}^2=d{\rho }^2+{\sinh }^2\rho d{\mathrm{\Omega }}_{n-2}^2}$ 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

${\displaystyle \{\begin{array}{l}{X}_1=\alpha \cos (t/\alpha )\\ X_2=\alpha \sin (t/\alpha )\cosh \rho \\ X_i=\alpha \sin (t/\alpha )\sinh \rho {\hat{x}}_{i}3\le i\le n+1\end{array}}$

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

De Sitter slicing

Let

${\displaystyle \begin{array}{cc}{X}_1& =\alpha \sinh \left(\frac{\rho }{\alpha }\right)\sinh \left(\frac{t}{\alpha }\right)\cosh \xi ,\\ X_2& =\alpha \cosh \left(\frac{\rho }{\alpha }\right),\\ X_3& =\alpha \sinh \left(\frac{\rho }{\alpha }\right)\cosh \left(\frac{t}{\alpha }\right),\\ X_i& =\alpha \sinh \left(\frac{\rho }{\alpha }\right)\sinh \left(\frac{t}{\alpha }\right)\sinh \xi {\hat{x}}_i,4\le i\le n+1\end{array}}$

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

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

where

${\displaystyle d{s}_{dS,\alpha ,n-1}^2=-d{t}^2+{\alpha }^2{\sinh }^2\left(\frac{t}{\alpha }\right)d{H}_{n-2}^2}$

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:

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

Geometric properties

AdS_n metric with radius ${\displaystyle \alpha }$ is one of the maximally 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

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

• Scalar curvature

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

Generalization

Any member of the family of maximally symmetric spaces of nonzero curvature can be isometrically embedded in the space ${\displaystyle {\mathbb{R}}^{p+1,q}}$ with coordinates (x₀, ..., x_p, y₁, ..., y_q) and with the metric

${\displaystyle d{s}^2=\sum _{i=0}^{p}d{x}_i^2-\sum _{j=1}^{q}d{y}_j^2}$

as the quasi-sphere

${\displaystyle \sum _{i=0}^p{x}_i^2-\sum _{j=1}^q{y}_j^2={\alpha }^2,}$

where α is a nonzero constant of metrological dimension length (the radius of curvature).

The metric is that induced from the ambient metric, has signature (p, q) and is nondegenerate.

The following subfamilies of these spaces my be identified:

• q = 0: the hyperspherical space S^p
• p = 0: (a double cover of) the hyperbolic space H^q‍
• q = 1: the de Sitter space dS_p+1
• p = 1: the anti-de Sitter space AdS_q+1

A space constructed in this way and has O(p + 1, q) as its isometry group. It is a symmetric space, and may be constructed by using quotient space construction.