We study the simplest two-body problem in asymptotically anti-de Sitter spacetime: two, infinitely thin, concentric spherical shells of matter. We include only gravitational interaction between the two shells, but we show that the dynamics of this system is highly nontrivial. We observe prompt collapse to a black hole, delayed collapse and even perpetual oscillatory motion, depending on the initial location of the shells (or their energy content). The system exhibits critical behavior, and we show strong hints that it is also chaotic.
Born-Infeld-AdS black holes in extended phase space may possess phase structures resembling that of van der Waals fluid in four-dimensional spacetime. We study dynamics of its state, which is in the unstable spinodal region initially on phase space, under time-periodic thermal perturbations. By applying the Melnikov method, it is found that there exists a critical amplitude $γ_c$ of the perturbations, which depends on the Born-Infeld parameter $b$ and the black hole charge $Q$, such that chaos occurs for $γ> γ_c$. We found that larger $b$ or $Q$ makes the onset of chaos easier. Space-periodic …
We investigate the conjectured bound on the Lyapunov exponent for a charged particle with angular motion in the Kerr-Newman-AdS black hole. The Lyapunov exponent is calculated based on the effective Lagrangian. We show that the negative cosmological constant reduces the chaotic behavior of the particle, namely, it decreases the Lyapunov exponent. Hence, the bound is more effective in the AdS spacetime than in the flat spacetime. Nevertheless, we find that the bound can be violated when the angular momenta of the black hole are turned on. Moreover, we show that in an extremal black hole, the bo…
Holographic theories with classical gravity duals are maximally chaotic; i.e., they saturate the universal bound on the rate of growth of chaos. It is interesting to ask whether this property is true only for leading large $N$ correlators or if it can show up elsewhere. In this Letter we consider the simplest setup to tackle this question: a Brownian particle coupled to a thermal ensemble. We find that the four-point out-of-time-order correlator that diagnoses chaos initially grows at an exponential rate that saturates the chaos bound, i.e., with a Lyapunov exponent $λ_L=2π/β$. However, the sc…
Extremal black holes have vanishing Hawking temperatures. In this article, we argue that for asymptotically AdS black holes, at extremality, a particular class of correlators in the dual CFT can exhibit exponential, maximally chaotic growth with a non-vanishing temperature. Our approach, at extremality, is two-fold. First, we geometrically investigate the modes that are responsible for chaos. Secondly, we study the dynamics of a probe string to capture chaos in worldsheet correlators. For rotating BTZ at extremality, the corresponding Lyapunov exponent is determined by the left-moving temperat…
Chaos Rules out Integrability of Strings in AdS_5 x T^{1,1}
我们证明了AdS_5 x T^{1，1}中的某些经典弦结构是混沌的。这就否定地回答了弦在这种背景下的可积性问题。我们考虑一个定域在AdS_5中心的弦，它绕着T^{1，1}的两个圆。相应的动力系统等价于两个耦合的引力摆，并允许一个非常直观的理解。通过系统地分析KAM定理的工作原理，我们找到了混沌行为的确凿证据。我们还表明，最大的李雅普诺夫指数是积极的。
To find the origin of chaos near black hole horizon in string-theoretic AdS/CFT correspondence, we perform a chaos analysis of a suspended string in AdS black hole backgrounds. It has a definite CFT interpretation: chaos of Wilson loops, or in other words, sensitive time-evolution of a quark antiquark force in thermal gauge theories. Our nonlinear numerical simulation of the suspended Nambu-Goto string shows chaos, which would be absent in pure AdS background. The calculated Lyapunov exponent $λ$ satisfies the universal bound $λ\leq 2πT_{\rm H}$ for the Hawking temperature $T_{\rm H}$. We also…
The scrambling time and its delay are calculated using holography in an asymptotically AdS black hole solution of the gauged Einstein-Maxwell-Dilaton-Axion (EMDA) theory, the dyonic Kerr-Sen-AdS$_4$ black hole, perturbed by rotating and charged shock waves along the equator. The leading term of the scrambling time for a black hole with large entropy is logarithmic in the entropy and hence supports the fast scrambling conjecture for this black hole solution, which implies that the system under consideration is chaotic. We also find that the instantaneous minimal Lyapunov index is bounded by $κ=…
