\mathrm{AdS}_n is an
n-dimensional vacuum solution for the theory of gravitation with
Einstein–Hilbert action with negative
cosmological constant \Lambda, (\Lambda ), i.e. the theory described by the following
Lagrangian density: : \mathcal{L} = \frac{1}{16 \pi G_{(n)}}(R - 2\Lambda) , where is the
gravitational constant in -dimensional spacetime. Therefore, it is a solution of the
Einstein field equations: : G_{\mu\nu} + \Lambda g_{\mu\nu} = 0, where G_{\mu\nu} is the
Einstein tensor and g_{\mu\nu} is the metric of the spacetime. Introducing the radius \alpha as \Lambda = \frac{-1}{\alpha^2}\frac{(n - 1)(n - 2)}{2}, this solution can be
immersed in a (n + 1)-dimensional flat spacetime with the metric \mathrm{diag}(-1, -1, +1, \ldots, +1) in coordinates (X_1,X_2,X_3,\ldots,X_{n+1}) 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,\ldots,\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, and in terms of the coordinates \varphi_i they are \hat{x}_1 = \sin\theta \sin\varphi_1 \ldots \sin\varphi_{n-3}, \hat{x}_2 = \sin\theta \sin\varphi_1 \ldots \cos\varphi_{n-3}, \hat{x}_3 = \sin\theta \sin\varphi_1 \ldots \cos\varphi_{n-2} and so on. The \mathrm{AdS}_n metric in these coordinates is: : ds^2 = \alpha^2\left(-\cosh^2\rho \, d\tau^2 + \, d\rho^2 + \sinh^2\rho \, d\Omega_{n-2}^2\right) 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 spacetime 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: : ds^2 = -f(r) \, dt^2 + \frac{1}{f(r)} \, dr^2 + r^2 \, d\Omega_{n-2}^2 where 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 t_1 and t_2 to
polar coordinates where the radial coordinate is x_0 and the angular coordinate is \varphi, such that t_1 = x_0\cos \varphi and t_2 = x_0\sin \varphi, where the space is periodic in \varphi with the period 2\pi, the quasi-sphere becomes : \sum_{i=1}^p x_i^2 - x_0^2 = -\alpha^2 , and the metric becomes : ds^2 = \sum_{i=1}^p dx_i^2 - dx_0^2 - (x_0 d\varphi)^2. We see that the quasi-sphere takes on the same equation as the manifold in the
hyperboloid model of -dimensional hyperbolic space, while the metric also takes on the equation of the metric in the hyperboloid model, except for the extra term -(x_0 d\varphi)^2. This tells us that the quasi-sphere is
translationally symmetric in the \varphi direction, and that curves in the quasi-sphere with a fixed \varphi-value are spacelike, while \varphi itself is timelike, where the proper time \tau evolves faster the greater x_0 is as \varphi increases if all other coordinates are fixed, i.e. : \left.\frac{d\tau}{d\varphi}\right|_{x_0,x_1,\dots,x_p} = x_0. If we write \varphi as a function of t_1 and t_2, we see that the
branch point is at t_1 = t_2 = 0, but since the constraint posed by the quasi-sphere requires that t_1^2 + t_2^2 \geq 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 \varphi, and in that way avoid
closed timelike curves (CTC).
Poincaré coordinates By the following parametrization: : \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 \mathrm{AdS}_n metric in the Poincaré coordinates is: : 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 0 \leq r. The codimension 2 surface r = 0 is the Poincaré Killing horizon and r \to \infty approaches to the boundary of \mathrm{AdS}_n spacetime. 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: : ds^2 = \alpha^2 \left( \frac{\,du^2}{u^2} + u^2 \, dx_\mu \, dx^\mu \right) where x^\mu = \left(t, \vec{x}\right). By the transformation z \equiv \frac{1}{u} also it can be written as: : 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 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 : ds^2 = -dt^2 + \alpha^2 \sin^2(t/\alpha) dH_{n-1}^2, where dH_{n-1}^2 = d\rho^2 + \sinh^2\rho d\Omega_{n-2}^2 is the standard metric on the (n-1)-dimensional hyperbolic plane. Of course, this does not cover all of AdS. These coordinates are related to the global embedding coordinates by : \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 \sum_i \hat{x}_i^2=1 parameterize the S^{n-1}.
De Sitter slicing Let : \begin{align} 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{align} where \sum_i \hat{x}_i^2=1 parameterize the S^{n-3}. Then the metric reads: : ds^2 = d\rho^2 + \sinh^2\left(\frac{\rho}{\alpha}\right) ds_{dS,\alpha,n-1}^2, where : 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 n - 1 dimensional de Sitter space with radius of curvature \alpha in open slicing coordinates. The hyperbolic metric is given by: : dH_{n-2}^2 = d\xi^2 + \sinh^2(\xi) d\Omega_{n-3}^2.
Geometric properties AdS
n metric with radius \alpha is one of the maximally 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{-1}{\alpha^2} (n - 1)g_{\mu\nu} •
Scalar curvature • : R = \frac{-1}{\alpha^2} n(n - 1) == Generalization ==