BLACK HOLES AND THE BUTTERFLY EFFECT A DISSERTATIONxc388jb9020/... · black holes and the butterfly effect a dissertation submitted to the department of physics and the committee

BLACK HOLES

AND THE BUTTERFLY EFFECT

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Douglas Stanford

July 2014

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/xc388jb9020

© 2014 by Douglas Stanford. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/xc388jb9020

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Leonard Susskind, Primary Adviser


Patrick Hayden


Stephen Shenker

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

What happens if you perturb a small part of a large system, and then you wait a while? If the

system is chaotic, one expects the butterfly effect to push the state far from its original trajectory. I

will present an analysis of this phenomenon in the setting of a strongly interacting quantum gauge

theory, using the tools of gauge-gravity duality. The original state corresponds to a black hole

geometry, and the perturbation is represented by a particle falling through the horizon. As time

passes, the boost of the particle grows exponentially, creating a shock wave that implements the

butterfly effect. Building on this framework, I will relate and explore the dynamics of chaos and the

region behind the horizon of a black hole.

This thesis is based on the papers [1] and [2], written with Stephen Shenker. It should not be

cited without also referencing those papers.

iv

Acknowledgments

I am deeply grateful to both Lenny Susskind and Steve Shenker for sharing some of their intuitions,

tools, and enthusiasm for physics. I started learning from Lenny during my first class as a college

freshman, and I can’t see the top yet.

I am also grateful to my other collaborators, including Daniel Harlow, Patrick Hayden, Nima

Laskhari, Dan Roberts, Ahmed Almheiri, Don Marolf, Joe Polchinski, and Jamie Sully. My work

was supported by the Stanford Institute for Theoretical Physics, by the NSF Graduate Research

Fellowship Program, and by a Graduate Fellowship from the Kavli Institute for Theoretical Physics.

v

Contents

Abstract iv

Acknowledgments v

1 Introduction 1

2 Black holes and the butterfly effect 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 A qubit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 A holographic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Unperturbed BTZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 BTZ shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Mutual information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.5 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 String and Planck scale effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Appendix A: Haar scrambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Appendix B: Geometrical generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7.2 Solutions with localized sources . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Multiple Shocks 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Wormholes built from shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 One shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Two shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Many shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vi

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Appendix A: Recursion relations for many shock waves . . . . . . . . . . . . . . . . . 41

3.6 Appendix B: Vaidya matching conditions . . . . . . . . . . . . . . . . . . . . . . . . 42

Bibliography 44

vii

List of Figures

2.1 Mutual information (upper, blue) and spin-spin correlation function (lower, red) in

the perturbed state |Ψ′〉, as a function of the time of the perturbation tw. The delay

is a propagation effect; if the perturbation at site five is sufficiently recent, sites one

and two are unaffected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The Kruskal diagram (center) and Penrose diagram (right) for the BTZ geometry. . 10

2.3 The Kruskal and Penrose diagrams for the geometry with a shock wave from the left,

represented by the double line. The dashed v = 0 and v = 0 horizons miss by an

amount α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 In the unperturbed BTZ geometry (left), a smooth horizon requires the black mode

on the left to be highly entangled with the blue mode on the right. By contrast, in the

shock wave geometry (right) the black and blue modes are far apart and unentangled.

Instead, the black mode is entangled with the green mode coming out of the white

hole. The arguments of [65] suggest that the green mode may be complicated in the

CFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 The geometry dual to Eq. (3.4One shockequation.3.2.4) consists of a perturbation

that emerges from the past horizon and falls through the future horizon (left). If t1 is

sufficiently early, the boost relative to the t = 0 slice generates backreaction in that

frame (right). Note that the horizons no longer meet. . . . . . . . . . . . . . . . . . 29

3.2 The dual to a two-W state is constructed from the one-W state by adding a pertur-

bation near the boundary at time t2 and then evolving forwards and backwards. . . 30

3.3 As t2 shifts earlier, the time at which the original shock reaches the boundary shifts

later, eventually moving onto the singularity (right). . . . . . . . . . . . . . . . . . . 32

3.4 The thermofield double and the first six multi-W states are drawn. In each case, the

next geometry is obtained from the previous by adding a shock either from the top

left or bottom left corner. The gray regions are sensitive to the details of a collision,

but the white regions are not. Using the time-folded bulk of [64], these states can be

combined as different sheets of an “accordion” geometry. . . . . . . . . . . . . . . . . 33

viii

3.5 A geodesic passes across a portion of the wormhole. It intersects the null boundaries

of the central regions halfway across their width. . . . . . . . . . . . . . . . . . . . . 34

3.6 The large-α four-W geometry is shown. Notice that the post-collision regions are small

and isolated near the singularities. The Kruskal diagram at the bottom emphasizes

the kinkiness of the geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7 The wormhole created from a large number of weak shocks (top) becomes a smooth

geometry in the α→ 0 limit (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.8 Bob falls in from the boundary at tB = 0 and experiences a mild interaction with

the stress energy supporting the solution. If he jumped in at a much earlier time

tB ∼ −t∗, he would experience a dramatic interaction. . . . . . . . . . . . . . . . . . 39

3.9 A candidate for the geometrical dual to a typical state? . . . . . . . . . . . . . . . . 39

3.10 The size of the S1 at the vertices is labeled rn, and the R parameter of the BTZ

geometry forming each plaquette is labeled Rn. . . . . . . . . . . . . . . . . . . . . . 41

ix

Chapter 1

Introduction

Consider a warm bath of water. Or, better, consider a warm bath of water in a physicist’s thought

laboratory, where we are free to run time backwards and forwards, and we are able to record the

precise positions of all of the water molecules. Suppose that we are particularly interested in one of

the 1028-odd molecules in the tub. We will refer to this molecule as A. As time passes, A is buffeted

by the other molecules, and, under their influence, it traces out a rather random path through space.

What determines the shape of this path? In fact, the detailed shape is sensitively dependent on the

initial conditions for every other molecule in the system.

To reveal this dependence, let us consider the following experiment. First, starting with the

configuration at t = 0, we run the system backwards in time for an interval tw. Second, we add a

new particle W at some location in the bath. Third, we evolve the system forwards in time again

to t = 0. What do we get? In particular, where does A end up? The answer depends on the time

tw and the distance between A and W . If tw is short enough, or if the distance is large enough,

causality ensures that adding W won’t have time to affect A, so the forwards evolution will cancel

out the backwards evolution, and A will end up where it started. On the other hand, if tw is large

enough, one expects that A will be buffeted by particles that themselves were buffeted by particles

that themselves were buffeted... by a particle that was buffeted by W . Under this circumstance,

the new momentum of A at t = 0 will be randomly re-oriented with respect to its original t = 0

momentum.

This phenomenon is one manifestation of the “butterfly effect:” changing the state of a single

particle at time −tw changes the states of all particles a short while later. This effect is best

understood in the context of classical mechanics, but a very similar phenomenon is expected for

many quantum systems. The trouble is that it is difficult to analyze, almost by definition. Think

about the bath of water: to calculate what happens to particle A, we would have to be able to

accurately study the fine-grained dynamics of 1028 water molecules. To solve the problem quantum

mechanically, we’d have to evolve forward a system of 101028

equations. Such direct approaches are

1

CHAPTER 1. INTRODUCTION 2

clearly impractical. However, this doesn’t mean that it is impossible to make progress. The field

of classical chaos is full of instructive and beautiful model systems (recursion relations, motion on

hyperbolic space, small-scale numerical studies) and clear general features (exponential divergence

in phase space controlled by Lyapunov exponents). By comparison, quantum chaos is rather poorly

understood.

In this thesis, we will provide a computable model system for quantum chaos. In order to do so,

we will set up a thought experiment very similar to the one described above, but with the tub of water

replaced by a particular type of strongly interacting quantum field theory. We will then analyze

the experiment using the AdS/CFT correspondence [3].1 This duality is a conjectured equivalence

between certain quantum field theories (systems of particles interacting non-gravitationally) and

gravitational systems in one higher dimension. The utility of this framework is that the strongly

interacting (and hard to study) limit of the quantum field theory is associated with a weakly-coupled

(easy to study) limit of the gravitational system.

In order to mimic the warm bath of water, we would like to start out with a thermal state of

the quantum field theory. It has been understood for some time that the AdS/CFT dual to such

a configuration is a black hole [5, 6].2 Perturbing the state by adding W corresponds to throwing

a particle into the black hole. By carefully studying the effect of this particle on the geometry of

the black hole, we will be able to read off the sensitivity of A to the addition of W . As we will see

in chapter 2Black holes and the butterfly effectchapter.2, the key feature is the boost-like nature

of time evolution in the black hole geometry. As we make the time of the perturbation earlier and

earlier, the added particle gets more and more boosted. Eventually, the energy of the perturbation

is large enough that it creates a gravitational shock wave, disrupting the geometry of the original

black hole and disturbing the degree of freedom A.

The analysis of this shock wave provides a clear picture of the butterfly effect in a strongly coupled

field theory, but it also provides quantitative insight. For example, for a quantum field theory with

an Einstein gravity dual, we can compute the time t∗(x, y) it takes for a W perturbation at location

x to affect the degree of freedom A at location y. One finds

t∗(x, y) =β

2πlogN2 +

√d

2(d− 1)|x− y|. (1.1)

In the first term, β is the inverse temperature of the quantum field theory, and N2 is the number of

degrees of freedom per site. In the second term, d is the spacetime dimension of the quantum field

theory, and |x− y| is the distance between the points. Taken together, we can interpret the formula

as follows. The small perturbation has very little effect on any other system for a time β2π logN2.

(The time is longer if there are more degrees of freedom in the system, but only logarithmically

1For a review, see [4].2This is more exotic than it sounds: black holes are the simplest nontrivial objects in the gravitational system,

and the fact that they are thermal has been understood since the 1970s [7].

CHAPTER 1. INTRODUCTION 3

so; this is characteristic of “fast scrambling” [8, 9, 10].) After this time, the effect spreads out

ballistically at a speed√d/2(d− 1), measured in units of the speed of light.3

Most of chapter 2Black holes and the butterfly effectchapter.2 is dedicated to the analysis just

described. In chapter 3Multiple Shockschapter.3, we will see a second application of the same tools.