We present an analytical study of chaos in a charged black hole in the extended phase space in the context of the Poincare-Melnikov theory. Along with some background on dynamical systems, we compute the relevant Melnikov function and find its zeros. Then we analyse these zeros either to identify the temporal chaos in the spinodal region, or to observe spatial chaos in the small - large black hole equilibrium configuration. As a byproduct, we derive a constraint on the Black hole charge required to produce chaotic behaviour. To the best of our knowledge, this is the first endeavour to understa…
We study the onset of chaos due to temporal and spatially periodic perturbations in charged Gauss-Bonnet AdS black holes in extended thermodynamic phase space, by analyzing the zeros of the appropriate Melnikov functions. Temporal perturbations coming from a thermal quench in the unstable spinodal region of P-V diagram, may lead to chaos, when a certain perturbation parameter $γ$ saturates a critical value, involving the Gauss-Bonnet coupling $α$ and the black hole charge $Q$. A general condition following from the equation of state is found, which can rule out the existence of chaos in any bl…
Lyapunov exponents, phase transition and chaos bound in Kerr-Newman AdS spacetime
School of Science, Xihua University（西华大学理学院）；Faculty of Engineering and Quantity Surveying, INTI International University（INTI国际大学工程与估价学院）；College of Physics and Electronic Engineering, Chongqing Normal University（重庆师范大学物理与电子工程学院）
Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time $\widehat{u}_*$. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes $AdS_2$ gravity and the low-energy dynamics of the SYK model. We identify a particular set of $2k$-point functions, characterized as being both "maximally braided" and "k-OTO", which exhibit exponential growth until progressively longer t…
In this paper, we investigate influences of the charge and angular momentum of a particle around a charged Einstein-Euler-Heisenberg AdS black hole on a Lyapunov exponent, and find spatial regions where the chaos bound is violated. Positions of circular orbits are gotten by fixing the charge and angular momentum of the particle, respectively. The positions gradually move away from the event horizon with the increase of the angular momentum when the charge is fixed and with the decrease of the charge when the angular momentum is fixed. For certain values of the charge, angular momentum and Eule…
From Confinement to Chaos in AdS/CFT Correspondence via Non-equilibrium Local States
Steklov Mathematical Institute, Russian Academy of Sciences（俄罗斯科学院列别杰夫数学研究所）；Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科罗蒙诺索夫国立大学理论与数学物理研究所）
This note is devoted to the investigation of Susskind's proposal(arXiv:1802.01198) concerning the correspondence between the operator growth in chaotic theories and the radial momenta of the particle falling in the AdS black hole. We study this proposal and consider the simple example of an operator with the global charge described by the charged particle falling to the Reissner-Nordstrom-AdS black hole. Different charges of the particle lead to qualitatively different behavior of the particle momenta and consequently change of the operator size behavior. This holographic result is supported b…
We reconsider the role of wormholes in the AdS/CFT correspondence. We focus on Euclidean wormholes that connect two asymptotically AdS or hyperbolic regions with $\mathbb{S}^1\times \mathbb{S}^{d-1}$ boundary. There is no solution to Einstein's equations of this sort, as the wormholes possess a modulus that runs to infinity. To find on-shell wormholes we must stabilize this modulus, which we can do by fixing the total energy on the two boundaries. Such a wormhole gives the saddle point approximation to a non-standard problem in quantum gravity, where we fix two asymptotic boundaries and constr…
Gravitational Collapse, Chaos in CFT Correlators and the Information Paradox
