Chaotic fast scrambling at black holes

Jose L. F. Barbon* and Javier M. Magan†

Instituto de Fısica Teorica IFT UAM/CSIC, Campus de Cantoblanco, E-28049 Madrid, Spain(Received 9 June 2011; published 29 November 2011)

Fast scramblers process information in characteristic times scaling logarithmically with the entropy, a

behavior which has been conjectured for black hole horizons. In this paper we use the AdS/CFT context to

argue that causality bounds on information flow only depend on the properties of a single thermal cell, and

admit a geometrical interpretation in terms of the optical depth, i.e. the thickness of the Rindler region in

the so-called optical metric. The spatial sections of the optical metric are well approximated by constant-

curvature hyperboloids. We use this fact to propose an effective kinetic model of scrambling which

behaves like a compact hyperbolic billiard, furnishing a classic example of hard chaos. It is suggested that

classical chaos at large N is a crucial ingredient in reconciling the notion of fast scrambling with the

required saturation of causality.

DOI: 10.1103/PhysRevD.84.106012 PACS numbers: 11.25.Tq, 05.30.Ch

I. INTRODUCTION

It was proposed in [1,2] that one defining characteristicof black holes as quantum systems is their very fast rate ofinformation scrambling. In systems with local degrees offreedom and local interactions, the scrambling of informa-tion can often be related to a diffusion process. In this case,we can say that locally coded information ‘‘randomlywalks’’ around the system, covering a volume Ld in atime proportional to L2. For ‘‘normal’’ states, with exten-sive entropy, S / Ld, we find scrambling times propor-

tional to S2=d, measured in units of some typical timescale characterizing the state, such as the inverse tempera-ture � ¼ T�1 in relativistic systems. In contrast, thescrambling time for black holes is conjectured to be

�s � T�1 logðSÞ;with T the Hawking temperature and S the black holeentropy. Roughly, it corresponds to the local scramblingfor a system with infinite spatial dimension, d ! 1. It isvery interesting to elevate this estimate to the rank offundamental law of nature, and regard black holes as thefastest possible scramblers [2].

The fast-scrambling rate of black holes can be argued ina number of ways. It turns out that �s is the time scale thatsaturates the no-cloning bound in black holes, i.e. it is ameasure of the minimum time for information retrieval inthe Hawking radiation (cf. [1,2]). The same time scale canbe identified in a kinematical effect characteristic of the‘‘membrane paradigm’’ [3–5], where a conserved chargedistribution on the stretched horizon, induced by the mo-tion of external charges, undergoes nonrelativistic Ohmicdiffusion with respect to the Schwarzschild time variable.It follows that this induced charge spreads exponentiallyfast, filling a given area of horizon in a time which scales

logarithmically with this area [2]. Finally, from the point ofview of hypothetical holographic duals of the black hole,both matrix theory [6] and AdS/CFT [7] suggest the con-sideration ofN � N matrix models with S� N2 degrees offreedom. The nonlocal character of the interactions ofmatrix elements in ‘‘index space’’ suggests that, preciselyfor these systems, we may get the maximal possible scram-bling rate.It is somewhat striking that the two heuristic determi-

nations of the �s time scale, based, respectively, on the no-cloning argument and the membrane paradigm, shouldlook so different at a superficial level. In this paper, westart by reinterpreting these arguments as causality boundsin a theory with a stretched horizon. In particular, it ispointed out that the no-cloning time is just the minimalreflection time from the stretched horizon, across thenear-horizon region. With this definition, we analyze acompletely general near-horizon system embedded in ageneralized AdS/CFT background and find a universalanswer for such a ‘‘reflection’’ time:

�� ¼ � logðScellÞ; (1)

where � ¼ T�1 and Scell is the entropy contained in asingle ‘‘thermal cell’’ of the dual CFT, a region with thesize of the thermal length, � ¼ T�1, measured in the CFTmetric.On general grounds, the idea that information should

scramble ‘‘as fast as the speed of light’’ seems counter-intuitive, motivating the view that no local or quasilocalmodel of the stretched horizon can describe such aphenomenon.In our main result in this paper we propose a refinement

in this conceptual tension. We will adopt a very basicnotion of ‘‘scrambling’’ by pursuing the random walkmodel of classical diffusion. To this end, we study a formalsystem specified by a classically pointlike probe, executinga null random walk in the near-horizon geometry, withinteractions confined to the stretched horizon. Such a

*jose.barbon@uam.es†javier.martinez@uam.es

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simple kinetic model behaves as a standard ‘‘slow scram-bler’’’ when the random walk is regarded as completelyembedded in the stretched horizon. However, we point outthat the ballistic glides between interactions will explorethe whole Rindler region, resulting in a random walk withrather different properties.

