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Journal ArticleDOI

Superluminal chaos after a quantum quench

23 Aug 2019-Journal of High Energy Physics (Springer Berlin Heidelberg)-Vol. 2019, Iss: 12, pp 1-41
TL;DR: In this article, out-of-time-order correlators (OTOCs) in BTZ-Vaidya spacetimes are computed to describe transitions between black holes at different temperatures.
Abstract: Thermal states holographically dual to black holes in Einstein gravity display maximal Lyapunov growth as well as “butterfly effect cones”. We study these effects in highly non-equilibrium states, obtained from an initial thermal state by the sudden injection of energy. We do this by computing out-of-time-order correlators (OTOCs) in BTZ-Vaidya spacetimes, which describe transitions between black holes at different temperatures. If both pairs of boundary operators appearing in the OTOC are inserted before the energy injection, we recover standard results, with butterfly effect cones displaying a light-cone structure. But when one pair of operators is inserted before and the other pair after the energy injection, the Lyapunov growth saturates the chaos bounds set by the local temperatures and the butterfly effect cones can “open up”, becoming superluminal, albeit in a way that does not violate causality. In the limiting case, in which the initial state is the vacuum, Lyapunov growth only starts after the energy injection. Our computations of the OTOCs are phrased in terms of gravitationally interacting particles, where fields are treated in a geodesic approximation and the eikonal phase shift is expressed in terms of stress tensors and shock waves associated to geodesics.

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Citations
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TL;DR: In this paper, small perturbations to a static three-dimensional de Sitter geometry are considered and the result is a shockwave geometry that leads to a time advance in the trajectory of geodesics crossing it, much like a traversable wormhole in Anti-de Sitter space.
Abstract: We consider small perturbations to a static three-dimensional de Sitter geometry. For early enough perturbations that satisfy the null energy condition, the result is a shockwave geometry that leads to a time advance in the trajectory of geodesics crossing it. This brings the opposite poles of de Sitter space into causal contact with each other, much like a traversable wormhole in Anti-de Sitter space. In this background, we compute out-of-time-order correlators (OTOCs) to asses the chaotic nature of the de Sitter horizon and find that it is maximally chaotic: one of the OTOCs we study decays exponentially with a Lyapunov exponent that saturates the chaos bound. We discuss the consequences of our results for de Sitter complementarity and inflation.

59 citations

Journal ArticleDOI
TL;DR: In this article, the inner horizons of rotating and charged black holes in anti-de Sitter space were studied, and it was shown that these black holes have a classical analytic extension through the inner horizon to additional asymptotic regions.
Abstract: We study the inner horizons of rotating and charged black holes in anti-de Sitter space. These black holes have a classical analytic extension through the inner horizon to additional asymptotic regions. If this extension survives in the quantum theory, it requires particular analytic properties in a dual CFT, which give a prescription for calculating correlation functions for operators placed on any asymptotic boundary of the maximally extended spacetime. We show that for charged black holes in three or greater dimensions, and rotating black holes in four or greater dimensions, these analytic properties are in- consistent in the dual CFT, implying the absence of an analytic extension for quantum fields past the inner horizon. Thus, we find that strong cosmic censorship holds for all AdS black holes except rotating BTZ. To further study the latter case, we insert classical perturbations near the boundary at late times, producing shockwaves traveling along the inner horizon. We holographically compute CFT correlators in this background that probe a high energy scattering process near the inner horizon and argue that the shockwave does not destabilize the inner horizon violently enough to prevent signaling between different asymptotic regions of the Penrose diagram. This provides evidence that the rotating BTZ black hole does violate the strong cosmic censorship conjecture.

31 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the effect of the cap region on the growth of an out-of-time-order correlator (OTOC) for a class of extremal black holes, namely maximally rotating BTZ black holes.
Abstract: Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.