研究了渐近Anti de Sitter时空中无质量标量场的引力坍缩。在AdS/CFT字典之后，我们通过在坍缩背景中探测标量场的Klein-Gordon方程进一步研究了场论方面的相关性。我们提出的证据表明，在一定的制度的探针标量场的行为混乱，从而支持霍金的论点在黑洞信息悖论提出，虽然信息可以检索的原则上，确定性的混乱损害，在实践中，从黑洞的单一提取信息的过程。我们强调，量子混沌将改变这一局面。
Entanglement, chaos, and complexity are as important for de Sitter space as for AdS and for black holes. There are similarities and great differences between AdS and dS in how these concepts are manifested in the space-time geometry. In the first part of this paper the Ryu-Takayanagi prescription, the theory of fast scrambling, and the holographic complexity correspondence are reformulated for de Sitter space. Criteria are proposed for a holographic model to describe de Sitter space. The criteria can be summarized by the requirement that scrambling and complexity growth must be "hyperfast." In…
We find an effective action for gravitational interactions with scalars in $AdS_3$ to first order in $G_N$ at the conformal boundary. This action can be understood as an action for the Brown-Henneaux modes and is given by the square-root product of right and left moving Schwarzian derivatives for conformal transformations of the boundary. We thus reproduce the result $λ_L=\frac{2π}β$ for OTOC computed first in arXiv:1412.6087 for a Schwarzchild black hole in $AdS_3$. Applying the same procedure to rotating BTZ we find the Lyapunov index to be $λ_L=\frac{2π}{β_+}>\frac{2π}β$ where $β_+=β(1-μ_L)…
Weakly nonlinear dynamics in anti-de Sitter (AdS) spacetimes is reviewed, keeping an eye on the AdS instability conjecture and focusing on the resonant approximation that accurately captures in a simplified form the long-term evolution of small initial data. Topics covered include turbulent and regular motion, dynamical recurrences analogous to the Fermi-Pasta-Ulam phenomena in oscillator chains, and relations between AdS dynamics and nonrelativistic nonlinear Schrodinger equations in harmonic potentials. Special mention is given to the way the classical dynamics of weakly nonlinear strongly r…
Particle experiments are difficult at weak coupling because interactions are rare and a huge number of collision attempts are needed to attain significant precision. One often hears that "one Higgs boson is produced in a billion of collisions at LHC." In this essay, we fantasize about possible advantages afforded in this regard by performing experiments in anti-de Sitter (AdS) spacetime instead of a usual collider in a nearly-flat spacetime. Being a perfectly resonant cavity, the AdS spacetime enhances all nonlinear interactions, which therefore produce effects of order one no matter how small…
We discuss $SL(N,\mathbb{C})$ Chern-Simons higher spin gauge theories in Euclidean AdS$_3$. With appropriate boundary term, we derive the higher spin generalization of Schwarzian on-shell action. We investigate gravitationally dressed bi-locals, and we study the soft higher spin mode expansion to obtain soft mode eigenfunctions. We also derive the spin-$s$ eigenfunction from $\mathcal{W}$-Ward identity and a recursion relation. Using the on-shell action, we evaluate the contributions of the soft higher spin modes to the out-of-time-order correlators, and the corresponding Lyapunov exponent of …
We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, $λ_L^{\pm}=\frac{2π}β\frac{1}{1\mp \ell Ω}$, where $Ω$ is the angular velocity and $\ell$ is the AdS radius. Since $λ_L^{-} \leq \frac{2π}β \leq λ_L^{+}$, there is an apparent contradiction with the chaos bound. We discuss…
We explore asymptotically AdS$_2$ solutions of a particular two-dimensional dilaton-gravity theory. In the deep interior, these solutions flow to the cosmological horizon of dS$_2$. We calculate various matter perturbations at the linearised and non-linear level. We consider both Euclidean and Lorentzian perturbations. The results can be used to characterise the features of a putative dual quantum mechanics. The chaotic nature of the de Sitter horizon is assessed through the soft mode action at the AdS$_2$ boundary, as well as the behaviour of shockwave type solutions.
We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the pl…
Melnikov方法是一种分析方法来证明由Smale马蹄铁产生的经典混沌的存在。这是一种强大的技术，尽管它的适用性有些有限。本文给出了一个适用于Melnikov方法的IIB型超引力解。这是AdS $_5\times$ S $^5$背景的膜波型变形。采用两种简化方法，研究了两类耦合谐振子系统。然后按照Holmes和Marsden的标准方法计算了每个系统的Melnikov函数，并分析了混沌的存在性。