The route to classical chaos in the near-horizon geome-try is actually found in the conformally related opticalmetric, which approximates any near-horizon region as aconstant-curvature hyperboloid. The main quantitative re-sult is an identification of the Lyapunov time of the nullrandom walk with the single-cell scrambling time (1). Thismodel has the added virtue of singling out the thermal cellas the maximal region with fast-scrambling properties, asthe null random walks always behave as slow scramblersover larger regions. Our final result is a combined estimatefor the scrambling time in a system much larger than athermal cell, given by

�s � � logðScellÞ�

S

Scell

�2=d

; (2)

where S is the total entropy and Scell is the entropy of asingle thermal cell. As indicated above, we shall argue thevalidity of this estimate in the regime S � Scell.

While Rindler space is metrically equivalent to flatMinkowski space, there are many dynamical properties,which depend on phase space rather than configurationspace, which are better characterized in terms of the opticalmetric. In particular, the noncompactness of the opticalmetric is directly related to the noncompactness of thephase space at black hole horizons. The stretching of thehorizon can be seen as a compactification of phase spacewhich renders black hole entropy finite. Our analysisshows that these ideas tile beautifully with the chaoticnature of scrambling, since a compact hyperboloid is aclassic paradigm of hard chaos.1

II. OPTICAL DEPTH AND CAUSALITY BOUNDS

In this section we interpret the time scale Rs logðSÞ in thenear-horizon physics of an ordinary Schwarzschild blackhole in terms of a causality bound. We begin with the no-cloning argument (see [1,2]) by noticing Fig. 1, whichillustrates the fact that the edge of the near-horizon region,given by the line XþX� ¼ �R2

s , is symmetric to thesingularity locus, XþX� ¼ R2

s , by a null reflection throughthe horizon. The points Q and Q0 are also symmetric andseparated a proper distance of order ‘P from the horizon.

If aQ-bit falls into the black hole from P toQwhile it iscloned at the horizon and returns within the Hawkingradiation at P0, a local verification of the cloning couldoccur inside the black hole ifQ and P0 can have a commonpoint S in the future. Notice that the points P and P0 are set

at height Rs because the cloned Q-bit must be efficientlydetected while it is emitted with a wavelength of OðRsÞ.This situation being incompatible with the linear evolutionof quantum mechanical states, it must be that such a localmeeting of the two copies is prevented by the occurrence ofthe singularity. In this way we obtain a lower bound on thetime delay between the two departures of the infallingQ-bit from P and P0.We compute this delay using the standard map between

Schwarzschild time and Kruskal coordinates, X� �Rs expð�t=2RsÞ. Since Q0 sits at the edge of the stretchedhorizon we have Xþ

Q0X�Q0 ¼ �‘2P. Furthermore, Xþ

P0X�P0 ¼

�R2s and, by the reflection conditions X�

S ¼ X�Q ¼

�X�P0 ¼ �X�

Q0 , we find

�tmin � 2Rs log

�XþP0

XþP

�¼ 2Rs log

�R2s

‘2P

�:

Noting that the entropy S� ðRs=‘PÞ2 we find, up to Oð1Þcoefficients and neglecting OðRsÞ additive contributions,

�tmin � �� ¼ Rs logðSÞ:The time-delay scale, ��, is equal to the ‘‘reflection time’’from the stretched horizon or, in order of magnitude, aminimal free-fall time to the stretched horizon across theRindler region. We shall denote this kinematical time scaleas the ‘‘optical depth’’.2 Since ‘P enters logarithmically,this time scale is very robust in order of magnitude. Forexample, the same result is obtained with the much lessconservative criterion that X�

Q >M�1, with M the mass of

FIG. 1 (color online). Diagram showing that theSchwarzschild time between P and P0 equals the reflectiontime of a photon from the stretched horizon, i.e. thepiecewise-null trajectory PQ0 [Q0P0, where Q and Q0 lie,respectively, at the inner and outer edges of the stretchedhorizon, i.e. on the hypersurfaces XþX� ¼ �‘2P.