The basic question is simple: if throwing a single particle into a black hole can create a highly

energetic shock wave, then what happens if we add many particles at various different times? This

investigation is partially motivated by the recent surge of interest in understanding the geometry

behind the horizon of a black hole in a typical state, stimulated by [11, 12]. Although we are not able

to say anything decisive about this important question, we will present a large collection of new black

hole geometries dual to perturbations of the thermofield double state. The general structure that

we identify includes a network of intersecting shock waves, one for each perturbation, supporting a

very long wormhole behind the horizon of the black hole. This structure is reminiscent of the ideas

in [13].

The main portion of this dissertation consists of two papers, both of them equal collaborations

with Stephen Shenker.

3Daniel A. Roberts has shown that both the coefficient of the logarithm and the coefficient of |x − y| receive α′

corrections. He has also checked, however, that the general form is preserved, at least in Gauss-Bonnet gravity.

Chapter 2

Black holes and the butterfly effect

This chapter consists of a paper [1] written in collaboration with Stephen Shenker, and published

as “Black holes and the butterfly effect,” JHEP 1403, 067 (2014) [arXiv:1306.0622 [hep-th]]. The

original abstract is as follows:

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.

2.1 Introduction

Entanglement is a central property of quantum systems. It plays a crucial role in the theory of

quantum information, quantum many body systems and quantum field theory. Two subsystems

A and B of a quantum system are entangled in the state |ψ〉 if the total Hilbert space H can

be decomposed into subfactors, H = HA ⊗ HB and the density matrix ρA obtained by tracing

out HB , ρA = trHB[|ψ〉〈ψ|], is not pure. This can be diagnosed using the von Neumann entropy

SA = −trHA[ρA log ρA] which is greater than zero if and only if |ψ〉 is entangled.

Entropy of entanglement of the ground state can be used as a diagnostic of topological order

in gapped quantum systems [14]. In conformal quantum field theories (CFTs) defined on a sphere,

the entropy of entanglement between hemispheres of the vacuum state has been shown to be the

correct measure of the number of degrees of freedom which decreases under renormalization group

flow, encompassing the c, a and F theorems [15, 16].

Entanglement in highly excited states is also of great importance. If |ψ〉 is a typical state and A

4

CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 5

is a small subsystem then ρA describes a thermal distribution. B serves as a heat bath for A. An

exactly thermal density matrix can be obtained from a pure entangled state using the thermofield

double construction. Consider two identical subsystems, L and R. Write a pure state |Ψ〉 in the total

Hilbert space:

|Ψ〉 =1

Z1/2

∑n

e−βEn/2 |n〉L|n〉R. (2.1)

Tracing over the R Hilbert space leaves a precisely thermal density matrix for the L system:

ρL =1

Z

∑n

e−βEn |n〉〈n|. (2.2)

These ideas have a holographic realization in the AdS/CFT correspondence [6]. If the L and R

systems are CFTs with AdS duals, and the temperature is sufficiently high, then |Ψ〉 describes a

large eternal AdS Schwarzschild black hole with Hawking temperature TH = 1/β. In this context

|Ψ〉 is referred to as the Hartle-Hawking state. The UV degrees of freedom of the L and R CFTs

describe dynamics at the disconnected large radius asymptotic regions of the eternal black hole

geometry. The entropy of entanglement SL = −tr[ρL log ρL] is the Bekenstein-Hawking entropy of

the black hole given by the area of the event horizon, SL = Ah/4GN .

Entanglement entropy has a more general holographic interpretation. It was proposed by Ryu

and Takayanagi [17] (RT) that the entanglement entropy of a region A in a CFT in a state |ψ〉 is

given by the area (in Planck units) of the minimal area codimension two spacelike surface whose

asymptotic boundary is the boundary of A in the geometry dual to |ψ〉. This proposal was first

proved in the case of spherical boundaries in [16] and recently explained in the most general static

case in [18]. The RT proposal has been extended to nonstatic geometries in [19].

Many thermal systems share another basic property–chaos. Starting from rather special states

these systems evolve to much more disordered typical states. There is sensitive dependence on

initial conditions, so that initially similar (but orthogonal) states evolve to be quite different. In the

subject of quantum information and black holes, such chaotic behavior has come to be referred to

as “scrambling,” and it has been conjectured that black holes are the fastest scramblers in nature

[8, 9, 10]. The time it takes such fast scramblers to render the density matrix of a small subsystem

A essentially exactly thermal is conjectured to be t ∼ β logS where S is the entropy of the system.

Scrambling can disrupt certain kinds of entanglement. In particular, if the pattern of entangle-

ment is characteristic of an atypical state, scrambling, which takes the state toward typicality, can

destroy it. This interplay is at the heart of the firewall proposal [11]. These authors argue that

the existence of a smooth region connecting the outside and inside of the horizon requires special

entanglement of degrees of freedom on the two sides. But during the evaporation of the black hole

the system scrambles, and these delicate correlations are destroyed. No smooth region can remain.1

1A related argument was provided in [20], along with a claimed resolution that relies on a non-standard model of


In this paper we will study the interplay of entanglement and scrambling using holographic

tools, assuming the validity of the classical bulk geometry. We will use a fine grained measure of

the correlation between two subsystems called mutual information. If A and B are subsystems then

the mutual information I is defined to be I = SA + SB − SA∪B .

This quantity has been studied holographically using RT surfaces in a number of papers. Mutual

information and entanglement entropy have been used to diagnose thermalization after a quantum

quench, in both conventional [21, 22] and holographic setups [19, 23, 24, 25, 26, 27, 28, 29]. A

common feature in the evolution of I is a sharp transition in which the connected A ∪ B minimal

surface exchanges dominance with the union of the disconnected A and B surfaces. At this point, I

goes to zero and stays there in a continuous but non differentiable way.2

Here we will focus on the eternal black hole setup discussed above, with regions A in the L

system and B in the R system. I has been studied for this situation in [30, 31]. The surface

determining SA∪B may pass behind the horizon, giving some information about that region. In

particular, Hartman and Maldacena [31] studied I between two regions as both boundary times are

increased.3 The Hartle-Hawking state |Ψ〉 is not invariant under this time evolution and I rapidly

decreases, going to zero linearly in a thermal time β.

Van Raamsdonk et. al. [32, 33] made the important point that while an arbitrary unitary

transformation applied to the left handed CFT leaves the density matrix describing right handed

CFT observables unchanged, it will change the relation between degrees of freedom on both sides

and hence the geometry behind the horizon. Certain unitaries correspond to local operators, which

can create a pulse of radiation propagating just behind the horizon which in some ways resembles a

firewall.

The new feature that we will explore is sensitivity to a very small initial perturbation. We

imagine choosing regions A and B in the L and R CFTs at time t = 0. Because of the atypical local

structure of entanglement in the thermofield double state, A and B may be highly entangled, even

if they are small subsystems of L and R. The state at an earlier time, −tw does not have these

correlations but is carefully “aimed” to give them at t = 0. We then consider the effect of injecting a

small amount of energy E into the L system, by throwing a few quanta towards the horizon at time

−tw. One expects that the CFTs dual to black holes have sensitive dependence on initial conditions,

and this small perturbation should touch off chaotic behavior in the L theory, disturbing the careful

aiming. The resulting Schrodinger picture state at t = 0, |Ψ′〉, should be more typical than the

thermofield double state. In particular, it should have less entanglement between A and B. 4

Hawking radiation.2When we say I is zero we mean the coefficient of 1

GN, or in the large N field theory context the coefficient of N2,

vanishes. There will continue to be a nonzero value of subleading strength.3In this paragraph and the one below, we are referring to the physical time conjugate to HR + HL, which runs

forwards on both CFTs. In the rest of the paper t will refer to the Killing time, which is conjugate to HR −HL andso runs forwards on the right CFT but backwards on the left.

4The entanglement between L and R as a whole will remain unchanged. The entropy SL is invariant under anyunitary operator, no matter how chaotic, applied to HL.


At first, this presents a puzzle: entanglement is determined by geometrical data, and, naively,

the geometry is unaffected by the addition of a few quanta. However, the boundary time t = 0

defines a frame in the bulk, and relative to this frame, the quanta released a time tw in the past

will have exponentially blue-shifted energy. Their backreaction must be included. The relevant bulk

geometry can be described as a shock wave [34], a limiting case of a Vaidya metric. Closely related

configurations have been discussed in a context very similar to ours by [32, 35]. 5 In the 3D BTZ

case that we focus on, the dimensionless effect of the quanta on RT surfaces passing through the

horizon at t = 0 is proportional to EM e2πtw/β , where M is the mass of the black hole. Eventually

this effect becomes of order one, RT surfaces exchange dominance, and I drops to zero. This begins

when tw becomes of order t∗ ∼ β2π log M

E . Assuming E takes the smallest reasonable value, the

energy in one quantum at the Hawking temperature E ∼ TH , the time t∗ is

t∗ ∼β

2πlogS (2.3)

which is the fast scrambling time. This is our central result. Flat space stringy effects will not

change t∗. However, as we will emphasize in Section 2.4String and Planck scale effectssection.2.4,

we are unable to reliably exclude the possibility that stringy effects in the presence of the black hole

will be parametrically stronger and lead to a smaller t∗.

The logarithmic behavior arises as in [9] from the relation between Rindler time evolution and

Minkowski boosts. The connection between fast scrambling and large boosts has also been empha-

sized recently in [13]. This importance of this time scale in black hole physics was pointed out in

earlier work, including [36].

The outline of our paper is as follows: In Section 2.2A qubit modelsection.2.2 we will illustrate

the basic idea of scrambling destroying mutual information in a simple qubit system. In Section 2.3A

holographic modelsection.2.3 we will describe the basic geometrical constructions used and calculate

the mutual information holographically, assuming Einstein gravity. We also discuss correlation

functions as probes of entanglement. In Section 2.4String and Planck scale effectssection.2.4 we will

address string- and Planck-scale corrections to the results from § 2.3A holographic modelsection.2.3.

In Section 3.4Discussionsection.3.4, we will discuss various issues, including the connection to other

notions of scrambling and the possible relevance to firewall ideas.

