Anti-de Sitter space

In mathematics and physics, n-dimensional anti-de Sitter space (AdS_n) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analogue of hyperbolic space, just as Minkowski space is the analogue of Euclidean space and de Sitter space is the analogue of elliptical space.

It is best known for its role in the AdS/CFT correspondence. 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.

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

In mathematics, anti-de Sitter space is sometimes defined more generally as a space of arbitrary metric signature (p, q). In physics, often only the case of one timelike dimension is considered. This corresponds to the equivalent metric signatures (n−1, 1) and (1, n−1), where the choice is by a sign convention.

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 corresponds to a general relativity-like spacetime in which time and space in all directions are mathematically equivalent.

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. A negative curvature corresponds to an attractive force, while a positive curvature such as a sphere corresponds to a repulsive force.

The AdS/CFT (anti-de Sitter space/conformal field theory) correspondence is an idea originally proposed by Juan Maldacena in late 1997. The AdS/CFT correspondence is the idea that it is possible in general 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.

A quantum field theory is a set of equations and rules for using them of the kind used in quantum mechanics to describe forces (such as electromagnetism, the weak force and the strong force) in a way that is mathematically unstable (see stability theory).

A conformal field theory is basically a quantum field theory that is scale invariant. Thus, the equations work the same way provided inputs have consistent units, even if the unit in question is unknown. In contrast, a scale variant quantum field theory would describe a force that behaves in a qualitatively different way at short distances compared to long distances.

The AdS/CFT correspondence is notable because it is not obvious that quantum field theories can be represented geometrically. Quantum field theories involve quantities that when explained to non-experts are commonly described as representing intangible ideas like probabilities and possible paths that a quantum could take to get from one place to another. The connection of quantum field theories to a physical geometric description is less obvious than the connection between the classical equations (i.e. non-quantum mechanical descriptions of gravity and electromagnetism) and geometry. There are no non-quantum mechanical equations for the weak nuclear force and the strong nuclear force, the other two fundamental forces.

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 $\textstyle F = G \frac{m_1 m_2}{r^2}\$ (i.e. gravitation 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-like effects seen in general relativity. However this approximation becomes inaccurate in extreme physical situations. For example, in general relativity, objects in motion have a slightly different gravitation effect than objects at rest.

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 impossible to detect without precise instruments.

de Sitter space distinguished from spacetime in general relativity

Fundamentally, the key concept behind the idea of de Sitter space is that it involves a variation on the spacetime of general relativity in which spacetime is itself slightly curved even in the absence of matter or energy.

The relationship of the normal idea of the spacetime in which general relativity operates to the de Sitter space is analogous to the relationship between Euclidean geometry (i.e. in two dimensions, the geometry of flat surfaces) and non-Euclidean geometry (i.e. in two dimensions, the geometries of surfaces that are not flat).

An inherent curvature of spacetime even in the absence of matter or energy is another way of thinking about the idea of the cosmological constant in general relativity. An inherent curvature of spacetime and the cosmological constant are also equivalent to the idea that a vacuum (i.e. empty space without any matter or energy in it) has a fundamental energy of its own.

In the common analogy of an object causing a dip in a flat cloth, normal de Sitter space has a curvature analogous to a flat cloth sitting atop a sphere with a very slight curvature because it is so large. Empty de Sitter space is slightly repulsive; it has a slight natural curvature in the opposite direction of the curvature in spacetime created by a massive object. It is a way of saying that gravity plays out against the background of a slightly anti-gravitational empty space.

Normal de Sitter space corresponds to the positive cosmological constant that is observed in reality, with the size of the cosmological constant being equivalent to the curvature of the de Sitter space.

de Sitter space can also be thought of as a general relativity-like spacetime in which empty space itself has some energy, which causes this spacetime (i.e. the universe) to expand at an ever greater rate.

anti-de Sitter space distinguished from de Sitter space

An anti-de Sitter space, in contrast, is a general relativity-like spacetime, where in the absence of matter or energy, the curvature of spacetime is naturally hyperbolic.

In the common analogy of an object causing a dip in a flat cloth, anti-de Sitter space has a curvature analogous to a flat cloth sitting on a saddle, with a very slight curvature because it is so large. This would correspond to a negative cosmological constant (something not observed in the real life cosmos). Anti-de Sitter space can also be thought of as a general relativity like spacetime in which empty space itself has negative energy, which causes this spacetime (i.e. the universe) to collapse in on itself at an ever greater rate.

In an anti-de Sitter space, as in a de Sitter space, the extent of inherent spacetime curvature corresponds to the magnitude of the negative cosmological constant to which it is equivalent.

de Sitter space and anti-de Sitter space as five-dimensional geometries

As noted above, the analogy used above describes 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. More generally, a geometrical approach to general relativity describes the effect of gravity as a curvature of the four dimensions of spacetime in a fifth dimension that corresponds to gravity and gravity-like effects in general relativity. When this five-dimensional superspace describes a version of general relativity without a cosmological constant, it is called Minkowski space.

The concepts of de Sitter space and anti-de Sitter space describe the effects of the cosmological constant 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 time and space that is produced by gravity and gravity-like effects in general relativity such as the cosmological constant.

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) described via string theory. This is called the AdS/CFT correspondence.

Note also that while an anti-de Sitter space would describe general relativity with a negative cosmological constant in five dimensions (four for spacetime and one for the effect of the cosmological constant), the idea is actually more general. One can have an anti-de Sitter space (or a de Sitter space) in an arbitrary number of dimensions. The generality of the concepts of de Sitter space and anti-de Sitter space make them useful in theoretical physics, particularly in string theory, that often assume a world with more than four dimensions.

Caveats

Naturally, as the remainder of this article explains in technical detail, the general concepts described in this non-technical explanation of anti-de Sitter space have 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 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.