1See [8] for another analysis of chaos in AdS/CFT, from quitea different point of view.

2The coincidence between the conjectured scrambling timeand the free-fall time was recently noticed in [9].

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the black hole, so that the infallingQ-bit is not heavier thanthe original black hole.

The actual relevance of these arguments for the issue ofinformation retrieval is not clear. First, it must be noted thatthe ‘‘same bit’’ can be extracted from a few Hawkingquanta provided one knows the black hole microscopicstate with precision [1], something that requires monitoringthe black hole for a time of the order of the evaporationtime (cf. [10]). Even then, the complete decoding of theinformation contained in a single Q-bit is likely to requiretime scales in excess of OðRse

SÞ, the Heisenberg time ofthe system controlling the detailed recurrences of the fullquantum state (cf. [11–16]). It is perhaps more appropriateto interpret �tmin as the causality bound on the minimumtime for the information to be returned to the edge of theRindler region, i.e. the minimum time for the informationto ‘‘resurface’’ in the outer layers of the black hole’sthermal atmosphere. Notice that, while the causality boundis saturated by a ‘‘mirror’’ boundary condition at thestretched horizon, the no-cloning heuristic argumentallows the ‘‘processing’’ of the Q-bits at the stretchedhorizon to introduce further delays of OðRsÞ, or even ofOð��Þ, without affecting the bound in order of magnitude.

A second argument introducing the time scale �� ¼Rs logðSÞ uses the dynamics of induced charge densitieson the stretched horizon according to the membrane para-digm [2–5]. These charges are nothing but rescaled ver-sions of the normal electric field at the stretched horizon,and their spacetime variations cannot violate local cau-sality. The fastest possible variations of such electromag-netic fields can be induced by letting a moving sourcecharge above the horizon approach the speed of light,leading naturally to the time scale ��, since that is thecharacteristic time for the source charge to cross theRindler region. To see that an exponentially fast spread,as measured in asymptotic time, is compatible with cau-sality we may recall a simple argument from [17] showingthat such a behavior is precisely saturating the causalitybound, rather than contradicting it. Let us define thestretched horizon in locally flat coordinates ðXþ; X�; X?Þas the Rindler surface

XþX� � �ðX0Þ2 þ Z2 ¼ ‘2P:

The future light-cone bounding causal events originating ata point of coordinates (X0 ¼ 0, X? ¼ 0, Z ¼ ‘P) on thestretched horizon is given by

� ðX0Þ2 þ ðZ� ‘PÞ2 þ X2? ¼ 0;

which intersects the stretched horizon itself along thehypersurface

X2? ¼ 2‘PðZ� ‘PÞ ! 2‘2Pe

t=2Rs ;

where we have inserted the longtime asymptotics of Z�Xþ � X�, with respect to the asymptotic Schwarzschildtime variable, showing the exponential rate that fills an area

of order X2? � R2

s in a time of order ��. Notice that the lightrays between two points on the stretched horizon do notpropagate confined to the stretched horizon, but actually goabove it through the bulk of the Rindler region, a simplefact that will be crucial in what follows.These arguments leave us in a quandary regarding the

interpretation of �� as a scrambling time. Since any processmeasured by scales of order �� is saturating causality inorder of magnitude, this means that the purported scram-bling must be not only fast but in fact as fast as it can be.Hence the suggestion that fast scramblers cannot behavelike standard local diffusion at all: a fast scrambler mustwork at ballistic speed. Meanwhile, interactions of probeQ-bits with the Hawking radiation are small at least for atime of order ��, since the ‘‘fall’’ is well approximated as afree fall.

