Questions tagged [anti-de-sitter-spacetime]

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

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Momentum Operator in Anti de Sitter spacetime

Im studying the dirac equation in two dimensional anti de sitter space with metric \begin{align*} g_{\mu\nu}= -\frac{L^2}{\cos^2(\theta)}\left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\...
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Radial Null Geodesics in AdS space

Consider AdS spacetime in coordinates such that the metric is $$ ds^2 \enspace = \enspace - f(r) dt^2 + f(r)^{-1} dr^2 + r^2 d\Omega^2 \quad ,$$ where $$f(r) \enspace = \enspace 1 + \frac{r^2}{R^2} \...
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Is the non-simply connected version of AdS space a maximally symmetric spacetime?

A common construction of anti-de Sitter space is the following: Start with the flat five-dimensional manifold with metric $ds_5^2 = -du^2 - dv^2 + dx^2 + dy^2 + dz^2$. Consider the hyperboloid ...
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Boundary condition on wave in Schwarzschild-Anti-deSitter spacetime at event horizon

I am trying to model the behaviour of a spherical symmetric wave near a black hole in asymptotically Anti-de-Sitter spacetime. The metric for such a spacetime in Eddington-Finkelstein-coordinates is $$...
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Stress- energy tensor in AdS

I´m trying to reproduce some of the equations from the paper -- A Stress Tensor For Anti-de Sitter Gravity, by Balasubramanian and Kraus, https://arxiv.org/abs/hep-th/9902121 -- and I keep getting one ...
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Infinite value of fields

I was reading this article Scalar propagators on AdS Space by Harold Erbin, where the author attempts to find the classical solution of the massive scalar field in AdS space. To discard one of the ...
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BTZ black hole and thermal $AdS_3$

Why is Euclidean $BTZ$ black hole at temperature $T$, equivalent to thermal $AdS_3$ at temperature $1/T$? I'm not really familiar with those objects, but I have used the modular transformation $$\tau_{...
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Euclidean BTZ black hole dual to thermal CFT state

I'm studying $\textbf{Holographic Derivation of Entanglement Entropy from AdS/CFT}$ by Shinsei Ryu and Tadashi Takayanagi. I'm trying to compute the entanglement entropy in $CFT_2/AdS_3$ at finite ...
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How can can we show that a metric is asymptotically AdS?

Given any metric, for example $$ ds^2=d\tau^2+L^2\cosh(H\tau)d\vec{x}^2 $$ how can we show that this metric is asymptotically Euclidean AdS? Specicifally, when $\tau\rightarrow\pm\infty$ is it ...
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Notation for the metric of $\rm dS_4$ and/or $\rm AdS_4$

4D de Sitter and anti-de Sitter spaces may have their metrics inferred from the induced metric on a hyperboloid embedded in 5D Minkowski space: $$ -( x^0)^2+( x^1)^2+( x^2)^2+( x^3)^2+( x^4)^2=\pm \...
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AdS-CFT correspondance from 1D to 4D

From what I understand the AdS-CFT correspondence states that the bulk dynamics of a $n$-dimensional gravitational theory are encoded in the degrees of freedom of its dual CFT in the $(n-1)$ ...
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Different AdS metrics in global coordinates

In David Tong's lecture notes I came across the folloing AdS metric in global coordinates \begin{equation} ds_3^2 = \left( \frac{dr^2}{\frac{r^2}{R^2} + 1 } - \left(\frac{r^2}{R^2} + 1 \right)dt^2 + ...
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AdS$_4$ and $\mathbb{H}^4$: What is the difference between them?

This figure (source) shows the embedding of 4D hyperbolic space $\mathbb{H}^4$ and 4D de Sitter space dS$_4$ in 5D Minkowski space $\mathbb{M}^5$. $\mathbb{H}^4$ is a hyperboloid of two sheets and dS$...
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Scalar field theory boundary term in AdS

In AdS we can have the following action $$ S=\frac{1}{2}\int_\mathcal{M}{d^{d+1}x\sqrt{-g}\phi(g^{\mu\nu}\partial_\mu\partial_\nu+m^2)\phi}-\frac{1}{2}\int_{\mathcal{M}}d^{d+1}x\sqrt{-g}\partial_\mu(g^...
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Anti-de Sitter superalgebra from preserved symmetry

Following Freedman & Van Proeyen's "Supergravity", the $\mathcal{N}=1,\, D=4$ AdS supergravity action is $$ \dfrac{1}{2\kappa^2}\int d^4x \; e \left[R(\omega) - \bar{\psi}_\mu \gamma^{\...
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How to obtain the metric(s) from an action?

So I am confused about dimensional reduction. Because most litterature that I have found so far jump over the steps where they obtain the corresponding metric(s) from some action. In particular, I ...
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AdS$_D$ relation killing vector and boost

I am following "Supergravity" from Freedman and Van Proeyen and I am working on problem 22.8 therein. They embedded AdS$_D$ in $\mathbb{R}^{D+1}$ as $$f(Y)=-(Y^0)^2+\sum_{i=1}^{D-1} (Y^i)^2 -...
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How to take the boundary limit in AdS

So I am reading these notes on AdS/CFT. In section 5, I am trying to understand the boundary of the AdS. They start with the metric in global coordinates \begin{equation} ds^2 = \frac{1}{\cosh(\frac{\...
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Embedding metric (projection) on AdS and Casimir operator

Recently, I am trying to follow up the appendix of arXiv:2106.10822, Embedding space formalism on $AdS_{d+1}$. Basically, I am trying to prove equation A.8 in the above figure. Let $X$ be a ...
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How is AdS/Schwarzschild asymptotically AdS?

A simple GR question. Consider planar Schwarzschild-AdS solution $$ds^2=\frac{R^2}{z^2}\left(-fdt^2+dx^2+\frac{dz^2}{f}\right)$$ where $f=1-(z/z_0)^d$ for constant $z_0$. I've heard this referred to ...
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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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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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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 Hollowood & Kumar paper

I Was re-deriving Hollowood & 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 ...
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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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