2.2 A qubit model

Directly following the thermalization of a chaotic system is challenging, almost by definition. Our

primary tool in this paper, holography, is powerful but somewhat indirect, and we would like to

illustrate the effect of scrambling on entanglement in a simpler context. One tractable approach is

5In particular [32] discussed the effect of a perturbation at large tw as a firewall candidate. [35] discussed, in theone sided black hole context, highly boosted horizon hugging branes.


to study a system with Haar random dynamics, which powerfully disrupt local two-sided mutual

information. We pursue this in appendix 2.6Appendix A: Haar scramblingsection.2.6. In the present

section, we will consider a more physical system, by numerically evolving a collection of thermal

qubits. Although we are limited to a rather small system, the basic effect will be visible.

Using sparse matrix techniques, it is possible to time-evolve pure states of twenty to thirty

qubits. We will be less ambitious, studying a system (L) made up of ten qubits, plus another ten

for the thermofield double (R). We will use an Ising Hamiltonian, with both transverse and parallel

magnetic fields:

HL =

10∑i=1

σ(i)z σ(i+1)

z − 1.05 σ(i)x + 0.5 σ(i)

z

. (2.4)

The coefficients -1.05 and 0.5 are chosen, following [37], to ensure that the Hamiltonian is far from

integrability.

0 2 4 6 8 10tw

0.0

0.5

1.0

1.5

2.0

2.5

Figure 2.1: Mutual information (upper, blue) and spin-spin correlation function (lower, red) in theperturbed state |Ψ′〉, as a function of the time of the perturbation tw. The delay is a propagationeffect; if the perturbation at site five is sufficiently recent, sites one and two are unaffected.

Our procedure is to prepare the thermofield double state |Ψ〉, as in Eq. (2.1Introductionequation.2.1.1),

at a reference time t = 0, and with dimensionless temperature set equal to 4.0. We then apply a

perturbation σ(5,L)z to the fifth qubit of the L system at a time tw in the past. In other words, we

consider the perturbed state

|Ψ′〉 = e−iHLtwσ(5,L)z eiHLtw |Ψ〉. (2.5)

Notice that the applied operator acts trivially on the R system. In the state |Ψ′〉, we then compute

the mutual information between sites one and two and their thermofield doubles. The result is the

blue curve in Fig. 2.1Mutual information (upper, blue) and spin-spin correlation function (lower,


red) in the perturbed state |Ψ′〉, as a function of the time of the perturbation tw. The delay is

a propagation effect; if the perturbation at site five is sufficiently recent, sites one and two are

unaffectedfigure.2.1.

In the unperturbed state |Ψ〉, the mutual information is near-maximal. For small tw, this con-

tinues to be true in the perturbed state. However, as tw increases and the perturbation is moved

farther into the past, I(A;B) drops sharply before leveling off at a floor value. By studying the same

problem for eight or nine qubits instead of ten, we note that the floor of the mutual information

appears to decrease with the total size of the system.

Although mutual information is a particularly thorough measure of AB correlation, the same

basic phenomenon is visible in simpler quantities. A useful example is the spin-spin two point

function 〈Ψ′|σ(1,L)z σ

(1,R)z |Ψ′〉, between spin one in the L system and spin one in the R system. This

quantity is also plotted as a function of tw in Fig. 2.1Mutual information (upper, blue) and spin-

spin correlation function (lower, red) in the perturbed state |Ψ′〉, as a function of the time of the

perturbation tw. The delay is a propagation effect; if the perturbation at site five is sufficiently

recent, sites one and two are unaffectedfigure.2.1, and we see that it exhibits the same qualitative

behavior as the mutual information: the special local correlations of the thermofield double state

are destroyed by a small perturbation applied sufficiently long in the past.

2.3 A holographic model

In this section we will present our main result, a bulk geometry that illustrates the sensitivity

of specific entanglements in the thermofield double state to mild perturbations long in the past.

We will use RT surfaces and correlation function probes to analytically follow the loss of local

correlation between the L and R sides. We will work with Einstein gravity in 2+1 bulk dimensions

in this section, deferring comments about string- and Planck-scale effects to section 2.4String and

Planck scale effectssection.2.4, and deferring comments about higher dimensional Einstein gravity

to appendix 2.7.1Higher dimensionssubsection.2.7.1.

2.3.1 Unperturbed BTZ

Let us begin by reviewing the geometrical dual of the unperturbed thermofield double state of two

CFTs [6]. This is an AdS-Schwarzschild black hole, analytically extended to include two asymptot-

ically AdS regions. We think of the CFTs as living at the boundaries of the respective regions. In

2+1 bulk dimensions, the black hole solution is a BTZ metric, which can be presented as

ds2 = −r2 −R2

`2dt2 +

`2

r2 −R2dr2 + r2dφ2 (2.6)

φ ∼ φ+ 2π R2 = 8GNM`2 β =2π`2

R, (2.7)


where we use ` to denote the AdS radius, and R to denote the horizon radius. In what follows,

it will often be more convenient to use Kruskal coordinates, which smoothly cover the maximally

extended two-sided geometry. In these coordinates, the metric is

u v

tt

Figure 2.2: The Kruskal diagram (center) and Penrose diagram (right) for the BTZ geometry.

ds2 =−4`2dudv +R2(1− uv)2dφ2

(1 + uv)2. (2.8)

We will use the standard u, v convention so that the right exterior has u < 0 and v > 0. The two

boundaries are at uv = −1, and the two singularities are at uv = 1.

Below, we will be interested in computing geodesic distances between points in the BTZ geometry.

Since BTZ is a quotient of AdS, we can use the formula for geodesic distance in pure AdS2+1:

coshd

`= T1T

′1 + T2T

′2 −X1X

′1 −X2X

′2, (2.9)

where we’ve used the embedding coordinates

T1 =v + u

1 + uv=

1

R

√r2 −R2 sinh

Rt

`2

T2 =1− uv1 + uv

coshRφ

`=

r

Rcosh

Rφ

`(2.10)

X1 =v − u1 + uv

=1

R

√r2 −R2 cosh

Rt

`2

X2 =1− uv1 + uv

sinhRφ

`=

r

Rsinh

Rφ

`.

These coordinates also allow us to relate (r, t) to (u, v). Note, in particular, that the left asymptotic

region can be reached in the (r, t) coordinates by adding iβ/2 to t.


2.3.2 BTZ shock waves

Having set up the bulk dual of the thermofield double state of the two CFTs, we would like to very

mildly perturb it. As an example, we might add a few particles at the left boundary, and let them

fall into the black hole. Naively, this would seem to have an insignificant effect on the geometry.

However, as is familiar from Rindler space, translation in the Killing time t is a boost in the (u, v)

coordinates, and if we release a perturbation with field theory energy E from the boundary at a

time tw long in the past,6 it will cross the t = 0 slice with proper energy

Ep ∼E`

ReRtw/`

2

(2.11)

as measured in the local frame of that slice. In this frame, the perturbation will be a high energy

shock following an almost null trajectory close to the past horizon.

If tw is sufficiently large, we must include the backreaction of this energy. For the simplest case

of spherically symmetric null matter, the backreacted metric is a special case of the AdS-Vaidya

solution.7 Closely related metrics have been previously studied in [38, 39, 40, 41], following the

original Schwarzschild analysis of [42, 34]. We will construct the geometry by gluing a BTZ solution

of mass M to a solution of mass M + E across the null surface uw = e−Rtw/`2

. Here, E is the

asymptotic energy of the perturbation, which we will take to be very small compared to M .

We will choose coordinates u, v to the right (past) of the shell and u, v to the left (future),

so that the metric is always of the Kruskal form (3.7One shockequation.3.2.7). Because of the

increase in mass, the radius is R to the right and R =√

M+EM R to the left. We will fix a relative

boost ambiguity in the relation between u, v and u, v by requiring the time coordinate t to flow

continuously at the boundary. This determines the location of the shell in terms of the tilded

coordinates as uw = e−Rtw/`2

. The other matching condition is the requirement that the radius of

the S1 be continuous across the shell. Inspecting the metric (3.7One shockequation.3.2.7), we find

the condition

R1− uwv1 + uwv

= R1− uwv1 + uwv

. (2.12)

For small E/M , the solution is a simple shift

v = v + α , α ≡ E

4MeRtw/`

2

. (2.13)

This matching condition is exact if we take E/M → 0 and tw → ∞ with α fixed. In this limit,

6We emphasize that t is the Killing time coordinate. In our convention, it runs forward on the right boundaryand backwards on the left (see Fig. 2.2The Kruskal diagram (center) and Penrose diagram (right) for the BTZgeometryfigure.2.2). In particular, a perturbation released at time tw from the left boundary is in the past of thet = 0 slice if tw > 0.

7This metric corresponds to a boundary source adjusted to make all particles fall through the horizon at the sametime. We comment further on this choice in the discussion. The use of a classical metric is simplest to justify if weconsider a perturbation that corresponds to a large but fixed number of quanta in the small GN limit. However, webelieve that our conclusions are also accurate for small numbers of quanta.


which is relevant for the small but early perturbations we wish to consider, R = R, and the metric

can be written

ds2 =−4`2dudv +R2 [1− u(v + αθ(u))]

2dφ2

[1 + u(v + αθ(u))]2 . (2.14)

The corresponding geometry is shown in Fig. 2.3The Kruskal and Penrose diagrams for the geometry

with a shock wave from the left, represented by the double line. The dashed v = 0 and v = 0 horizons

miss by an amount αfigure.2.3. For computations, it is sometimes useful to use discontinuous

coordinates U = u, V = v + αθ(u), so that the metric takes a more standard shock wave form

ds2 =−4`2dUdV + 4`2αδ(U)dU2 +R2(1− UV )2dφ2

(1 + UV )2. (2.15)

Either way, the geometry of the patched metric is continuous but its first derivatives are not: there

Figure 2.3: The Kruskal and Penrose diagrams for the geometry with a shock wave from the left,represented by the double line. The dashed v = 0 and v = 0 horizons miss by an amount α.

is an impulsive curvature at the location of the shell. One can check that the Einstein equations

imply a stress tensor

Tuu =α

4πGNδ(u), (2.16)

corresponding to a shell of null particles symmetrically distributed on the horizon.