Optical depth and the thermal cell

In this section we give an estimate of the optical depthfor a general near-horizon geometry with a putative holo-graphic dual. The main purpose of this exercise is toexpress the result in terms of dual quantum field theory(QFT) variables in a situation where the AdS/CFT loremay be applied.Consider a general ðdþ 2Þ-dimensional background

that may admit a holographic interpretation in terms of aðdþ 1Þ-dimensional field theory at finite temperature:

ds2 ¼ Fð�Þð�hð�Þdt2 þ d‘2Þ þ d�2

hð�Þ ; (3)

where � is the holographic radial coordinate, parametriz-ing the UV regime at large � and the IR regime at small �.If the model has the luxury of being defined as a perturba-tion of a well-defined UV fixed point, the warp factor hasthe asymptotic behavior Fð�Þ ! expð2�=R1Þ as � ! 1,with R1 the asymptotic AdS radius of curvature. Themetric of the dual QFT is defined as the asymptotic inducedmetric after removing the Fð�Þ warp factor, i.e.

ds2QFT ¼ �dt2 þ d‘2; (4)

where ‘ parametrizes the d-dimensional spatial manifoldon which the QFT is defined. The function hð�Þ modelsthermal effects, having a simple zero at the horizon,hð�0Þ ¼ 0, and approaching unity at large values of �,far from the horizon. The near-horizon, or Rindler region,reaches out to radii of order ��, defined by hð��Þ � 1. In

the linear approximation we have hð��Þ � h00ð�� � �0Þ �1, which gives an order-of-magnitude estimate for ��. The

Hawking temperature is given by

T ¼ h004�

ffiffiffiffiffiffiF0

p;

where F0 � Fð�0Þ.

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Throughout our discussion, we assume that the tempera-ture is much larger than any other energy scale in the dualQFT state, so that we can neglect finite-size effects in thegeometrical setup. The stretched horizon is set at Planckdistance from the horizon, at coordinate �� defined by

‘P ¼Z ��

�0

d�ffiffiffiffiffiffiffiffiffiffihð�Þp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� � �0

h00

s:

Using h� ¼ hð��Þ � h00ð�� � �0Þ we find the relation

�ffiffiffiffiffiffiffiffiffiffiffiF0h�

p � ‘P;

which expresses the fact that the local blue-shifted tem-

perature, proportional to ðFðzÞhðzÞÞ�1=2, grows from OðTÞat the edge of the Rindler region, to Planckian orderOðmPÞat the stretched horizon.

Null radial trajectories are better described in terms ofthe Regge–Wheeler coordinate, z defined by

dz ¼ � d�

hð�Þ ffiffiffiffiffiffiffiffiffiffiFð�Þp ;

where the minus sign is meant to indicate that z grows‘‘inwards’’. The metric in this frame reads

ds2 ¼ FðzÞhðzÞð�dt2 þ dz2Þ þ FðzÞd‘2; (5)

so that null radial trajectories satisfy dz=dt ¼ �1, and theoptical depth can be measured directly by the extent of thez coordinate. The same null trajectories occur in the con-formally related metric

ds2op ¼ �dt2 þ dz2 þ hðzÞ�1d‘2; (6)

so-called ‘‘optical metric’’ [18]. With an appropriatechoice of the additive normalization for the z coordinate,we find that the optical metric of the holographic back-ground (3) has the following universal structure: theasymptotic vacuum region is represented by a flat strip oflength �,

ds2op � �dt2 þ dz2 þ d‘2; 0< z < z� � �; (7)

with optical depth of order z� � �. The near-horizon or

Rindler region is well approximated by the metric

ds2op � �dt2 þ dz2 þ e4�Tðz�z�Þd‘2; z > z�; (8)

with the horizon sitting at z ¼ 1. Notice that the spatialsections of this metric are, locally in ‘-space, ðdþ1Þ-dimensional hyperboloids with radius �=2�, i.e. under

the change of variables y ¼ ð2�TÞ�1e�2�Tðz�z�Þ we have,on ‘-distance scales where the QFT metric (4) is approxi-mately flat,

ds2op � �dt2 þ��

2�

�2ds2

Hdþ1 ; (9)

where

ds2Hdþ1 ¼ dy2 þ d‘2

y2(10)

is the metric of the unit radius hyperbolic space in dþ 1dimensions. In these coordinates, the near-horizon regionis defined by 2�y �, with the horizon itself sitting aty ¼ 0. Therefore, trajectories of photons in the near-horizon optical metric trace a geodesic flow on the hyper-bolic spaceHdþ1. This conformal map between the opticaland the physical metric is nonsingular except at the hori-zon, a fact that will be obviated by the presence of thestretched horizon cutoff.It should be stressed that the asymptotic forms of the

optical metric (7) and (9) are approximations to the exactsolutions. Similarly, the precise value of the ‘‘mergingpoint’’ z� is ambiguous by a numerical multiplicative

factor of Oð1Þ. Since we are only interested in order-of-magnitude estimates of the time scales, this crude specifi-cation of the geometry is enough for our purposes. For thisreason, our estimates below are subject toOð1Þ ambiguitiesby additive and multiplicative factors.The optical depth of the Rindler region acquires the

simple interpretation of an ‘‘optical distance’’ to thestretched horizon:

