Questions tagged [anti-de-sitter-spacetime]

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

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Derivation of Isometries of $AdS_3$ in Poincare Coordinates

We know that $SO(d,2)$ is the isometry group of $AdS_{d+1}$. Let's only consider $AdS_3$ in this question. In Poincare coordinates ($r,t,x)$, these can be grouped as follows : Two translations $$r'=...
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Given in de Sitter space time is frozen on the horizon, can we say that in anti-de Sitter space time goes infinitely fast at the space boundary?

Particularly, in de Sitter space everything on the cosmic event horizon is redshifted, while in anti-de Sitter space everything on the horizon is blueshifted.
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How to find a coordinates transformation on $A\,\mathrm dS_2$?

I have the following 2D metrics (describing the $AdS_2$ spacetime), which are supposed to be the same in different coordinates: \begin{align} \mathrm ds^2 &= \mathrm dt^2 - \sin^2{\!\omega t} \, \...
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Which AdS/CFT correspondences have been found so far?

When I read about AdS/CFT correspondence, there always comes the most famous example of conjectured correspondence, which is the one between type IIB string theory (AdS side) and $\mathcal{N}=4$ ...
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How does AdS/CFT enact and not just be static geometry?

I understand the duality between the two regions of phase space (as Maldacena described it) that are Anti-de Sitter geometry and conformal field theory as an asymptotic grafting on of scale-invariant ...
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Why can AdS/CFT correspondence be applied to condensed matter systems when their space is not anti-deSitter?

The AdS/CFT correspondence postulates a duality between string theory of gravity and a CFT on an AdS background. This duality is employed in some condensed matter systems. I was wondering why it is ...
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The sphere $S^d$ is Euclidean space $E^d$ with infinity identified as a single point

I'm reading about anti de Sitter spacetime, and I found the following statement: $$ds^2 = \frac{1}{\cos^2 \psi} \big( -dt^2 + d\psi^2+ \sin^2 \psi d\Omega_{d-2}^2 \big).$$ Thus, the spatial ...
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$dS^d$ and $AdS^d$ are conformally equivalent

I have seen in the book by Zee (Einstein Gravity in a Nutshell) the following metric for $dS^4$: $ds^2 = \frac{1}{\cos^2 \tau} \big( - d\tau^2 + d\psi^2 + \sin^2 \psi \, d\Omega_2^2 \big) $, (Eq. IX....
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Geodesics of anti-de Sitter space

It is said that (p. 9), given the anti-de Sitter space $\text{AdS}_2$, let's say in the static coordinates $$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1+x^2)} dx^2$$ Every timelike geodesic will cross the ...
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What condition do global coordinates fulfil?

This may be a dumb or vague question: Is there any criterion that a metric tensor needs to fulfill such that coordinates it is expressed in can be called global. Or alternatively what is the ...
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BTZ black hole as an identification of $AdS_3$

I heard in a talk that a BTZ black hole can be obtained as an identification of the $AdS_3$ spacetime. Why exactly is the statement true? Also is it true that in (2+1)D all solutions of the Einstein ...
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Lorentz transformations in non-Euclidean geometries

I have an assessment/investigation coming up in my math class and I plan to investigate Lorentz transformations in geometries other than Euclidean such as spherical or hyperbolic. For spherical/polar, ...
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Metric of a Multipartite AdS-Schwarzschild Black Hole

In section 1 of Susskind's article (see https://arxiv.org/abs/1604.02589) on ER-EPR duality and its connection to the Everett and Copenhagen interpretation of quantum mechanics, he briefly studied ...
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Maximally symmetric spaces

In GR, what is the most precise definition of a maximally symmetric spacetime? Also, we study about the temporal boundary of dS space, and a spatial boundary of AdS space, but aren't these spaces ...
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Form of Light-Cone in pure $AdS_3$

There is a picture in the article of Harlow which (in particular) represents light cone in $AdS_3$. Note that point X has nonzero radial coordinate. If point X has zero radial coordinate it is quite ...
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Can a massive particle in AdS be dropped from the boundary?