2.3.3 Geodesics

Since we can boost to a frame in which the shock wave has very little stress energy, the patched

solutions described above do not give rise to any large local invariants. The scalar curvature, for

example, is regular at u = 0. However, there are large nonlocal invariants that distinguish the


shock wave geometry from unperturbed BTZ. Geodesic distance, which we will relate holographi-

cally to field theory quantities in § 2.3.4Mutual informationsubsection.2.3.4 and § 2.3.5Correlation

functionssubsection.2.3.5, is an important example of such an invariant.

Let us consider a geodesic connecting a point at Killing time tL on the left boundary with a

point at time tR on the right boundary. We will take both points to be located at the same value

of φ. Any real geodesic between them will pass through the shock at u = 0 at some value of v. We

can use the embedding coordinates (2.9Unperturbed BTZequation.2.3.9) to compute the distance,

d1, from the left boundary to this intermediate point and, d2, the distance from the intermediate

point to the right boundary:

coshd1

`=

r

R+

1

R

√r2 −R2 e−RtL/`

2

(v + α) (2.17)

coshd2

`=

r

R− 1

R

√r2 −R2 e−RtR/`

2

v. (2.18)

To find the total geodesic distance, we extremize d1 + d2 over v. For large r, the result is

d

`= 2 log

2r

R+ 2 log

[cosh

R

2`2(tR − tL) +

α

2e−R(tL+tR)/2`2

]. (2.19)

Setting α = 0, we recover the distance in the unperturbed BTZ geometry. The contribution of α

represents an increase in this distance due to the shock wave.

It is clear from Eq. (2.19Geodesicsequation.2.3.19) that the impact of the shock wave on the

geodesic distance is insignificant if tL + tR is sufficiently large. Indeed, if tL ∼ tw and tR ∼ tw,

then the frame in the bulk defined by the geodesic approximately agrees with the frame natural for

the infalling shell. Clearly, the effect on the geometry caused by adding a few quanta should be

negligible in this frame, and we don’t expect a significant change in the geodesic distance. However,

if we fix tL = tR = 0 and take Rtw `2, then there is a large relative boost between the frame of

the quanta and the frame of the geodesic. In the frame of the geodesic, we have highly blueshifted

quanta that significantly increase the distance.

We will also record the geodesic distance between two equal-time points on the same boundary,

with angular separation φ. This is unaffected by the shock wave, and is given at large r by

d

`= 2 log

2r

R+ 2 log sinh

Rφ

2`. (2.20)

2.3.4 Mutual information

So far in this section, we have constructed the bulk dual to the mildly perturbed thermofield double

state. We will now use this geometrical data to understand the behavior of correlations between

regions A ⊂ L and B ⊂ R in the two CFTs. One useful measure of correlation is the mutual

information I(A;B) = SA + SB − SA∪B . Employing the RT proposal [17] and its time-dependent


extension [19], we can compute the entropy SΩ of the density matrix associated to a boundary region

Ω as Amin/4GN , where Amin is the area of the smallest extremal codimension-two bulk surface that

shares a boundary with Ω.8 In a 2+1 dimensional bulk, extremal codimension-two surfaces are

geodesics, and the “area” is the length of the geodesic.

Following [30, 31], we will consider a spatial region at t = 0 consisting of two disconnected

components, A ⊂ L in the left asymptotic region, and B ⊂ R in the right asymptotic region. For

simplicity, we will take them to be of equal angular size φ < π, and we will center them at the same

angular location on their respective boundaries. The only subtlety in the calculation arises from the

fact that a given spatial region can be bounded by different extremal surfaces. RT instruct us to

use the one of minimal area.

First, let us consider SA, or equivalently SB . There are two choices of extremal surface. The first

choice is a geodesic that connects the endpoints of the A interval. The other choice is a geodesic

that connects one endpoint to the image of the other by the BTZ identification, plus a contribution

from the horizon of the black hole required by the RT homology condition. When φ < π, the former

always has smaller area, and we use (2.20Geodesicsequation.2.3.20) to obtain

SA = SB =`

4GN

(2 log

2r

R+ 2 log sinh

Rφ

2`

). (2.21)

Next, consider SA∪B . When φ < π, we have two possible choices of extremal surface. First, we

have the union of the two geodesics used to compute SA and SB . This gives S(1)A∪B = SA + SB .

Second, we have a pair of geodesics connecting the endpoints of A to the endpoints of B. Using

(2.19Geodesicsequation.2.3.19), we find that the second gives

S(2)A∪B =

`

GN

[log

2r

R+ log

(1 +

α

2

)]. (2.22)

For small regions with sinh Rφ2` < 1, we have S

(1)A∪B < S

(2)A∪B , so that I(A;B) = 0 for all values of

α [30]. However, for larger regions, S(2) wins for sufficiently small α, and we find positive mutual

information. Substituting for α using (2.13BTZ shock wavesequation.2.3.13), and rewriting M and

R in terms of the Bekenstein-Hawking entropy S and the inverse temperature β, we obtain

I(A;B) =`

GN

[log sinh

πφ`

β− log

(1 +

Eβ

4Se2πtw/β

)]. (2.23)

This mutual information is a decreasing function of tw. For high temperature, I reaches zero when

tw is equal to

t∗(φ) =φ`

2+

β

2πlog

2S

βE. (2.24)

8More precisely, this expression gives the contribution proportional to N2 in the entropy. There may be numericallylarge but subleading terms, as well as finite λ corrections. The RT prescription also requires that the bulk surfacemust be homologous to Ω.


When the string coupling gs (which is ∼ 1/N in terms of the large N gauge theory) is small, so

S ∼ N2 is large, and E assumes its smallest reasonable value E ∼ T = 1/β then

t∗ =β

2πlogS. (2.25)

as announced in the introduction.

Similar formulas can be obtained for the case where φ > π. There, the mutual information reaches

a floor with a finite positive value, rather than zero. One can check that the mutual information

between regions with φ = π takes the longest to relax.

2.3.5 Correlation functions

Compared to mutual information, two point functions are a very crude measure of correlation.9

However, the effect of scrambling on local entanglement is not subtle, and we saw in the spin

system that two point functions and mutual information have a qualitatively similar response to a

perturbation of the thermofield double state. In this section, we will use the shock wave geometry

to obtain an understanding of this response, using the approximation of free field theory on the

perturbed background. We first observe that we are interested in computing the following matrix

element:

〈ϕLϕR〉W ≡〈Ψ|W †ϕLϕRW |Ψ〉〈Ψ|W †W |Ψ〉

, (2.26)

where W is an operator on the left boundary that creates a few particles at a time tw in the past,

and ϕL, ϕR are the field operators in the L and R theories being correlated, at time t = 0. W is

assumed to have no one-point function in the thermofield double state.

For geometries with a real Euclidean continuation, such as the unperturbed BTZ metric, spacelike

correlation functions (in the associated Euclidean vacuum) of CFT operators dual to heavy bulk

fields of mass m can reliably be related to the (renormalized) geodesic distance as

〈ϕ(x)ϕ(y)〉 ∼ e−md(x,y). (2.27)

This fact has been previously exploited for the purposes of studying black hole interiors [44, 45, 46,

47, 48, 49].10 The BTZ shock wave metric is nonanalytic, and analytic approximations do not have

real Euclidean continuations. However, in the regime where the shock wave is a small perturbation

of the metric, we expect that the the saddle point represented by the perturbed geodesic continues to

give the dominant contribution to the two point function, and we can estimate two point functions

in the shock wave background using spacelike geodesics that pass through the black hole interior.

9The mutual information is lower-bounded by two-point correlation functions of bounded operators. See e.g. [43].10There has been some discussion about whether such two point functions actually diagnose behind-the-horizon

physics. The following analysis shows that the two point function is directly sensitive to dynamics that is extremelydifficult to interpret solely in terms of supergravity degrees of freedom outside the horizon.


In fact, an exact calculation of the free field two point function in the BTZ shock wave background

has been previously carried out in [41], and matches our geodesic estimates below up to expected

multiplicative corrections of order 1/m`.11

Let us therefore proceed to use Eq. (2.27Correlation functionsequation.2.3.27) to estimate cor-

relation functions. We will focus on the correlator with tL = tR = φR = φL = 0, and study the

dependence on tw. Using the geodesic distance Eq. (2.19Geodesicsequation.2.3.19), and subtracting

the UV-divergent first term, we obtain the expression

〈ϕLϕR〉W ∼

(1

1 + E8M eRtw/`2

)2m`

. (2.28)

This correlator is unaffected by the perturbation until tw becomes of order t∗. For larger tw, the

correlator tends to zero exponentially as we make the perturbation earlier, as e−mR(tw−t∗)/`. We

emphasize that the value of tw at which the correlators start to be significantly affected is the

same value that we obtained by studying the RT prescription for mutual information. This is not

surprising, since the two quantities are determined by the same geodesic data. However, there is

a significant difference: the mutual information has a sharp feature where it becomes zero shortly

after t∗, whereas correlators computed in the geodesic approximation merely start to exponentially

decay. This discrepancy results from the free field approximation to the scalar correlator. We will

see in the next section that inelastic interaction effects turn off the correlation function more sharply

after t∗.

2.4 String and Planck scale effects

The analysis of the previous section relies on Einstein gravity. But, as noted above, a single thermal

quantum released at time tw carries enormous energies in the rest frame of the t = 0 slice, Ep ∼1` eRtw/`

2

. When tw is large enough this energy can exceed string or even Planck scales12. The effect

on the mutual information comes from string- or Planck-suppressed corrections to the RT formula.