�� � z� � z� � � log

��� � �0

�� � �0

���� logðh00‘PÞ

� � log

� ffiffiffiffiffiffiF0

pT‘P

�; (11)

where we have neglected numerical factors of Oð1Þ.In order to translate (11) into dual QFT variables, we

compute the entropy, or horizon volume, in Planck units:

S� 1

‘dP

Zdd‘ð ffiffiffiffiffiffi

F0

p Þd � V

� ffiffiffiffiffiffiF0

p‘P

�d;

with V ¼ Rdd‘ the volume in the QFT metric. With

these conventions, the entropy in a QFT volume of onethermal length, Vcell � �d, is given by Scell ¼ S=ncell �ðT‘P=

ffiffiffiffiffiffiF0

p Þ�d, where ncell ¼ VTd is the number of ‘‘ther-mal cells’’. Returning now to (11), we find, up to Oð1Þnumerical coefficients, the following expression in dualQFT variables

�� � � log

�S

ncell

�¼ � logðScellÞ; (12)

as claimed in the introduction. We find that the no-cloningtime scale �� is sensitive to the parameters of a singlethermal cell. This is actually natural, since the emissionof a typical Hawking bit can be localized spatially on thehorizon within a region of size � in ‘-space. Therefore, thebehavior of regions larger than a thermal cell have a lessdirect relevance to the issue of information retrieval by asingle Hawking bit.The entropy per thermal cell, Scell, estimates the

number of microscopic QFT degrees of freedom partici-pating on a thermal state at temperature T, and supportedon length scales of order�, so that we may use the notation

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Scell ¼ NeffðTÞ. For a conformal field theory Neff is asymp-totically independent of the temperature and proportionalto the central charge, of orderNeff � N2 for CFTs based onYang–Mills theories. For nonconformal theories it gives asort of ‘‘running’’ central charge with nontrivial scaledependence. This running central charge also controlsother observables related to counting of degrees of free-dom, such as the entanglement entropy computed in thegravity prescription [19] (see [20] for such a discussion).

III. SCRAMBLING AND HARD CHAOS

The heuristic arguments leading to the �� ¼ � logðScellÞtime scale, essentially causality bounds, are based on aphenomenological treatment of the horizon. Standardlarge-N formulations of the AdS/CFT rules are not sensitiveto the finiteness of a horizon entropy, implying infinitevalues for the scrambling time scale, as well as the(Heisenberg) time scale of Poincare recurrences. The onlyhorizon time scale that is visible at N ¼ 1 is the quasinor-mal behavior of local correlation functions, i.e. the expo-nential decay at large times

hOðtÞOð0Þi � e�t=�� ; t � �� � �: (13)

This behavior is dual to the familiar no-hair property ofbulk horizons and characterizes the loss of information inthe N ¼ 1 system.3 In terms of the bulk description, thequasinormal behavior is associated with the continuousspectrum of bulk fields in the vicinity of the horizon, inparticular, the single-particle frequency spectrum ! is ob-tained from the eigenvalues of the operator (cf. [22,23])

! 2 ¼ � d2

dz2þ VeffðzÞ; (14)

where the effective potential has a universal form near thehorizon of (3),

VeffðzÞRindler /�F0m

2 þ c

"2

�expð�4�TzÞ; z� z� ��;

(15)

featuring an exponential barrier controlled by the tempera-ture T. Herem is the mass of the bulk field andp2

‘ � c="2 isthe effective momentum-squared introduced by the UVlocalization of the particle probe within a cutoff length ",as measured by the ‘ coordinate. Local CFT correlationfunctions are defined as boundary correlation functions ofsuch bulk fields, and the quasinormal behavior (13) isrelated to tunneling through the exponential barrier seen in(15).

