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Questions tagged [anti-de-sitter-spacetime]

Anti-de-Sitter (AdS) spacetime is a spacetime with a constant negative Ricci Scalar.

Anti-de-Sitter (AdS) spacetime is a spacetime with a constant negative Ricci Scalar. It has a pseudo-metric:

$${\text{d}}{s^2} = \sum\limits_{i = 1}^p {{\text{d}}x_i^2 - \sum\limits_{j = 1}^{q + 1} {{\text{d}}t_j^2} } $$

AdS Spacetime has played an important role in the correspondence.

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Only the geodesic distance matters for maximally symmetric spacetimes

Any physical quantity $K(t,x,x')$ on a maximally symmetric spacetime only depends on the geodesic distance between the points $x$ and $x'$. Why is this so? N.B.: This statement is different from the statement that The geodesic distance on any…
nightmarish
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Boundary conditions on conical deficit AdS geometry

The conical deficit global AdS$_3$ geometry is given by $$ds^{2} = - \cosh^{2}\rho\ dt^{2} + d\rho^{2} + \sinh^{2}\rho\ d\varphi^{2}, \qquad 0 \leq \varphi < 2\pi(1-4Gm'),$$ where $0 \leq 1-4Gm' < 1$, and $0 \leq \rho < \infty$. Does a finite…
nightmarish
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Explicit form of the Poincare-AdS$_3$ geometry

The Poincare-AdS$_3$ geometry is given in the Wikipedia article on Anti-de Sitter space as $$ds^{2} = \frac{dr^{2}}{r^{2}} + r^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}$$ $$=\frac{dr^{2}}{r^{2}} + r^{2}(-dt^{2} + dx^{2}),$$ where $(t,x)$ are the…
nightmarish
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Symmetries of a radially-cut-off AdS$_3$ cylinder

The isometry group of the anti-de Sitter spacetime is $SO(d-1,2)$, which has a total of $\frac{1}{2}d(d+1)$ isometries. For the three-dimensional anti-de Sitter spacetime, these are $6$ isometries. Would the number of isometries change for an…
nightmarish
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Range of the radial coordinate for the Poincare-AdS$_3$ geometry

The metric of the Poincare-AdS$_3$ geometry is given in the Wikipedia article on the Poincare coordinates of AdS$_3$ geometry: $$ds^{2} = \alpha^{2}\left(\frac{du^{2}}{u^{2}} + u^{2}g_{\alpha\beta}dx^{\alpha}dx^{\beta}\right),$$ where $\alpha$ is…
nightmarish
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