These corrections are not completely understood (but see [50, 51]) and so it is difficult to evaluate

their effect in the shock wave background. But we have seen above that the two point correlation

function diagnoses similar information. So we will try to assess in a qualitative way the effect of

such corrections on the two point function (2.26Correlation functionsequation.2.3.26). 13 We are

11In making the comparison, note that the shift function h in [41] should be identified with twice our shift in v.12We thank Eva Silverstein for valuable discussions about the significance of this situation.13We might consider the spacelike geodesic method of calculating the correlation function. The worldline action

should in general contain terms with higher derivatives of the coordinates with respect to proper time, possiblymultiplied by curvatures. These would be multiplied by the appropriate powers of the string mass ms and the Planckmass mp. In 3 dimensions these are related by ms ∼ g2smp where gs is the string coupling. Here GN ∼ 1/mp. Theresults of the previous section show that the high energy in the shock wave causes large derivatives with respect toproper time along the geodesic world line. These would seem to cause the 1/ms suppressed terms to become orderone at times tw far smaller than t∗. Because there are no powers of gs involved, the time when this would happen


interested in the time tw when this quantity starts to differ substantially from the two point function

〈Ψ|ϕLϕR|Ψ〉.The expectation value (2.26Correlation functionsequation.2.3.26) computes spacelike correlations

in the state W |Ψ〉, not scattering information. So it is difficult to evaluate it in an S-matrix theory

like flat space perturbative string theory. Nonetheless AdS/CFT teaches us that this quantity is

well posed in quantum gravity, so there should be some way of understanding it in the region where

perturbative string theory is valid. This seems technically difficult, even in BTZ, but some insights

might be gained from a string scattering calculation in pure AdS. The methods of [52, 53] could be

helpful.

On the other hand, in a situation like the shock where interactions are localized, if we know the

spatial correlations at a given time then we can propagate them forward using scattering data. So we

expect that when scattering is weak the change of spatial correlations will be small. Concretely, flat

space field theory and Einstein gravity calculations in AdS/CFT [41] indicate that when scattering

is weak the disturbance of spacelike correlations is also weak. So we proceed by estimating the

strength of flat space string scattering in the relevant energy and coupling regime.

The basic features of closed string scattering in flat space in the region of interest are discussed in

[54, 55, 56, 57, 58]. The largest scattering amplitude occurs in the Regge region, large Mandelstam

s and fixed t (as opposed to to the highly suppressed fixed angle region). Here the amplitude is

small when the dimensionless quantity ε = g2ssl

2s is small. Putting in s = EpT , appropriate for a

thermal quantum sourced by ϕR, one finds that ε becomes of order one at a time tε somewhat prior

to t∗, by an additive S-independent amount proportional to β log `/ls.14 So we find that flat-space

stringy effects would not change the logS dependence of the time t∗ at which the correlator starts

decreasing.

Stringy effects do become important, however. The phase shift obtained from the tree level

Virasoro-Shapiro scattering amplitude at large s, as a function of impact parameter b, agrees with

the result of Einstein gravity down to a value b ∼ bI where

bI = ls√

log sl2s (2.29)

describes the famous logarithmic spreading of strings at high energy. For b < bI , there are substantial

corrections to the Einstein gravity calculation of the elastic part of the phase shift, summarized by a

metric with a transverse profile of size bI that grows logarithmically with s. There are also inelastic

processes, that give an imaginary part to the phase shift. The magnitude of the imaginary part of

the phase shift is suppressed relative to the real part by (ls/b)2.

We now turn to scattering15 in the black hole, whose characteristic lengths are the horizon size

would be much sooner, of order logms/T rather than logmp/T ∼ logR/GN ∼ logS. The following remarks aboutflat-space scattering show that this argument is incorrect.

14In D > 4 dimensions.15By “scattering” behind the horizon we mean an off shell process that resembles scattering with a finite energy


R, the curvature length ` and the geodesic time from the horizon to the singularity, `. The flat

space results above are easily applicable only if these lengths are larger than the scales relevant to

the flat-space string scattering problem. This not the case for t ∼ t∗.String spreading causes the string to expand in directions transverse to its motion. Naively it

covers the horizon bI/R times which is roughly√

log(ε/g2s). Interaction effects should be at most

ε√

log (ε/g2s), which give a log log correction to t∗, which we ignore.16

So far, we have assumed that the scattering takes place far from the singularity and that the

string spreading is purely transverse. This may not be the case. If the string spreads significantly in

the longitudinal directions,17 the singularity may become important. For this reason, we are unable

to reliably exclude the possibility that singularity effects might dramatically enhance the scattering

rate. This could have the effect of making t∗ much shorter than β logS.

For tw > tε, several effects come into play. First, there is the increasing effect of the Einstein

gravity scattering, which becomes of order one at the time t∗ = β2π logS. In the string scattering

problem, this is a purely elastic effect, and should be accurately captured by the homogeneous

metric discussed in § 2.3A holographic modelsection.2.3. The inelastic phase shift is suppressed by

l2s/`2, and becomes important at a time tw ∼ t∗ + (const.) log `/ls. This causes the correlator to

decay schematically like exp (−s). At even larger values of tw, and correspondingly larger energies,

a variety of inelastic nonperturbative effects should occur. For instance, black holes may form. A

rough estimate suggests this occurs when the Schwarzschild radius RS , RD−3S ∼ GN

√s becomes of

order R. This occurs at a time 2t∗.

Although our estimates have not been conclusive, it is clear that there is an interesting connection

between high energy scattering in the black hole background and sensitive dependence on initial

conditions in the boundary field theory. This interplay deserves further attention.

2.5 Discussion

In the context of Einstein gravity, we have exhibited a bulk holographic dual to the sensitive de-

pendence on initial conditions in the boundary field theory. Small perturbations at early times

create highly blueshifted shock waves that disrupt measures of correlation between the L and R field

theories. The original gravitational interpretation of scrambling as charge spreading on the horizon

[9] is very much in the spirit of our calculation. In particular, the large boost is the source of the

logarithmic time dependence. The similarity of the bulk calculations suggests a relation between

sensitive dependence on initial conditions and scrambling, and it would be interesting to understand

the connection further.

and momentum resolution.16On a target space torus, this mild enhancement is completely absent, and therefore may not be present in the

black hole problem either.17Longitudinal spreading in the string ground state has been computed in light-cone gauge in Ref. [59]. This is a

large effect, but it appears to be gauge-dependent [60] and we are unsure of its significance to our setup.


The shock wave solutions we have used in this paper correspond to boundary sources that are

carefully constructed so that all particles launched from the boundary fall into the black hole at

the same time. This allows an exact analytic treatment of the nonlinear general relativity effects at

large boost. A simple local boundary perturbation of the type familiar in field theory would source

particles that would fall into the black hole over a band of times, with the probability of staying

outside of the black hole decreasing exponentially with the time after the perturbation as exp (−Rt).In this more general situation each particle still blue shifts after it falls into the black hole and the

shock wave metric gives an accurate picture of the disturbance of correlation. But there are some

situations where this spread of infall times becomes important, as we will now discuss.

The observations of the previous section identify inelastic effects that make the correlator behave

schematically like exp(−s) ∼ exp(−et). This is an extraordinarily rapid turnoff, dropping to almost

zero at t ∼ t∗, but is in keeping with the expectations from random dynamics, as in appendix

2.6Appendix A: Haar scramblingsection.2.6. But because of the spread in infall times we do not

actually expect the correlator to go to zero so rapidly. In the CFT, this corresponds to some

amplitude for the perturbation to remain in the ultraviolet degrees of freedom for some time and

not to touch off scrambling. If we fold the exp (−Rt) spread against the exp (−et) turnoff we expect

to recover an ordinary exponential decay of the 〈ϕLϕR〉W correlator as a function of tw. The double

exponential effect should only leave a subtle imprint, albeit an interesting one.

The shock wave solutions do not display any of the hydrodynamical effects in same side correlators

that have been extensively explored in AdS/CFT calculations. These depend on the nontrivial field

profiles connected to the spread in infall times. As usual the decay of quasinormal modes and the

related hydrodynamical dissipation are related to the infall of particles through the horizon. 18

The interplay of hydrodynamical behavior, in particular diffusive spreading [9], and the scram-

bling behavior discussed here raises a number of interesting questions for further study. In particular

it would interesting to study the spatial propagation of the disturbance of correlations by analyzing

the appropriate localized gravity solutions, in contrast to the spherically symmetric perturbations

discussed in this paper. We give a set of such solutions in appendix 2.7.2Solutions with localized

sourcessubsection.2.7.2 but they are adjusted to not give a spread in infall times and so are too

specialized to give full insight into this problem.

Although section 2.3A holographic modelsection.2.3 focused on the three-dimensional BTZ ge-

ometry, one can consider similar perturbations to higher dimensional black holes. We give a prelim-

inary analysis in appendix 2.7.1Higher dimensionssubsection.2.7.1, where we find that the leading

dependence of t∗ is universal, t∗ = β2π logS.

Finally we turn to firewalls. The driving force behind the firewall proposal of [11] is a conflict

between chaos and specific entanglement [12]. Although our work is closely related to this issue,

and to its recent treatment by Maldacena and Susskind [13], we are not able to offer any decisive

18The perturbation at tw can also affect conserved quantities, such as the energy. This will give rise to small butnon-decaying terms in the correlation function.


insight. However, we will make a few comments.

1. Our results provide a new example of an emerging pattern: after a scrambling time, there do

not seem to be any simple probes of the behind-the-horizon region. 19 The RT surfaces disconnect,

and the correlator goes to zero.

2. The shock wave geometry defeats a naive argument for firewalls. Smoothness of the left

horizon requires entanglement between modes b and b shown in Fig. 2.4In the unperturbed BTZ

geometry (left), a smooth horizon requires the black mode on the left to be highly entangled with the

blue mode on the right. By contrast, in the shock wave geometry (right) the black and blue modes

are far apart and unentangled. Instead, the black mode is entangled with the green mode coming

out of the white hole. The arguments of [65] suggest that the green mode may be complicated in

the CFTfigure.2.4. In the unperturbed geometry, b and b are related to smeared CFT operators ϕL

and ϕR [62, 63, 64], so the bulk correlation 〈bb〉 can be viewed as arising from the CFT correlation

〈Ψ|ϕLϕR|Ψ〉 characteristic of the thermofield double state |Ψ〉. One might worry that the smallness

of 〈Ψ|W †ϕLϕRW |Ψ〉 implies de-correlation of b and b and a firewall in the state W |Ψ〉.The geometry shown in Fig. 2.4In the unperturbed BTZ geometry (left), a smooth horizon re-

quires the black mode on the left to be highly entangled with the blue mode on the right. By contrast,

in the shock wave geometry (right) the black and blue modes are far apart and unentangled. Instead,

the black mode is entangled with the green mode coming out of the white hole. The arguments of

[65] suggest that the green mode may be complicated in the CFTfigure.2.4 gives an alternate ex-

planation.20 Although b = ϕL holds in any geometry that approximates AdS-Schwarzschild outside

the horizon, the relationship b = ϕR is not valid in the shock wave metric. Indeed, it is clear from

Fig. 2.4In the unperturbed BTZ geometry (left), a smooth horizon requires the black mode on the

left to be highly entangled with the blue mode on the right. By contrast, in the shock wave geom-

etry (right) the black and blue modes are far apart and unentangled. Instead, the black mode is

entangled with the green mode coming out of the white hole. The arguments of [65] suggest that

the green mode may be complicated in the CFTfigure.2.4 that ϕR represents a mode c that is far

from b and therefore naturally uncorrelated with it.