The dynamical time scales at the horizon are renderedfinite by specifying a phenomenological ‘‘stretching,’’whose crudest version is the brick-wall model [22],consisting on a sharp mirror at Planckian distance fromthe horizon. This simple boundary condition discretizes thespectrum of fields in the right amount to account for thefiniteness of the entropy, even for a thermal state of freequantum fields. Such a finite entropy turns out to beextensive in the optical metric,

S� VopTdþ1;

with the optical volume of the region outside the stretchedhorizon scaling as Vop � �VNeff (cf. [18,23]). This simple

argument shows that the optical metric provides a goodgeometrical intuition for the ‘‘phase space size’’ of thethermal atmosphere, i.e. the number of states supportedby the system. In the conformally related frame of theoptical metric, the Hartle–Hawking state of quantum fieldsis a homogeneous thermal state of constant temperature Tover a cutoff version of the hyperboloid (10).This very crude model reproduces the right time scale

of recurrences [14]. It also introduces a ballistic free-falltime scale of order ��, which is not obviously related to‘‘scrambling’’ since interactions of localized probes withthe thermal atmosphere are suppressed by powers of1=Neff . On the other hand, low-energy effective field theorymust be violated at the stretched horizon, so that the 1=Neff

estimate for the interactions breaks down at the wall. Thisis one of the basic tenets of black hole complementarity[24] and can be seen in this context by noticing that aPlanck-mass field m�mP does make a contribution oforder T2 to the effective potential (15) when dragged tothe stretched horizon z� z�. Hence, we can scramble theprobe by letting it interact strongly at the wall, after aballistic glide of �� duration.A naive treatment of such a wall-scrambling seems to

suggest a behavior of slow scrambler type. In the optical-frame description, the state of the bulk quantum fields isthermal with uniform temperature T. Once the probereaches the stretched horizon, if confined there, one cansee it scatter on time scales of order�, exercising a randomwalk through the stretched horizon with step �. Since the

QFT thermal cell has optical size �ðNeffÞ1=d, the time toscramble the whole QFT thermal cell is of order

�ðNeffÞ2=d � �ðScellÞ2=d, i.e. the characteristic behavior ofa slow scrambler.4

3The quasinormal time scale �� � � also controls the large-Ntime-dependence of certain nonlocal (but sharp) operators, suchas the entanglement entropy (cf. [21]).

4The same result follows by thinking in terms of local staticobservers at the stretched horizon, who measure local Plancktemperatures. In this frame, the Planck-size probe will execute arandom walk of Planck step. Since there are Scell Planckian cellsin a single QFT thermal cell, the random walk will cover it in a(local) time ‘PðScellÞ2=d. Upon redshifting back to QFT-time, t,we see that a QFT thermal cell is scrambled in a time of order�ðScellÞ2=d.

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In the next section we argue that this estimate is not quitecorrect. The reason is that, while the collisions are happen-ing at the stretched horizon, the ballistic glide betweencollisions is not necessarily confined to the stretched hori-zon hypersurface. In fact, as already pointed out in Sec. II,the faster free-fall motion between two points on thestretched horizon takes a path above the stretched horizon.This fact, combined with the peculiar geometry of theRindler space, will suffice to provide a kinetic model ofthe thermal atmosphere with fast ‘‘scrambling’’ features.

Optical metric and the chaotic billiard model

Let us consider a Planck-localized probe following nullgeodesics in the Rindler region, punctuated by collisions ofOð1Þ strength at the stretched horizon [we neglect thecollisions of Oð1=NeffÞ strength within the bulk of thethermal atmosphere]. In order to benefit from the geomet-rical intuition, we notice that any such piecewise-nulltrajectory defines a corresponding piecewise-null trajec-tory on the conformally related optical manifold (8) withhyperbolic spatial sections. Therefore, we can use theoptical metric to discuss the dynamical time scales ofsuch an ideal probe motion. The advantage of this descrip-tion is the uniform proportionality between time scales fornull propagation and ‘‘optical’’ length scales. Of course,the time coordinate t is the same in the physical Rindlermetric and the optical metric, so that no rescaling of timevariables is necessary in interpreting the dynamical timescales of the random walk, provided it is really piecewisenull.5

The optical box supports a thermal state of bulk fieldsat uniform temperature T, which can be approximated asan ideal gas (up to small 1=Neff effects,) except at thestretched horizon, where the thermal fields are regardedas strongly interacting, with a bulk thermal cell of (optical)size �.