My question revolves around the geodesics of massive point particles in fixed background AdS spacetime. Expressed in global coordinates $(\tau,\rho,\Omega_{i})$, the metric of $AdS_{d+1}$ (actually ...
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Symmetries of asymptotically AdS spacetimes

For asymptotically flat spacetimes, the group of symmetry tranformations of flat space, the Poincare group, is enhanced to the Bondi-Metzner-Sachs (BMS) group. Is there a similar enhancement for ...
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Representation theory of AdS and its classical field theory

I am looking for any resources which review the representation theory of the (Anti)-de Sitter group and its Lie algebra and its application to (classical) field theory. I am familiar with how this is ...
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Why isn't Anti de Sitter space taken seriously as a model of reality?

I asked a few questions here earlier regarding a physics model which could possibly point to Eternal Return, and was pointed towards Anti De Sitter space. Arguments in theories for Eternal Return ...
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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 ...
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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. ...
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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^{...
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Anti de Sitter space

While writing the metric for AdS Space, why are we starting with a five dimensional Flat space and embedding a hyperboloid in it? Does it have to do with the fact that the cosmological constant being ...
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Anti-De Sitter Space

Anti - De Sitter Space is the maximally symmetric solution to field equations with negative cosmological constant. The negative cosmological constant also shows that the spacetime has negative ...
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Does AdS$_3$ geometry have an angular momentum?

Consider the following hierarchy of masses for three-dimensional Einstein geometries with a locally AdS$_3$ metric: $$\text{global AdS}_3: M = -1/8G$$ $$\text{conical-defect-AdS}_3: -1/8G < M < ...
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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 $0 \leq 1-4Gm' &...
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Range of the radial coordinate for the Poincare-AdS$_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^...
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Why Can't $AdS_5$ SUSY Extremal Black Holes be Large?

When we put the BPS condition and the extremality condition together on the most general black hole solution in $AdS_5$ (with minimally gauged supergravity), we get that the relation between the ...
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The AdS in AdS/CFT correspondence is really a class of spacetimes which asymptotes to a subclass of spacetimes with the same causal structure as AdS

On page 131 of these notes, a precise formulation of the AdS/CFT correspondence is given by the GKPW dictionary $$Z_{\text{grav}}[\phi_{0}^{i};\partial M] = \langle \exp \left( - \frac{1}{\hbar} \...
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Anti-de Sitter space in an embedding space

The Wikipedia page on anti-de Sitter space defines such a space as the hypersurface $$- X_{1}^{2}-X_{2}^{2} + \sum_{i=3}^{n+1}X_{i}^{2} = \frac{(n-1)(n-2)}{2\Lambda}$$ in an $n+1$-dimensional flat ...
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Boundary conditions due to local and global diffeomorphisms

Consider the following extract from page 2 of this paper. $AdS_3$ is the $SL(2, \mathbb{R})$ group manifold and accordingly has an $SL(2, \mathbb{R})_{L} \times SL(2, \mathbb{R})_{R}$ isometry ...
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Large $D$ limit of (Anti) de Sitter Space is Minkowskian Space?

As is well known, the solution of the vacuum Einstein equations with a non-zero cosmological constant, $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$, is an asymptotically (anti) de Sitter space based on the sign ...
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Anti de Sitter and FLRW metrics

Perhaps this is a naive question, but I have been reading about conformal invariance and conformally related metrics and I would like to know if someone can clarify me some concepts on this. Anti de ...
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Deriving the Poincare patch from global coordinates in AdS$_{3}$

I have been reading Thomas Hartman's lecture notes on Quantum Gravity and Black Holes. In page 97, he derives (9.4), which is the metric of AdS$_{3}$ in global coordinates: $$ds^{2} = \ell^{2}(-\...
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Global coordinate system and universal covering space of AdS$_3$