3. A stronger argument for firewalls in the state W |Ψ〉 can be made [12]. If we assume (i)

that the perturbation at sufficiently early tw acts like a random unitary,21 (ii) that b can always

be represented by the same (perhaps very complicated) linear operator, and (iii) that this operator

commutes with b, then the counting arguments of AMPSS imply a firewall on both the left and

right horizons. To make this argument, one considers the operator eiθNb , where Nb is the number

operator. This rotates the phase of 〈bb〉, but leaves the ensemble generated by random unitaries

invariant, so we conclude that the ensemble average of 〈bb〉 must be zero.

19But see [61].20A similar situation arises if we consider simultaneous forward time evolution of both CFTs [31], and related

comments were made by those authors.21In this discussion we will abuse notation and refer to a random unitary that approximately commutes with the

Hamiltonian as a “random” unitary.


b

φL

φR

b~

b~

b

φL φR

c

Figure 2.4: In the unperturbed BTZ geometry (left), a smooth horizon requires the black mode onthe left to be highly entangled with the blue mode on the right. By contrast, in the shock wavegeometry (right) the black and blue modes are far apart and unentangled. Instead, the black modeis entangled with the green mode coming out of the white hole. The arguments of [65] suggest thatthe green mode may be complicated in the CFT.

We are unable to determine which, if any, of (i-iii) should be relaxed, but we note that the CFT

representation of the green b mode emerging from the white hole (Fig. 2.4In the unperturbed BTZ

geometry (left), a smooth horizon requires the black mode on the left to be highly entangled with the

blue mode on the right. By contrast, in the shock wave geometry (right) the black and blue modes

are far apart and unentangled. Instead, the black mode is entangled with the green mode coming

out of the white hole. The arguments of [65] suggest that the green mode may be complicated in

the CFTfigure.2.4) is rather mysterious. At higher energies (earlier tw), when the evolution across

the shock can no longer be described as a shift, the correct description of the mode b becomes even

less clear.

4. The right horizon is not smooth, and the shock would affect an observer falling in from that

side [32]. In the regime where the shock wave metric is an accurate description, the observer’s world

line will be abruptly shifted over and the proper time before he hits the singularity reduced. At

higher energies, the infaller will experience a painful inelastic collision. Note, however, that for fixed

tw, the strength of all such effects decreases as we make the infall time tR later. In the regime where

Einstein gravity is valid the entanglement of high energy modes is unaffected. On the other hand,

for fixed tR, we can always make the experience extremely painful by making tw earlier and earlier.

This suggests a connection between further increasing chaos and the more complete disruption of

smooth geometry. It is clear that this shock wave has many of the attributes of a firewall.22

5. Finally, if “real” AMPS firewalls form in this system before the scrambling time, then our

bulk calculations would very likely be inaccurate statements about the CFT dynamics. We view

22One might have thought that by making an early perturbation in both CFTs one might have created shock waveson both horizons. At least for spherical shock waves this is not the case. The future horizons in the resulting geometryare well beyond the location of the collision and do not coincide with the shock waves [34].


this as a feature, not a bug. CFT quantities that are straightforward to formulate (albeit not to

calculate!) would differ from expectations.

2.6 Appendix A: Haar scrambling

In the main text of the paper, we’ve considered the effect of an operator OL(tw) = e−itwHOLeitwH

on the the entanglements between local subsystems A ⊂ L and B ⊂ R in a thermofield double state

|Ψ〉LR. If the Hamiltonian is sufficiently chaotic, and we take very large values of tw, we might

model such a perturbation as a random unitary matrix, so that the perturbed thermofield double

state is

|Ψ′〉 =1

|L|1/2

|L|∑n,m=1

Unm|m〉L|n〉R. (2.30)

In this appendix, we will study the mutual information I(A;B) and correlation functions in this

state, using the tool of Haar integrals. The result will not be surprising to the reader familiar with

Page’s analysis of random states [66]. However, our setup is not identical to that of Page, and we

will include the discussion for completeness, following the computationally efficient norm approach

of [8].23

Specifically, in order to study the mutual information I(A;B) in this state, we will consider the

distance d1 = ‖ρAB−ρA⊗ρB‖1, where the 1-norm of a matrix M is defined as ‖M‖1 = tr[√M†M ].

For d1 ≤ 1/e, this quantity lower-bounds the mutual information via the Pinsker inequality, and

upper-bounds it via the Fannes inequality [67]:

1

2d2

1 ≤ I(A,B) ≤ d1 log |A| − d1 log d1. (2.31)

Unfortunately, because of the square-root, the distance d1 is difficult to average over U , so we will

use a further inequality (derived from Cauchy-Schwarz applied to the eigenvalues), that ‖M‖1 ≤√rankM‖M‖2, where the 2-norm is defined as ‖M‖2 =

√tr[M†M ]. The benefit here is that we

can compute the average over unitaries of the square of the 2-norm exactly. Using the fact that for

any U , the density matrix for A obtained from |Ψ〉 is maximally mixed, ρA(U) = ρB(U) = 1/|A|,we compute

‖ρAB(U)− ρA(U)⊗ ρB(U)‖22 = tr[(ρAB(U)− 1/|A|2)2] (2.32)

= tr[ρAB(U)2]− 2

|A|2tr[ρAB ] +

1

|A|4tr[1] (2.33)

= tr[ρAB(U)2]− 1

|A|2. (2.34)

23This approach has the benefit of emphasizing that Haar randomness is not essential to the calculation, and thata 2-design would lead to identical results.


We would now like to take the expectation value over U of this quantity, using the Haar measure.

The most direct way to do this computation is to break up them and n indices of Umn intom→ (i, I),

where i runs over the A Hilbert space, and I runs over the Hilbert space of Ac, the tensor complement

of A in L. The operator U is then represented as UiI jJ , and one can check that

tr[ρAB(U)2] =1

|L|2∑

iIjJi′I′j′J′

UiI jJU∗i′I j′JUi′I′ j′J′U

∗iI′ jJ′ . (2.35)

We can now take the expectation value using∫dU Ui1j1Ui2j2U

∗i′1j′1U∗i′2j′2 =

1

|L|2 − 1

(δi1i′1δi2i′2δj1j′1δj2j′2 + δi1i′2δi2i′1δj1j′2δj2j′1

)(2.36)

− 1

|L|(|L|2 − 1)

(δi1i′1δi2i′2δj1j′2δj2j′1 + δi1i′2δi2i′1δj1j′1δj2j′2

).

The terms on the bottom line are subleading and we will drop them, along with the “1” in

the first line. Summing as in Eq. (3.30Discussionequation.3.4.30), we find that tr[ρAB(U)2] =

|A|−2 + |Ac|−2. Using the convexity of the square root, the Cauchy-Schwarz inequality and Eq.

(2.34Appendix A: Haar scramblingequation.2.6.34), this implies∫dU‖ρAB(U)− ρA(U)⊗ ρB(U)‖1 ≤

|A||Ac|

. (2.37)

If A is less than half of the L system, the one-norm distance is suppressed by a ratio of Hilbert

space dimensions. This quantity is exponentially small in, e.g. the number of extra qubits in Ac

compared to A. Using Eq. (2.31Appendix A: Haar scramblingequation.2.6.31), we can bound the

mutual information ∫dU I(A,B) ≤ |A|

|Ac|log |Ac|. (2.38)

The large logarithmic factor is probably an artifact of our shortcut through the 1-norm, but in any

case, if A is significantly smaller than half of the total system, the above is exponentially small.

If the L and R systems are composed of qubits, we can also study correlation functions of a spin

in the L system and a corresponding spin in theR system. One can bound these correlations using the

computation of d1 above, but a direct calculation in the state (2.30Appendix A: Haar scramblingequation.2.6.30)

is simple enough. Averaging over U , one finds that the expected value of the spin-spin correlator is

zero, and the rms value is |L|−1.


2.7 Appendix B: Geometrical generalizations

2.7.1 Higher dimensions

Our analysis in this paper was largely restricted to three spacetime dimensions. In this appendix, we

will explore the effect of the shock wave for D-dimensional AdS black holes. D is the bulk spacetime

dimension, i.e. D = 3 for BTZ. We will not attempt to compute geodesic distances and RT surfaces.

As a simpler proxy, we will estimate how large tw has to be to make the shift in the v coordinate of

order one.24

We begin with the metric

ds2 = −f(r)dt2 + f−1(r)dr2 + r2dΩ2D−2. (2.39)

Assuming the existence of a horizon at r = R, we pass to Kruskal coordinates:

ds2 = − 4f(r)

f ′(R)2e−f

′(R)r∗(r)dudv + r2dΩ2D−2 (2.40)

uv = −ef′(R)r∗(r) u/v = −e−f

′(R)t, (2.41)

with dr∗ = f−1dr the usual tortoise coordinate. As in § 2.3.2BTZ shock wavessubsection.2.3.2, we

add a spherically symmetric null perturbation of asymptotic energy E M , at a time tw in the left

asymptotic region. We define coordinates u, v to the left of the perturbation, and continue to use

u, v to the right. The shell propagates on the surface

uw = ef′(R)

2 (r∗(∞)−tw) uw = ef′(R)

2 (r∗(∞)−tw), (2.42)

and the matching condition relates v to v via

uwv = −ef′(R)r∗(r) uwv = −ef

′(R)r∗(r). (2.43)

We would like to use these equations to find the shift v = v + α, to linear order in E and at large

tw. Small E allows us to approximate uw = uw. Large tw pushes us to a limit where r approaches

R, so we can expand f(r) = f ′(R)(r −R) + .... Evaluating r∗, we find ef′(R)r∗(r) = (r −R)C(r,R),

where C is smooth and nonzero at r = R. To linear order in E and at large tw, we therefore have

α =E

uw

d

dM

[(R− r)C(r,R)

]∣∣∣r=R

=E

uw

dR

dMC(R,R). (2.44)

We can relate R to SBH using the area formula, and use the first law of thermodynamics to evaluate

24The main point, that the coefficient of the logarithm in t∗ is dimension-independent, should already be clear fromthe discussion in § 2.4String and Planck scale effectssection.2.4.


dR/dM . Also using f ′(R) = 4π/β, we find that α becomes equal to one at time25

t∗ = r∗(∞) +β

2πlog

[(D − 2)ΩD−2

4C(R,R)

T

E

RD−3

GN

]. (2.45)

Fixing E,R, T and taking GN ∝ N−2 to zero, we have t∗ ∼ β2π logN2 in any spacetime dimension.