Consider a localized probe sent radially inwards with amaximal (Planck) bulk resolution at the top of the Rindlerregion, as measured in the physical metric (3), i.e. theinitial resolution �‘i in the ‘ coordinate satisfies

F0ð�‘iÞ2 � ‘2P:

The geometrical stretching of the optical metric impliesthat such a probe is smeared over a region whose opticalsize in the ‘ directions is of order

�sop � �‘iffiffiffiffiffih�

p � ‘PffiffiffiffiffiffiffiffiffiffiffiF0h�

p � �

on arrival to the stretched horizon. The thermal state ofstrongly interacting degrees of freedom at the stretched

horizon has thermal length of Oð�Þ (again, in the opticalmetric), which determines the effective interaction lengthof probe scattering at the stretched horizon. Since theeffective resolution in the impact parameter for these scat-tering events is also of Oð�Þ, we conclude that the out-going scattering angle at each collision has maximaluncertainty.In between scattering events the probe is assumed to

glide freely on the Rindler region. Maximizing the glidevelocity, we can relate the motion to a random walk on aðdþ 1Þ-dimensional hyperboloid with steps made of geo-desic arcs on the spatial sections of the optical geometry:

ds2op � �dt2 þ��

2�

�2 dy2 þ d‘2

y2: (16)

In the Poincare coordinates of (16), each free glidebetween collisions is a circular arc with radius �y��‘=2 if the collision points are separated a distance �‘in the QFT metric (cf. Fig. 2). The maximum extent of acircular glide is given by arcs of radius ymax � �. Thereason is that the scarce6 trajectories with larger radii hitthe end of the Rindler region and are effectively reflectedback by the confining wall of the asymptotic AdS region,the resulting complete trajectory returning to the stretchedhorizon at �‘ < �. We thus conclude that, after a fewscattering events, there is complete uncertainty about theposition of the probe within a distance scale of the order ofthe QFT thermal length. Since each maximal glide throughthe optical hyperboloid takes a time ofOð��Þ, we concludethat the ‘‘position’’ scrambling of the Planckian probewithin the thermal cell has been achieved in times of order�� � � logðScellÞ.

FIG. 2. Near-horizon random walk of a localized probe byscattering at the stretched horizon, pictured in the Poincarecoordinates of the spatial optical metric (16). Free paths betweensuccessive collisions are circular arcs, with maximal radius�ymax � �, since longer glides are reflected back by the asymp-totic AdS potential well which starts at the edge of the Rindlerregion, represented in this picture by a dashed line.

5If the random walk is not piecewise null, the correspondingscrambling times will be larger, so that the null-walk modelprovides the lower bound on the scrambling times.

6The proportion of solid angle enclosing those trajectorieswhich are ‘‘vertical’’ enough to hit the end of the Rindler regionis of the order of �d=VolðSdÞ � ðh�Þd=2 � 1=Neff .

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The kinetic model presented here regards the near-horizon region as a chaotic billiard over a negative-curvature space, one of the early examples of hard chaosin classical mechanics [25,26]. Indeed, if we substitute thestretched horizon by a random arrangement of hard reflect-ing spheres of size � in the optical metric, we get essen-tially the same picture. The Lyapunov exponent, definedlocally in terms of the divergence rate of geodesics, is oforder �L � ��1. The Lyapunov time, defined as the timefor complete erasure of localization information, from aninitial resolution �‘i to a final resolution of one thermallength �‘f � �, is of order

�L � ��1L logð�‘i=�‘fÞ � � logð‘P=�

ffiffiffiffiffiffiF0

p Þ � ��;

since complete ignorance over the position of the Planck-size probe on the extent of a thermal cell is achieved afterOð1Þ collisions, with characteristic collision time ofOð��Þ.

Our proposed ‘‘ballistic catalysis’’ of the scrambling isonly effective as long as the glides are large geodesic arcson the hyperbolic optical geometry (16). Since the finiteextent of the Rindler region imposes an effective cutoff tothe size of these arcs, on time scales larger than �� thesystem behaves as a slow scrambler with effective timestep ��. Therefore, the whole system with ncell thermal

cells is scrambled in a time �s � ��ðncellÞ2=d. In this waywe arrive at our main result of the paper: the scramblingtime of a general horizon as a function of the entropy andthe effective number of degrees of freedom

�s � � logðNeffÞðncellÞ2=d � � logðNeffÞ�

S

Neff

�2=d

: (17)

IV. CONCLUSIONS

We have discussed aspects of the fast-scrambling timescale, introduced in [1,2] as proportional to the logarithmof the entropy, from the viewpoint of the AdS/CFTcorrespondence.