28 citations

Posted Content
TL;DR: In this paper, it was shown that there is a soft limit in quantum gravity where the information paradox is readily resolved due to virtual soft graviton exchange on the black hole horizon.
Abstract: We show that there is a remarkable soft limit in quantum gravity where the information paradox is readily resolved due to virtual soft graviton exchange on the black hole horizon. This regime is where collision energies satisfy $\sqrt{s}\gg \gamma M_{Pl}$ (with $\gamma \sim M_{Pl}/M_{BH}$) near the horizon. We call this the black hole eikonal phase, in contrast to its flat space analogue where collisions are trans-Planckian. Hawking's geometric optics approximation neglects gravitational interactions near the horizon, and results in thermal occupation numbers in the Bogoliubov coefficients. We show that these interactions are mediated by graviton exchange in $2 \rightarrow 2$ scattering near the horizon, and explicitly calculate the S-matrix non-perturbatively in $M_{Pl}/M_{BH}$ and $\hbar$. This involves a re-summation of infinitely many ladder diagrams near the horizon, all mediated by virtual soft gravitons. The S-matrix turns out to be a pure phase \textit{only} upon this re-summation. The impact parameter $b$ satisfies $L_{Pl}\ll b \lesssim R_S$, where $R_S$ is the Schwarzschild radius; therefore, our results are agnostic of Planckian physics. Our calculation shows that non-renormalisability of gravity is irrelevant for a resolution of the information problem, and is agnostic of any specific ultraviolet completion. In contrast to the flat space eikonal limit, the black hole eikonal phase involves collisions of extremely low energy near the horizon, thereby avoiding firewalls for black holes much larger than Planck size.

20 citations

Journal ArticleDOI
TL;DR: In this paper, the eikonal phase associated with the gravitational scattering of a highly energetic light particle off a very heavy object in AdS spacetime has been computed to all orders in the ratio of the impact parameter to the Schwarzschild radius of the heavy particle.
Abstract: We consider the eikonal phase associated with the gravitational scattering of a highly energetic light particle off a very heavy object in AdS spacetime. A simple expression for this phase follows from the WKB approximation to the scattering amplitude and has been computed to all orders in the ratio of the impact parameter to the Schwarzschild radius of the heavy particle. The eikonal phase is related to the deflection angle by the usual stationary phase relation. We consider the flat space limit and observe that for sufficiently small impact parameters (or angular momenta) the eikonal phase develops a large imaginary part; the inelastic cross-section is exactly the classical absorption cross-section of the black hole. We also consider a double scaling limit where the momentum becomes null simultaneously with the asymptotically AdS black hole becoming very large. In the dual CFT this limit retains contributions from all leading twist multi stress tensor operators, which are universal with respect to the addition of higher derivative terms to the gravitational lagrangian. We compute the eikonal phase and the associated Lyapunov exponent in the double scaling limit.

17 citations

References
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Journal ArticleDOI
TL;DR: In this paper, a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom is given, based on plausible physical assumptions, establishing this conjecture.
Abstract: We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ L ≤ 2πk B T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

2,216 citations

Journal ArticleDOI
TL;DR: In this article, the authors used holography to study sensitive dependence on initial conditions in strongly coupled field theories and showed that the effect of the early infalling quanta relative to the t = 0 slice creates a shock wave that destroys the local two-sided correlations present in the unperturbed state.
Abstract: We use holography to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the t = 0 slice, creating a shock wave. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to the firewall controversy.

1,589 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that information can propagate in a quantum spin system only with a finite group velocity, where μ(ν) > 0, where ρ is the group velocity.
Abstract: It is shown that if Ф is a finite range interaction of a quantum spin system, τ t Ф the associated group of time translations, τ x the group of space translations, and A, B local observables, then $$ \mathop {{\text{lim}}}\limits_{\mathop {\left| t \right| \to \infty }\limits_{\left| x \right| > v\left| t \right|} } \left\| {\left[ {\tau _t^\varphi \tau \left( A \right),B} \right]} \right\|e^{\mu \left( v \right)t} = 0$$ (1) whenever v is sufficiently large (ν > V Ф ,) where μ(ν) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.

1,345 citations

Journal ArticleDOI
TL;DR: In this article, the problem of how fast a quantum system can scramble (thermalize) information, given that the interactions are between bounded clusters of degrees of freedom; pairwise interactions would be an example
Abstract: We consider the problem of how fast a quantum system can scramble (thermalize) information, given that the interactions are between bounded clusters of degrees of freedom; pairwise interactions would be an example Based on previous work, we conjecture: 1) The most rapid scramblers take a time logarithmic in the number of degrees of freedom 2) Matrix quantum mechanics (systems whose degrees of freedom are n by n matrices) saturate the bound 3) Black holes are the fastest scramblers in nature The conjectures are based on two sources, one from quantum information theory, and the other from the study of black holes in String Theory

925 citations

Journal ArticleDOI
TL;DR: In this article, a review of the mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime is presented.
Abstract: This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

910 citations