2.7.2 Solutions with localized sources

Additional insight into the process of de-correlation might be gained by considering solutions with

stress energy localized in the angular directions. The interpretation of such solutions is subtle, but

we will record their form here. We focus on the case where the shock is produced by a very low-

energy perturbation long in the past, so it lies entirely on the right horizon. We assume the infalling

source is at the north pole of the (D− 2)-sphere, and we make the ansatz v = v+ h(Ω). Evaluating

the Ricci tensor of the patched metric (see, e.g. appendix A of [39]) and plugging into the Einstein

equations with Tuu ∝ δ(u)δD−2(Ω), we find that h must satisfy an equation[∇2SD−2 −

D − 2

2Rf ′(R)

]h(Ω) ∝ δD−2(Ω). (2.46)

For a large AdS black hole with R `, we have f ′(R) ≈ (D − 1)R/`2. The shift h is the Green’s

function for a very massive field on the sphere, and it decays with angular distance from the north

pole as h ∝ e−√

(D−1)(D−2)2 Rθ/`. Comparing this to the rate at which the perturbation grows as we

push tw earlier, ef′(R)tw/2, we find that as we increase tw, the level sets of h expand outward with

a “speed of propagation”

vD =

√D − 1

2(D − 2), (2.47)

where D is the spacetime dimension of the AdS space.

25Note that the combination r∗(∞)− β2π

logC(R,R) is invariant under additive shifts in the definition of r∗(r).

Chapter 3

Multiple Shocks

This chapter consists of a second paper written [2] in collaboration with Stephen Shenker. It was

released on the arXiv as “Multiple Shocks,” arXiv:1312.3296 [hep-th]. The original abstract is as

follows:

Using gauge/gravity duality, we explore a class of states of two CFTs with a large degree of

entanglement, but with very weak local two-sided correlation. These states are constructed by

perturbing the thermofield double state with thermal-scale operators that are local at different times.

Acting on the dual black hole geometry, these perturbations create an intersecting network of shock

waves, supporting a very long wormhole. Chaotic CFT dynamics and the associated fast scrambling

time play an essential role in determining the qualitative features of the resulting geometries.

3.1 Introduction

The firewall [11] controversy has highlighted the conflict between the special local entanglements

required for smooth geometry and the randomness of typical states. Aspects of this tension become

especially clear in the two sided black hole [6, 68] context, as Van Raamsdonk has emphasized. The

two sided eternal AdS Schwarzschild black hole is dual to two copies of a CFT, L (left) and R (right),

in the thermofield double state

|TFD〉 =1

Z1/2

∑n

e−βEn/2|n〉L|n〉R. (3.1)

The particular LR entanglement in this state is highly atypical, as local subsystems of L are entangled

with local subsystems of R. This structure is closely related to the smooth geometry of the eternal

black hole. The primary goal of this paper is to explore how geometry can respond to operations

that delocalize the entanglement.

Van Raamsdonk [33] pointed out that a random unitary transformation applied to the left handed

26

CHAPTER 3. MULTIPLE SHOCKS 27

CFT leaves the density matrix describing right handed CFT observables unchanged, but will change

the relation between degrees of freedom on both sides and hence the geometry behind the horizon.

Certain unitaries correspond to local operators, which can create a pulse of radiation propagating

just behind the horizon [32].

We examined this situation in detail in our study of scrambling [1]. We showed that a local

operator on the left hand boundary that only injects one thermal quantum worth of energy, if applied

early enough, scrambles the left hand Hilbert space and disrupts the special local entanglement. This

happens when the time since the perturbation, tw, is of order the fast scrambling time [9, 8]1

t∗ =β

2πlogS (3.2)

where S is the black hole entropy and β is the inverse temperature. From the bulk point of view,

the perturbation sourced at an early time (large tw) is highly boosted relative to the t = 0 frame,

creating a shock wave, as illustrated in the right panel of Fig. 3.1The geometry dual to Eq. (3.4One

shockequation.3.2.4) consists of a perturbation that emerges from the past horizon and falls through

the future horizon (left). If t1 is sufficiently early, the boost relative to the t = 0 slice generates

backreaction in that frame (right). Note that the horizons no longer meetfigure.3.1. This shock

disrupts the Ryu Takayanagi surface [17, 19] passing through the wormhole [30, 31]. The area of this

surface is used to calculate the mutual information I(A,B) that diagnoses the special entanglement

between local subsystems A ⊂ L, B ⊂ R of the two CFTs. For subsystems smaller than half, one

finds that the leading contribution to I drops to zero when tw ∼ t∗.The two point correlation function 〈ϕL(t)ϕR(t)〉, with operators at equal Killing time on opposite

sides, also diagnoses the relation between degrees of freedom and should become small if |t− tw| is

of order the scrambling time. In the bulk it is related to geodesics and hence probes the geometry

[44, 45, 46, 47, 48, 49]. Using (2+1) Einstein gravity and ignoring nonlinear effects, the correlation

function was computed in [1], using the length of the geodesic connecting the correlated points.

Roughly, the result decreases like a power of 1/(1 + e2π(|t−tw|−t∗)/β

). The fact that this expression

depends only on (t− tw) is a consequence of the boost symmetry of the eternal black hole. It is clear

that, for any choice of tw, there is a time t ∼ tw at which the correlator 〈ϕL(t)ϕR(t)〉 is order one.

As pointed out in [1], when |t−tw| is large, the relative boost between the geodesic and the shock

wave is very large. This makes likely the possibility that nonlinear corrections to the correlation

function result are important. We are currently exploring these effects but in this paper we will ignore

them. We hope the Einstein gravity results will be a useful guide to the important phenomena. In

any event they should serve as a lower bound to the strength of these effects.

Marolf and Polchinski [69] analyzed the behavior of truly typical two sided states where the

average energy of the total Hamiltonian HL + HR is fixed. Using the Eigenvector Thermalization

Hypothesis [70], they showed that the two point correlator between local operators on the two sides is

1The importance of this time scale in black hole physics was pointed out in earlier work, including [36].


typically ∼ e−S , and is never larger than ∼ e−S/2, for any choice of times for the two operators. This

is in contrast with the behavior of correlators in the shock wave geometry discussed above. Marolf

and Polchinski interpreted their result as evidence for a “non geometrical” connection between the

two sides.

The work of Maldacena and Susskind [13] suggests a different potential interpretation. These

authors considered the time evolution of the thermofield double state2 as a family of states in which

the local entanglements present in |TFD〉 are disturbed. At late times, two-sided correlations become

small because of the increasing length of the geodesic threading the wormhole. This suggests that

the behavior found in [69] could be consistent with a smooth but very long wormhole linking the

two sides.

In fact, very little is known about more general states. To this end, we explore in §3.2Wormholes

built from shock wavessection.3.2 a class of geometries obtained by perturbing the left side of the

thermofield double state with a string of unitary local operators with order-one energy,

Wn(tn)...W1(t1)|TFD〉. (3.3)

If the time separations are sufficiently large, the boosting effect described above means that these

states are dual to geometries with n shock waves. We will outline an iterative procedure that builds

the geometry one shock wave at a time. Using this method, we will explore a small part of the

diverse class of metrics dual to states of this form. If the time separations and/or the number of

shocks is large, one finds that the wormhole connecting the two asymptotic regions becomes very

long in all boost frames, indicating weak local correlation between the two boundaries at all times.

The timescale t∗ plays a central role in the construction, indicating that the geometry is sensitive

to chaotic dynamics in the CFT. The application of aW operator creates a short-distance disturbance

in the CFT. The application of a second, at time separation greater than t∗, creates a second

disturbance and erases the first. This manifestation of scrambling is represented in the bulk by the

second shock wave pushing the first off the AdS boundary and onto the singularity.

The states (3.3Introductionequation.3.1.3) and their bulk duals provide examples of how Einstein

gravity can accommodate weak two-sided correlations, but they are not typical in the Hilbert space.

This is for multiple reasons. First, the W operators inject some energy into one of the CFTs, making

the energy statistics not precisely thermal. Second, the operators leave a distinguished time tn at

which a local perturbation is detectable in the left CFT. In order to make states with weak two-sided

correlation, we pay the price of an atypical ρL.

In general, the duals to (3.3Introductionequation.3.1.3) are geometrical, but they are not drama-

free. In particular, by boosting the geometry one way or another, one can always find a frame in which

an infalling observer collides with a high energy shock very near the horizon. In §3.3Ensemblessection.3.3,

we will emphasize that the class of truly typical states should be invariant under such boosts. This

2Here, we mean time evolution with HL +HR.


constrains the possible form of a smooth geometrical dual to a typical state.

We will conclude in §3.4Discussionsection.3.4. Certain technical details of the shock wave con-

struction are recorded in two appendices.

AdS/CFT applications of wide wormholes have previously been discussed in [71]. In [13], it was

noted that adding matter at the boundaries of the eternal black hole would make a wide wormhole

describing less than maximal entanglement. Our examples are similar, but we add a small amount of

matter, relying on the effect of [1] to amplify the perturbation, and leaving the total entanglement

near maximal. The length of the resulting wormhole is related to the absence of local two-sided

entanglement [31]. The paper [72] contains further discussion of the connection between chaos and

geometry described here.