In the space $\mathbb{M}_{2}\times$ $\mathbb{M}_{2}$ with metric \begin{align} H_{AB}dX^{A}dX^{B} = - dX_{0}^{2} + dX_{1}^{2} + dX_{2}^{2} - dX_{3}^{2}, \label{9.2} \end{align} AdS$_{3}$ ...
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Schwarzschild-anti-de-Sitter solution and the conformal boundary

The solution of the Einstein field equation with the negative cosmological constant $\Lambda = -3k^2$ or the Schwarzschild-anti-de-Sitter solution is: $$ds_4^2=-\left(1-\frac{2m}{r}+k^2r^2 \right)c^...
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Periodic motion of timelike geodesics in homogeneous AdS spacetime

All timelike geodesics that pass through the coordinate origin of AdS (under the standard parameterization) execute simple harmonic motion about the origin with a period proportional to the AdS radius,...
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Can an accelerated timelike trajectory reach the boundary of AdS space in finite time?

I understand why in anti-de Sitter (AdS) spacetime, null geodesics can reach spatial infinity in finite coordinate time, while timelike geodesics cannot reach spatial infinity at all, not even in ...
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Velocity of massless particles in $\mathrm{AdS}_4$ space-time

In spherical coordinates in a $(1+3)$-dimensional space-time one has the $\mathrm{AdS}_4$ metric $$ds^{2}=-\left(1+k^{2}r^{2}\right)dt^{2}+\left(1+k^{2}r^{2}\right)^{-1}dr^{2}+r^{2}d\Omega ^{2}.$$ ...
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The AdS-Schwarzschild black hole solution

My question is from an AdS/CFT review: http://arxiv.org/abs/1112.5403 The AdS_5 metric in the article is written $$ ds^2=\frac{l^2}{z^2}(-dt^2+dz^2+dx^2), $$ where I'm denoting collectively three ...
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Importance of AdS boundary

I was reading a chapter about Anti-de Sitter space-time, it was mentioned that it has a boundary and this boundary is its most striking feature. Note that they weren't taking about the AdS/CFT ...
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BTZ Black holes

I computed the two point function for two scalar fields in BTZ black hole that is defined as a local $AdS_3$ space time with discrete identification, as defined in many papers. Referring to this paper,...
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Question on $E_8$ and twistor space

The Kahler $4$ form constructed from two-forms $\{\alpha, \beta\} \in H^2(M,\mathbb Z)$, and $M$ a $4$-manifold, is induced by $\alpha\wedge\beta$ with the map $H^2(M, \mathbb Z)\otimes H^2(M, \mathbb ...
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AdS boundary global vs Poincare'

Is the global boundary of AdS the same of the boundary written in Poincare' coordinates?
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Volume form of the AdS_{4} Space

Regarding the unit radius $AdS_{4}$ space, the metric in global coordinates, is given by: $$ds^{2}_{AdS_{4}}=\frac{1}{\cos^{2}{\rho}}[dt^{2}-d\rho^{2}-\sin^{2}\rho d\Omega_{2}^{2}]$$ where $$d\...
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$AdS_5$ Schwarzschild Black hole Temperature

This question is an extension of my previous Phys.SE question, but now in $AdS$ spacetime. I am attempting to derive the Temperature of the Schwarzschild solution in this space, which is given by: $$-...
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Anti de-Sitter Geodesics

Timelike geodesics in anti de-Sitter space cannot reach infinity. I believe this has something to do with Clairaut's relation. I'm pretty sure it's true though as the analogy with conservation of ...
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Radiation collapse to black hole

I want to find the temperature at which radiation in AdS will collapse to form a black hole. I have even found a reference that gives the answer but I cannot understand it: http://srv2.fis.puc.cl/~...
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Light Ray in AdS

On p77 of these lecture notes (http://arxiv.org/pdf/0712.0689v2.pdf), we are asked to check that a light ray takes infinitely long to reach the centre of AdS. 1, Why doesn't the Penrose diagram for ...
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Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...