Image of (1 + 1)-dimensional anti-de Sitter space embedded in flat (1 + 2)-dimensional space. The t₁ and t₂ axes lie in the plane of rotational symmetry, and the x₁ axis is normal to that plane. The embedded surface contains closed timelike curves circling the x₁ axis, but these can be eliminated by "unrolling" the embedding (more precisely, by taking the universal cover).

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, ..., x_p, t₁, ..., t_q+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). 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 for this patch is

$ds^2=\frac{1}{y^2}\left(-dt^2+dy^2+\sum_idx_i^2\right),$

with $y>0$ giving the half-space. 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 $y\to 0$, this half-space metric is conformally equivalent to the Minkowski metric $ds^2=-dt^2+\sum_idx_i^2$. 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 "half-space" region of anti-deSitter space and its boundary.

Another commonly used coordinate system which covers the entire space is given by the coordinates t, $r \geqslant 0$ and the hyper-polar coordinates α, θ and φ.

$ds^2 = - \left( k^2r^2 + 1\right)dt^2 + \frac{1}{k^2r^2+1}dr^2 + r^2 d\Omega^2$

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.

As a homogeneous, symmetric space

In the same way that the 2-sphere

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

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

$\mathrm{AdS}_n=\frac{ \mathrm{O}(2,n-1) }{ \mathrm{O}(1,n-1) }$

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

$\frac{ \mathrm{Spin}^+(2,n-1) }{ \mathrm{Spin}^+(1,n-1) }$

of spin groups.

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

$\mathcal{H}= \begin{pmatrix} \begin{matrix} 0&0\\ 0&0 \end{matrix} & \begin{pmatrix} \cdots 0\cdots\\ \leftarrow v^t\rightarrow \end{pmatrix}\\ \begin{pmatrix} \vdots & \uparrow\\ 0 & v \\ \vdots & \downarrow \end{pmatrix} & B \end{pmatrix}$,

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

$\mathcal{Q}= \begin{pmatrix} \begin{matrix} 0&a\\ -a&0 \end{matrix} & \begin{pmatrix} \leftarrow w^t\rightarrow \\ \cdots 0\cdots\\ \end{pmatrix}\\ \begin{pmatrix} \uparrow & \vdots\\ w & 0\\ \downarrow & \vdots \end{pmatrix} & 0 \end{pmatrix}.$

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

A simple definition for anti-de Sitter space and its properties

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

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

Therefore, it is a solution of the Einstein field equations:

$G_{\mu\nu}+\Lambda g_{\mu\nu}=0$

where $G_{\mu\nu}$ is Einstein tensor and $g_{\mu\nu}$ is the metric of the space-time. Introducing the radius $\alpha$ as $\Lambda=\frac{-(n-1)(n-2)}{2\alpha^2}$ this solution can be immersed in a $n+1$ dimensional space-time with signature $(-,-,+,\cdots,+)$ by the following constraint:

$-X_1^2-X_2^2+\sum_{i=3}^{n+1}X_i^2=-\alpha^2$

Global coordinates

$\mathrm{AdS}_n$is parametrized in global coordinates by the parameters $(\tau,\rho,\theta,\varphi_1,\cdots,\varphi_{n-3})$ as:

$\begin{cases} X_1=\alpha\cosh\rho \cos \tau\\ X_2=\alpha\cosh \rho \sin \tau\\ X_i=\alpha \sinh \rho \,\hat{x}_i \qquad \sum_i \hat{x}_i^2=1 \end{cases}$

where $\hat{x}_i$ parametrize a $S^{n-2}$ sphere. i.e. we have $\hat{x}_1=\sin \theta \sin \varphi _1 \dots \sin\varphi _{n-3}$, $\quad\hat{x}_2=\sin \theta \sin \varphi _1 \dots \cos\varphi _{n-3}$ etc. The $\mathrm{AdS}_n$ metric in these coordinates is:

$\mathrm{d} s^2=\alpha^2(-\cosh^2 \rho \,\mathrm{d} \tau^2+\,\mathrm{d} \rho^2+\sinh ^2 \rho \,\mathrm{d} \Omega_{n-2}^2)$

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

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

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

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

Poincaré coordinates

By the following parametrization:

$\begin{cases} X_1=\frac{\alpha^2}{2r}(1+\frac{r^2}{\alpha^4}(\alpha^2+\vec{x}^2-t^2) ) \\ X_2=\frac{r}{\alpha}t \\ X_i=\frac{r}{\alpha}x_i \qquad i\in \{3,\cdots , n\}\\ X_{n+1}= \frac{\alpha^2}{2r}(1-\frac{r^2}{\alpha^4}(\alpha^2-\vec{x}^2+t^2) ) \end{cases}$

the $\mathrm{AdS}_n$ metric in the Poincaré coordinates is:

$\mathrm{d} s^2=-\frac{r^2}{\alpha^2}\,\mathrm{d} t^2+\frac{\alpha^2}{r^2}\,\mathrm{d} r^2+\frac{r^2}{\alpha^2}\,\mathrm{d} \vec{x}^2$

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

$\mathrm{d} s^2=\alpha^2 \left( \frac{\,\mathrm{d} u^2}{u^2}+u^2(\,\mathrm{d} x_\mu \,\mathrm{d} x^\mu) \right)$

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

$\,\mathrm{d} s^2=\frac{\alpha^2}{z^2}(\,\mathrm{d} z^2+\,\mathrm{d} x_\mu \,\mathrm{d} x^\mu)$

Geometric properties

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

Riemann curvature tensor:

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

Ricci curvature:

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

Scalar curvature:

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

References

• Bengtsson, Ingemar: Anti-de Sitter space. Lecture notes.
• Qingming Cheng (2001), "Anti-de Sitter space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• 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.