Questions tagged [anti-de-sitter-spacetime]

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

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Mapping coordinates of $EAdS_2$ space

Three possible parametrizations of the Euclidean $AdS_2$ space ($H^2=EAdS_2$) are $$ds^2=\frac{1}{y^2}\left(dx^2+dy^2\right),\tag{1}\label{1}$$ $$ds^2=dr^2+\sinh^2r\,d\phi^2,\tag{2}\label{2}$$ $$ds^2=...
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Isometries or Isometry direction of $AdS_5 \times S^5$

This is a consequent question with my previous question https://physics.stackexchange.com/q /610501/ I want to know the isometries (or isometry direction) of $AdS_5 \times S^5$. Usually, when we ...
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Metric form of $AdS_5 \times S^5$

I want to know the metric form of $AdS_5 \times S^5$. I know there are two forms (maybe more?) Poincare patch and global patch. And what is the difference between these two patches? Can you state the ...
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Embed anti-de Sitter space in Minkowski or Euclidean space? (mathematical differences)

I'm a mathematician considering some geometry problems and generalizing them from spherical and hyperbolic manifolds. I wanted to try considering them in anti-de sitter geometry. Consider the manifold ...
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Killing Vectors in $AdS^2 \times S^2$

The metric for the product spacetime $AdS^2 \times S^2$ is given by $$ ds^2 = \dfrac{-dt^2 + dy^2}{y^2} + d\theta^2 + \sin^2 \theta \, d\phi^2.$$ Writing out the Killing equations yields a set of 10 ...
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Anti-deSitter space vacuum, null and weak energy condition

Anti-deSitter space is characterized by a negative cosmological constant. This implies the energy momentum tensor = $T_{\mu\nu} = -\lambda g_{\mu\nu}$, where $\lambda$ is taken positive. This means ...
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Where is the observer in AdS-Schwarzschild coordinates?

for an AdS-Schwarzschild black hole in 4d, the metric is $$ ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2d\Omega^2 $$ where $f(r) = 1 + r^2/l^2 - C/r$. $l$ is the AdS length scale and $C$ is some ...
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How to view thermal spacetime in “Minkowski” picture?

For things such as the Page-Hawking phase transition, we perform a Wick rotation, and consider the Free energy of the metric of a Black Hole in AdS (which has a periodic time to avoid conical ...
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Euclidean anti-de Sitter space embedding

Let us take $\mathbb{R^{d+2}}$ with the cartesian coordinates $(X_0,\dots,X_{d+1})$ and the following metric : \begin{equation}\label{equ1} ds^2 = -dX^2_0-dX^2_{d+1}+\sum^d_{i=1}dX^2_i. \end{equation} ...
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Solving Schwarzian derivative differential equation in kumar paper

I Was re-driving Kumar paper (here is arxiv link of it) which is about Anti-De Sitter Black-Holes with JT Gravity,anyway I got a problem with solving a schwarzian derivative differential equation in 3....
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Isometries of $AdS_2$ space

So, in many places it is mentioned that isometries of pure $AdS_2$ space is the group $SL(2,R)$, defined by the transformation, $t = (at+b)/(ct+d)$ where $ad-bc=1$. Here, the boundary of $AdS_2$ is ...
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Anti-de Sitter Spacetime Properties

I learned from reading nLab (https://ncatlab.org/nlab/show/anti+de+Sitter+spacetime) that the anti-de Sitter Spacetime of dimension $d$, $AdS_d$, is homeomorphic to $\mathbb{R}^{d-1} \times S^1$. I ...
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Why should fields in AdS spacetime vanish at infinity, but not in Minkowski spacetime?

I was watching the following lectures by Prof. Ashoke Sen. Between 39:00 and 56:00, he was solving the equation of classical field in the AdS global coordinates, and says that the values of $\omega$ ...
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Does AdS space-time have global translation symmetry?

I couldn't find any papers/articles addressing this particular issue. To elucidate, my question is the following, Question: Given the maximally symmetric nature of AdS space-time, we also have that it ...
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Does there exist charged BTZ black hole analogue in 1+3D with a negative cosmological constant?

