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

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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: ...
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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 ...
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Quasilocal stress tensor

I have been reading through the paper hep-th/9902121 and have a few questions about the first five lines of the introduction: 1) "In a generally covariant theory, it is unnatural to assign a local ...
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de Sitter reviews?

I'm very interested in learning more about de Sitter and anti-de Sitter spaces and their applications in GR and cosmology. Can anyone recommend favorite review articles -- or, more likely, a series ...
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Infinite distance in finite time

It is shown in a previous thread (AdS Space Boundary and Geodesics) that it's possible for null rays to travel to infinity and back in AdS space in finite coordinate time. That is to say, an observer ...
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Solution of Dirichlet problem for scalar field in Ads

I am reading "Anti de Sitter space and holography" by Witten. In this article he derives the two-point function for CFT from theADS/CFT correspondence for a massless scalar field living in the bulk. ...
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Anti de Sitter Motion without matter

I've read that a non zero cosmological constant can lead to "motion without matter" in vacuum spacetimes such as AdS. Initially I didn't understand what the significance of this statement was since I ...
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Motion without matter

Suppose you have a vacuum spacetime with non zero cosmological constant then you can show that two test particles will move toward each other or apart depending on whether it is negative or positive ...
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Acceleration in AdS

I've been reading some notes ("Anti-de Sitter space" by Bengtsson) on anti-de Sitter space. It is shown in equation 152 that timelike observers at fixed radial distance from the origin experience a ...
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Geodesics on dS and AdS

I have been reading through the following paper "The dS and AdS Sightseeing Tour" (http://www.bourbaphy.fr/moschella.pdf) but I am not clear about why geodesics appear as hyperbola in the dS case (see ...
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Timelike Boundary

I was reading in a paper (see 1st paragraph of introduction section in http://arxiv.org/pdf/1510.00709.pdf) that in AdS space, waves can reach the boundary in finite time and, since said boundary is ...
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Norm of Dilatation operator [closed]

The dilatation operator is given by $$D=x^{a}\frac{\partial}{\partial x^{a}}+z\frac{\partial}{\partial z}$$ How the norm can be $$D^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}x^{\mu}x^{\nu}+z^{2})$$ where ...
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Why can only asymptotically flat and AdS black hole have the thermodynamics? What's about asymptoticaly dS black hole?

Almost all advanced GR textbooks will have the content of black hole thermodynamics for asymptotically flat black hole. And this paper solve the asymptotically AdS (Anti-de Sitter) black hole ...
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What is the status of the area theorem in AdS?

The area theorem, by Hawking (in "Gravitational Radiation from Colliding Black Holes", also found in Wald), states that the area of an event horizon cannot decrease with time. As I understand, there ...
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Holographic dual of a massive QFT?

A naive question about holographic dual of a massive QFT: The Ryu-Takanayagi formula for the entanglement entropy (see their paper) seem to suggest that the holographic dual of a massive QFT (e.g. a ...
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How does one show that asymptotically $AdS_3$ spacetimes are locally $AdS_3$?

Time and again I keep reading that any asymptotic $AdS_3$ spacetime is locally isomorphic to $AdS_3$. I tried to find proof of this by analyzing the Riemann tensor $R_{\rho\sigma\mu\nu} $ in Ricci ...
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Massive vector field in curved spacetime

Setup Consider a massive vector field in anti-de Sitter space AdS$_{d+1}$ with metric $$ ds^2=\frac{1}{z^2}\left(dx_\mu dx^\mu+dz^2\right) $$ where $dx_\mu dx^\mu$ is the line element in d+1 ...
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What are maximally dissipative boundary conditions?

I ran into this term when reading about the initial boundary value problem in general relativity. They seem to be relevant when you need to impose boundary conditions on a timelike boundary, for ...
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Analogy for the AdS/CFT Correspondence

Some time ago, I heard about a simple analogy for the AdS/CFT correspondence to something in everyday life. Consider a room filled with furniture, with the walls of the room covered in mirrors. The 2D ...
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Rotating conical singularities in three dimensional gravity

A conical singularity in three dimensional flat space is usually desrcibed by the metric $ds^2=-dt^2+dr^2+r^2\gamma^2d\phi^2$ In three dimensional gravity with a negative cosmological constant, we ...
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Simplifying effect of a hidden Weyl symmetry in a QFT on curved spacetime

We consider AdS$_{d+1}$ in Poincaré coordinates: $$ ds^2=\frac{1}{z^2}\left(-dt^2+dz^2+dx_{d-1}^2\right), $$ where we set the AdS radius to unity. We study a scalar in this background with action $$ ...
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Euclidean AdS space in Poincaré coordinates

I have read anti-de Sitter (AdS) space and its Euclidean version both in Global and Poincaré coordinates. For Lorentzian case it is clear how one Poincaré patch cover only one half of the whole AdS ...
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Penrose diagram for spacetime which flows to $AdS_{2}$ at infinity

Consider I have the following 2 dimensional spacetime $(t,z)$: $$ds^2=\frac{4}{z^{2}}\left(1+\frac{1}{z}\right)^{-1}(-dt^{2}+dz^{2}).\tag{1}$$ When $z\rightarrow \infty$ we have ...
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Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ ...
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Misner String Singularity

In correspondence to AdS black hole solutions, what does it mean by Misner string singularities? And when there are no Misner string singularities, what does this mean in terms of curvature ...
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Geodesics in AdS3

I'm having some trouble doing an easy computation with the AdS space. I'm considering $\text{AdS}_3$ space with the Poincaré coordinates, so the metric reads $$ds^2 = \frac{R^2}{z^2}(dz^2 - dt^2 + ...
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Role of the canonical ensemble and electric charge in AdS/CFT

If we consider a charged black hole in AdS spacetime, we can either do thermodynamics in the grand canonical or the canonical ensemble. In the former, we fix the electrostatic potential ...
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Confinement of charged tachyons in AdS spacetime

It is well known that the negative cosmological constant of AdS spacetime can act like a confining potential. That is, in contrast to asymptotically flat spacetime, in an asymptotically AdS spacetime ...
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Why are anti-de Sitter spaces so interesting when we believe the universe is expansionary?

Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS/CFT correspondence, ...
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Stability condition for AdS background (when gravity coupled to matter fields)

In finding the stability condition for AdS background (when gravity coupled to matter fields), why the conserved energy should be positive?
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Sign convention with the $AdS$ metric

One would say that $AdS_n$ satisfies the equations for the scalar curvature (R) and Ricci tensor ($R_{\mu \nu}$), $R = - \frac{n(n-1)}{L^2}$ and $R_{ab} = - \frac{n-1}{L^2}g_{ab}$. But do the signs ...
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AdS Space Boundary and Geodesics

I'm new to working with AdS space and am primarily concerned with black holes. I'm just playing round with the metric for AdS$_4$ $$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\zeta^2$$ for $f(r)=r^2+m $, ...
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How to draw the Poincaré patch of ${AdS_3}$?

My main reference for this question are these notes (maximally symmetric spaces.pdf) by Kurt Hinterbichler. I'm using Global Coordinates: \begin{align} x^0&=\sec{R}\cos\tau\\ ...
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Thermal AdS and the Hawking Page phase transition

I have some difficulty understanding the concept of pure thermal radiation, as described in Hawking and Page's paper on the Hawking-Page phase transition. The four-dimensional thermal AdS solution ...
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Warped AdS${}_3$ and symmetry breaking

In this article it is explained how on can (in suitable coordinate basis) get a so called warped AdS${}_3$ black hole, by introducing a warping factor. The original metric in 'Euler coordinates' for ...
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Changing vector basis in AdS$_3$

I have AdS${}_3$ given as a surface embedded in a 4 dimensional pseudo-Riemannian space $$x^2+y^2-u^2-y^2=-l^2$$ With metric: $$ds^2=dx^2+dy^2-du^2-dv^2$$ I have Killing vectors of that space ...
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Warped AdS geometry

I am having difficulty of finding more basic information on warped geometries. All the standard textbooks are not covering it. In the wiki article it's only said that warped geometry is the one which ...
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What is the radius of convergence of the Fefferman-Graham expansion?

There is this general result that for any metric $ds^2$ that is asymptotically $AdS_{d+1}$, then there is a coordinate system in which $$ ds^2 = \frac{1}{r^2}(dr^2 + g_{ij}(r,x^k)dx^i dx^j) $$ where ...
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Questions about Type HE Matrix String Theory

I was reading the heterotic string section of this thesis desertation by Luboš Motl, since I think I now understand the Type IIA Matrix String Theory. The only thing I knew about Type HE Matrix ...
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Getting diffeomorphisms from boundary conditions in $AdS_3$

As usual I'm asking a question about boundary conditions for AdS${}_3$, based on the thesis by Porfyriadis. He is solving equations $\mathcal{L}_\xi g_{\mu\nu}$ for AdS${}_3$ metric, with a given ...
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Defining Euclidean global AdS

How does one see that that the Euclidean AdS is the same as the hyperbolic space at the same dimension ie $EAdS_n = \mathbb{H}_n = SO_0(n,1)/SO(n)$? Or is this to be seen as the definition of ...
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Generalisations of AdS/CFT with string theory on both sides

From my previous post, I found out from the comments that there are various generalisations of AdS/CFT with different things replacing the CFT on the RHS; such as AdS/CMT, AdS/QCD, and also with ...
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Does an antiDeSitter space compensate the vacuum energy?

Sometimes I read here or there that a problem with the electroweak vacuum is that it implies a huge positive cosmological constant. On other side, I also read that AdS spaces are not useful anymore ...
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Mass of empty AdS$_5$

Five dimensional empty AdS$_5$ space has mass $$ E = \frac{3 \pi \ell^2}{32 G}. $$ Is the above equation correct? Let's do some dimensional analysis to confirm. In natural units, in 5 dimensions ...
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Classical theories and AdS/CFT

When I was editing the Physics.SE tag wiki for ads-cft, I initially wrote something on the lines of : The AdS/CFT correspondence is a special case of the holographic principle. It states that ...
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AdS/CFT and boundary translational invariance

I work in quantum information theory/condensed matter and have some very basic questions about AdS/CFT correspondence. For simplicity, I would like to restrict to 1+1 CFT <-> 2+1 AdS. I apologize ...
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What is the geometry of our universe?

I have some questions about the notion of space in current cosmological theories. I have been trying to decipher a few papers written on this topic such as this one, but I keep running into some ...
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Symmetry transformation in AdS space

In AdS/CFT papers the action of the SO(D,2) symmetry is usually given at the boundary where the transformations are just the conformal transformations (Poincare, scaling and special) for D+1 ...
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How scalar curvature of following spacetime can be equal to zero?

For an interval of this spacetime, $$ ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})), $$ scalar curvature is equal to zero. Also, Ricci ...