We first pointed out that a naive implementation of theno-cloning bound singles out the free-fall time scale acrossthe Rindler region, �� � � logðScellÞ, as the important fig-ure of merit. We have shown that there is a natural geo-metrical interpretation of this quantity in terms of the‘‘depth’’ of the stretched horizon in the optical metric foran arbitrary AdS/CFT background.

The dependence of �� on the entropy of a single thermalcell, Scell, deserves special comment. In particular, thisresult was derived under the assumption that the QFTthermal state is extensive on length scales larger than theinverse temperature. For very cold states it may happenthat the system is actually smaller than its thermal length.One interesting example is given by the Reissner–Nordstrom black hole, for which the entropy remains finitein the limit of divergent thermal length: � ! 1.

Computing the optical depth for such a near-extremalblack hole one finds �� � � logðSÞ, rather than (1) (noticethat both laws are equivalent for Schwarzschild blackholes, since those have a single thermal cell on theirhorizon). It turns out that the rules of the AdS/CFT corre-spondence ensure that it is the total entropy, rather than theentropy per thermal cell, the relevant quantity when thescrambler is smaller than its thermal length [27].The hyperbolic character of the near-horizon optical

metric suggests that fast scrambling may be compatiblewith a purely kinetic picture of information diffusion. Inparticular, local probes scattering at the stretched horizonwith null free paths do scramble within a single thermalcell in just a few collision times, because the negativecurvature amplifies any initial resolution error of the probe,making the near-horizon region into an effective chaoticbilliard with Lyapunov time of order ��. On scales largerthan a single thermal cell, the subsequent scrambling fol-lows the pattern of standard slow scramblers with time stepof Oð��Þ, leading to our main estimate

�s � ��ðncellÞ2=d (18)

of the scrambling time of a general fast scrambler withncell 1.Our result indicates that classical chaos may be a key

element in making compatible the built-in large-N localityof bulk dynamics with the apparently gross violation oflocality implied by the very fast scrambling of informationat generic horizons. This includes very interesting physicalsituations of a different kind, such as de Sitter spacetime,for which there is no concrete holographic description.On the other hand, it is not obvious that the kinetic

notion of scrambling adopted here is the relevant onefrom the point of view of the issues of black hole infor-mation retrieval, as laid down in [1]. In particular, whileclassical chaos implies a certain notion of scrambling inposition space for local probes, over times of order �� onlya few scattering events can take place and thus only a smallsubspace of the complete Hilbert space has a chance toentangle with the probe. In addition, nonlocal probes suchas extended branes may behave in quite a different wayfrom the ‘‘billiard paradigm’’ outlined here, so thatstrongly nonlocal physics on the scale of the thermal lengthmay still be required to achieve scrambling in the sense ofthe quantum entanglement test [1].It is conceivable that the details of the scrambling will

depend on the structure of the state, such as the initialdegree of localization of the information. In this context,the mechanism proposed here is mainly a fast way ofhomogenizing the quantum state over the thermal cell,which may be subsequently scrambled nonlocally through-out the whole Hilbert space of dimension expðNeffÞ, as in amatrix model. Both processes could be accomplished intimes of Oð��Þ, so that the complete scrambling timewould still satisfy �s � �� within numerical factors.

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We leave these and other fascinating issues for futurework. It would be of great interest to independently check(17) in a toy model of the type studied in [28] (see therecent Ref. [29]), although the relevance of the Rindlerregion in the heuristics behind (17) suggests that CFTswith large gaps in the conformal dimensions (cf. [30]) areneeded to see the logðNeffÞ term. More generally, we canonly expect that quantum chaos will enter as a key idea inthe characterization of black hole information processing.

ACKNOWLEDGMENTS

We are indebted to C. Gomez, E. Lopez, and E.

Rabinovici for useful discussions. The work of J. L. F. B.

was partially supported by MEC and FEDER under Grant

No. FPA2009-07908, the Spanish Consolider-Ingenio 2010

Programme CPAN (CSD2007-00042), and Comunidad

Autonoma de Madrid under Grant No. HEPHACOS

S2009/ESP-1473. J.M.M. is supported by MICINN.

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