Anti de Sitter
1. History
2. Topology
The canonical Anti de Sitter space is modelled by a hyperquadric in $\mathbb{R}^{n+1}$, with topology $S^1 \times \mathbb{R}^{n-1}$. It admits a universal cover with topology $\mathbb{R}^{n}$ by unwrapping the time coordinate.
3. Metrics and coordinates
3.1. Stereographic coordinates
$$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1 + x^2)} dx^2$$3.2. Global coordinates
$$ds^2 = \alpha^2 (- \cosh^2(\rho) d\tau^2 + d\rho^2 + \sinh^2(\rho) d\Omega)$$3.3. Poincaré coordinates
$$ds^2 = -\frac{r^2}{\alpha^2} dt^2 + \frac{\alpha^2}{r^2} dr^2 + \frac{r^2}{\alpha^2} d\vec{x}^2$$4. Tensor quantities
5. Symmetries
Anti-de Sitter space is maximally symmetric and thus admits the full $\dfrac{n(n+1)}{2}$ Killing vectors. It admits as an isometry group the Lie group $\operatorname{O}(n-1, 1)$
6. Stress-energy tensor
7. Curves
8. Equations
9. Causal structure
Anti de Sitter space fails to be globally hyperbolic due to the unbounded widening of its light cones over a period of time $\pi$. It is otherwise geodesically complete and stably causal. If the definition of nakedly singular is used where any non-globally hyperbolic spacetime is nakedly singular, the naked singularities of AdS are the set of ideal points at infinity, from which information may come.
10. Asymptotic structure
11. Energy conditions
12. Limits and related spacetimes
Anti-de Sitter space admits Minkowski space in the limit of infinite radius (particularly clearly if we're dealing with the $\mathbb{R}^n$ topological one). In particular we have that the group of isometry of AdS $\text{O}(n-1,2)$ reduces to the Poincaré group $O(n-1,1) \ltimes \mathbb{R}^n$ under the group contraction by the radius.