Questions tagged [anti-de-sitter-spacetime]

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

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Why is AdS spacetime like a "saddle"?

When the shape of the universe is discussed, the three cases are flat, closed and open. Where AdS spacetime with a negative cosmological constant describes the open spacetime, as in the middle in the ...
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How to derive the AdS spacetime metric

So I have been working on AdS/CFT for a while now and realized that I have never actually seen the derivation for the metric. In every literature, introductory or advanced, they just give you the AdS ...
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Two-twistors formulation vs twistors formulation

I have seen in some research work that the classic formulation using twistors (introduced by Penrose) is replaced with a formulation that considers two-twistors. For example the linking article says ...
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Is there any place in an AdS Universe with positive spacial curvature?

An AdS space has constant negative curvature, gravitational objects introduce more negative curvature. Therefor it seems like it is possible that everywhere in the Universe has negaative curvature (...
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Directly solving equation of motion in embedding space coordinates

If you're given a manifold as an embedding, how does one solve an equation directly in the embedding space coordinates? Specifically, I am trying to solve the equation $(\nabla_{\mu}\nabla_{\nu}-g_{\...
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Are quadratic gravity's equations of motion just regular gravity with some minimal length?

So, I came to this while doing some calculations in quadratic gravity with the following action: \begin{equation} S = \int d^4x \sqrt{-g}\left[ \frac{1}{2}m^4+\frac{1}{6}m^2 R +\frac{1}{72}R^2+\frac{1}...
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Breitenlohner-Freedman bound in AdS coupled theory

In this review of the AdS/CFT correspondence in section 2.2.2 we are given the scalar field equations of a quantum field theory in AdS spacetime. The energy-momentum tensor is given by $$ T_{\mu\nu}=2\...
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Stress-energy tensor of a scalar theory in AdS

In this review of the AdS/CFT correspondence, going through section 2.2.2 I am not quite sure where the stress-energy tensor $$ T_{\mu\nu}=2\partial_\mu\phi\partial_\nu\phi-g_{\mu\nu}\left((\partial\...
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Ricci scalar of AdS in $D$ spacetime dimensions from structure equations

Starting from the AdS metric in $D$ spacetime dimensions in Poincare coordinates $ds^2 = \frac{R^2}{(x^3)^2}\eta_{\mu\nu}dx^\mu dx^\nu$ (R here is the AdS radius), I would like to compute the ...
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Anti de Sitter & de Sitter Spacetime [duplicate]

Can anyone please explain what are the Anti de Sitter and de Sitter spacetime and what is special about them? I am learning general relativity and I stumbled upon them a few times, even on the subject ...
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Can I just add a cosmological constant to an on-shell supergravity action?

Suppose one has some supergravity Lagrangian $\mathcal{L}_\text{sugra}$ in flat (Minkowski) space. Moreover, assume that this Lagrangian is on-shell, in the sense that the auxiliary fields have been ...
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Anti de Sitter metric with signature +---

Most of the textbooks write EFE $$ R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\Lambda g_{\mu\nu} $$ With metric signature $-+++$. However, I am using the other signature $+---$, but when I solve for Anti de ...
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Derivation of $zz$-component of Einsteins Equations in AdS

I am trying to understand how we get the Einsteins equations in here section 4.1 equation 4.2 where we use the metric $$ ds^2 = a^2(z)(dz^2+dx^\mu dx_\mu) $$ to derive the $zz$-component of Einstein's ...
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Difference between asymptotically AdS and asymptotically locally AdS spacetime

In the literature, there is often a distinction made between spacetimes that are asymptotically or asymptotically locally some other spacetime. For example, in holography, referring to some spaces ...
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AdS/CFT spacetime visualization

I was inspired by comments under this answer to ask this question. In the context of AdS/CFT, one often finds an embedding diagram of the $10d$ spacetime that I don't find particularly enlightening, ...
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Misunderstanding of $\mathrm{AdS}_3$ spacetime

In my research I deal with $\mathrm{AdS}_3$ spacetime. It is convenient for me to use Poincare coordinates, which means that interval is given by $$ds^2 =\frac{1}{z^2}(-dt^2+dx^2+dz^2).\quad \tag{1}$$ ...
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Flat space limit of the $AdS$ metric: Very basic question

Suppose I am given the following global coordinates in empty $AdS_n$: $$ds^2 = \alpha^2\left(-\cosh^2\rho \, d\tau^2 + \, d\rho^2 + \sinh^2\rho \, d\Omega_{n-2}^2\right)$$ where the length scale $\...
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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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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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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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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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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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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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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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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 ...