BTZ black holes are defined for the case of 1+2D gravity theory because of closed form computation. I’m wondering if there exist a $1+3D$ analogue black hole? Edit: What I'm actually looking for is ...
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A question about the recent study that supports the Holographic Principle

First off, I agree with Susskind. There's no way to get around the Holographic universe. Maybe there will be one day, but each year it gets stronger and stronger. There's simply a limit to the ...
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Does one observe light reach infinity in finite time in AdS spacetime?

I have a question about the following passage from this article: Moschidis imagined standing in the middle of AdS space-time, which would be like standing inside a giant ball whose edge or boundary ...
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What's the right way to include the characteristic length in the metric of a maximally symmetric space in General Relativity?

Given a maximally symmetric space with characteristic length $L$, I have seen different conventions when writting the factor explicitly in the metric, and I wonder if any interpretation is more ...
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The range of the angular coordinates in (asymptotically) AdS$_5$ spacetime

In the papers I’ve seen with GR solutions in (asymptotically) AdS$_5$ spacetimes, when Boyer-Linquist-like coordinates $(t,r,\theta,\phi,\psi)$ are used, the ranges of the angular coordinates is as ...
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$AdS_3$ in complex coordinates

I am looking for a 2d manifold parametrized by $z$ and $\bar z$ such that $z \bar z = 1$. Now, to see what manifold it leads to I write $$z = \frac{x+iy}{1+iz}$$ so that by imposing $z\bar z = 1$, I ...
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Boundary terms in AdS space

Given the metric in AdS space $$ ds^2=\frac{r^2}{L^2}(-dt^2+d\vec{x}^2)+\frac{L^2}{r^2}dr^2 $$ I am trying to calculate the variation of the action of the KG equation in this metric. What would be ...
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KG on AdS space-time

In his TASI notes Oliver DeWolfe starts with the KG equation on the Poincaré patch metric $$ ds^2=\frac{r^2}{L^2}(-dt^2+dx^2)+\frac{L^2}{r^2}dr^2. $$ When we use the ansatz$$ \phi(r\rightarrow\infty,x,...
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I need help computing the effect of curvature on the FRW metric

Apparently there are different forms of the FLRW metric. I'm focusing on Anti-de Sitter space, so I'll just give the hyperbolic version of the function. $$ds^2=-c^2dt^2+a^2(t)\left[dr^2+R_0\space \...
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Deriving the expression for the AdS metric in global coordinates

I'm trying to work through the derivation of the coordinate expression for the AdS metric in global coordinates. I've found some resources for this, but they all leave out a few calculations, and when ...
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Superluminal speed in anti de sitter

This a bit of an elementary question, but I would like to understand how one correctly computes velocities in anti de sitter. It is well known that photons, traveling on null geodesics, will actually ...
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Confusion between Green function and solution of equation of motion in Witten's paper on holography and AdS

I was going through Witten's paper on AdS and holography , and am confused in section 2.4. He starts by considering a massless scalar action in Euclidean AdS spacetime, with a boundary value $\phi_0$. ...
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Criterion for a black hole in Anti-de Sitter background

Consider a Schwarzschild-Anti de Sitter (SAdS) metric $$ds^2=-(1-\frac{2M}{r}+ k\, r^2 )\, dt^2+\frac{dr^2}{1-\frac{2M}{r}+k \,r^2}+r^2 d\Omega_2^2, $$ with $M,k>0$. This solution has only ...
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Conformal properties of AdS space

My current research is connected with anti-de Sitter ṣpace, which is why I am interested in the following question. It is well known that the metric in Poincare patch is conformally equivalent to that ...
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Null Geodesics in Anti-de Sitter space time [closed]

Would anyone be able to explain how the step was taken in getting the final equation with $R \tan(t/R)$ I understand the steps before where we are finding the null geodesic equation for the AdS space ...
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Symmetry v.s. isometry of Minkowski and AdS or dS spacetime

We know some nice spacetime have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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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 sections ...
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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. ...