3.2 Wormholes built from shock waves

3.2.1 One shock

Let us begin by reviewing the geometrical dual to a single perturbation of the thermofield double

[1]. We consider a CFT state of the form

W (t1)|TFD〉, (3.4)

where the operator W acts unitarily on the left CFT and raises the energy by an amount E. The

scale E is assumed to be of order the temperature of the black hole, much smaller than the mass M .3

To keep the bulk solutions as simple as possible, we will assume that W acts in an approximately

spherically symmetric manner. We will also assume that W is built from local operators in such a

way that it acts near the boundary of the bulk AdS space.

One can think about the expression (3.4One shockequation.3.2.4) in different ways. One option

would be to understand it as a thermofield double state that was actively perturbed by a source at

time t1; the W operator would then be time-ordered relative to other operators in an expectation

value. Another option is to understand it as the state of a system evolving with a strictly time-

independent Hamiltonian. We will occasionally use language appropriate to the first interpretation,

but where it makes a difference (i.e. for expectation values involving operators before t1) we will

stick to the second, ordering the W operator immediately after the state vector.

With this understanding, the bulk dual to the state (3.4One shockequation.3.2.4) consists of a

perturbation that emerges from the past horizon of the black hole, approaches the boundary at time

t1, and then falls through the future horizon, as shown in the left panel of Fig. 3.1The geometry dual

to Eq. (3.4One shockequation.3.2.4) consists of a perturbation that emerges from the past horizon

3For a large AdS black hole dual to a state with temperature of order the AdS scale, we have E ∼ 1 in AdS units,while M ∼ 1/GN , which is proportional to N2 in the large-N gauge theory.


MMM+E

M+E

M+E

t1

t1

α

Figure 3.1: The geometry dual to Eq. (3.4One shockequation.3.2.4) consists of a perturbation thatemerges from the past horizon and falls through the future horizon (left). If t1 is sufficiently early,the boost relative to the t = 0 slice generates backreaction in that frame (right). Note that thehorizons no longer meet.

and falls through the future horizon (left). If t1 is sufficiently early, the boost relative to the t = 0

slice generates backreaction in that frame (right). Note that the horizons no longer meetfigure.3.1.

Since the energy scale of the perturbation is order one, backreaction on the metric is negligible.

However, if we increase the Killing time t1, the perturbation is boosted relative to the original

frame, and the energy relative to the horizontal t = 0 surface increases as4

E(t=0)p ∼ Ee2πt1/β , (3.5)

where β is the inverse temperature of the black hole. Once t1 ∼ t∗, backreaction must be included.

The resulting geometry is sketched in the right panel.5 Details of the shock wave metric are given

in [1], following earlier work by [42, 38, 39, 40]. For the remainder of this section, we will work in

the (2+1) dimensional setting of the BTZ black hole. This is for technical convenience; the essential

features generalize to higher dimensions. For small E and large t1, a good approximation to this

metric consists of two pieces of the same BTZ geometry, glued together across the u = 0 surface,

with a null shift in the v coordinate by amount

α =E

4Me2πt1/β ∼ e2π(t1−t∗)/β . (3.6)

Here, we are using Kruskal coordinates for each of the patches, with metric

ds2 =−4`2dudv +R2(1− uv)2dφ2

(1 + uv)2. (3.7)

4In our conventions, the Killing time t increases downwards on the left boundary.5Notice that we have represented the matter as a thin-wall null shell. Physical perturbations will have some spatial

width, and they might follow massive trajectories. However, because of the highly boosted kinematics that we willconsider in this paper, it will be permissible to treat all matter in this way.


3.2.2 Two shocks

Next, we consider a state of the form

W (t2)W (t1)|TFD〉. (3.8)

To construct the bulk dual, we simply need to act with W (t2) on the single-shock geometry con-

structed above. In order to do this, it is helpful to generalize our problem slightly, and understand

how to construct the bulk dual to a state

W (t)|Φ〉, (3.9)

assuming that we already know the geometry for |Φ〉. In general, the prescription is as follows: we

start with the geometry for |Φ〉 and select a bulk Cauchy surface that touches the left boundary at

time t. We record the data on that surface, add the perturbation corresponding to W (t) near the

boundary, and evolve the new data forwards and backwards.

In Fig. 3.2The dual to a two-W state is constructed from the one-W state by adding a perturba-

tion near the boundary at time t2 and then evolving forwards and backwardsfigure.3.2, we use the

above procedure to build the two-W geometry. The left panel represents the state W (t1)|TFD〉,and the dashed blue line is the Cauchy surface that touches the left boundary at time t2. We add

the second perturbation and evolve forwards and backwards in time, producing the geometry shown

on the right.

M

M+E

M

M+E

M+2E

MM+E

M+2E

t1

t2

Mt

Figure 3.2: The dual to a two-W state is constructed from the one-W state by adding a perturbationnear the boundary at time t2 and then evolving forwards and backwards.

We can understand this prescription in terms of the “folded” bulk geometries discussed in [64].

The two-shock geometry corresponds to a folded bulk with three sheets. On the first sheet, we

evolve from −∞ to t1. On the second sheet (a portion of the left panel of Fig. 3.2The dual to a

two-W state is constructed from the one-W state by adding a perturbation near the boundary at

time t2 and then evolving forwards and backwardsfigure.3.2), we add a perturbation at t1 and evolve

backwards in time from t1 to t2. On the final sheet (a portion of the right panel of Fig. 3.2The dual

to a two-W state is constructed from the one-W state by adding a perturbation near the boundary

at time t2 and then evolving forwards and backwardsfigure.3.2), we add a perturbation at t2 and


evolve forwards to +∞. Our prescription to order the W operators immediately after the state

means that we focus on the final fold of the bulk, extending it in time from −∞ to +∞, however

we use each of the sheets in our iterative construction procedure.

It is clear from the figure that the two shells collide on the final sheet. Our assumptions of

spherical symmetry and thin walls make it possible to construct the full geometry by pasting together

AdS-Schwarzschild geometries with different masses. There are two conditions: first, we require r,

the size of the sphere, to be continuous at the join. Second, we have the DTR regularity condition

[34, 73, 74]

ft(r)fb(r) = fl(r)fr(r), (3.10)

where t, b, l, r refer to the top, bottom, left and right quadrants, and f is the factor in the metric

ds2 = −fdt2 +f−1dr2 +r2dΩ2. Explicitly, for the 2+1 dimensional BTZ case, f(r) = r2−8GNM`2,

where M is the mass of the black hole and ` is the AdS length. The DTR condition becomes

[r2 − 8GNMt`

2][r2 − 8GN (M + E)`2

]=[r2 − 8GN (M + 2E)`2

][r2 − 8GNM`2

]. (3.11)

If the collision takes place at large r, the evolution is nearly linear and this equation implements

conservation of energy of the shells. However, even beyond the linear regime, the equation plays a

similar role, fixing the mass Mt of the Schwarzschild solution in the post-collision region in terms of

the other masses and r, the radius of the collision. In turn, r is set by the time difference (t2 − t1).

To find the precise relation, it is simplest to use Kruskal coordinates. By matching the size of the

S1 in the two coordinate systems, we find that r is determined by u and v as

r

R=

1− uv1 + uv

, (3.12)

where the radius of the horizon, R, is determined by R2 = 8GNM`2, with M is the mass of the black

hole and ` the AdS length. The u and v coordinates are conserved, respectively, by right-moving

and left-moving radial null trajectories. Using the Kruskal conventions in [1], we can determine the

value of u or v using the time coordinate at which the trajectory hits the left boundary:

u = e−Rt/`2

, v = −eRt/`2

. (3.13)

In particular, in the Kruskal system of the bottom quadrant, the v coordinate of the left-moving

shock is −eRbt1/`2

, while the u coordinate of the right-moving shock is e−Rbt2/`2

.6 This determines

the r value of their collision asr

Rb=

1 + eRb(t1−t2)/`2

1− eRb(t1−t2)/`2. (3.14)

6Rb is the BTZ radius in the lower quadrant, defined by R2b = 8GN (M + E)`2.


Plugging this value of r into Eq. (3.11Two shocksequation.3.2.11), we find

Mt = M + E +E2

M + Esinh2 Rb(t2 − t1)

2`2. (3.15)

The final, exponentially growing term begins to dominate the first term when (t2 − t1) ≈ 2t∗.

Given that a W (t) operator creates a perturbation in the UV at time t, one might have expected

a two-W state to have perturbations near the boundary both at t1 and at t2. In fact, if the time

difference is greater than scrambling, this is not the case. In the bulk, we can understand this by

going back to the left panel of Fig. 3.2The dual to a two-W state is constructed from the one-

W state by adding a perturbation near the boundary at time t2 and then evolving forwards and

backwardsfigure.3.2. In this one-W state, the W (t1) perturbation approaches the boundary at time

t1, but at much earlier times it is very close to the horizon. If we add the second perturbation W (t2)

sufficiently early, then the outward jump of the horizon due to the increase in mass will be enough

to capture the first shock, as shown in the right panel of Fig. 3.3As t2 shifts earlier, the time at

which the original shock reaches the boundary shifts later, eventually moving onto the singularity

(right)figure.3.3.

t1t1

t2

t2

|t1-t2| < t* |t1-t2| > t*

Figure 3.3: As t2 shifts earlier, the time at which the original shock reaches the boundary shiftslater, eventually moving onto the singularity (right).

To analyze this effect in detail, it is again helpful to use Kruskal coordinates. The key is to

determine the v coordinate of the trajectory of the W (t1) shell in the Kruskal system of the left

quadrant. If v is negative, then the shell hits the boundary at time eRt/`2

= −v. If v is positive, then

the shell runs from singularity to singularity. To find the v coordinate, we can use Eq. (3.12Two

shocksequation.3.2.12), plugging in the r coordinate in Eq. (3.14Two shocksequation.3.2.14), and

the u coordinate in the left Kruskal system e−Rlt2/`2 ≈ e−Rt2/`2 . We find

v ≈ −eRt1/`2

+E

4MeRt2/`

2

. (3.16)

The coordinate becomes positive, indicating that the shock wave has moved off the left boundary

and onto the singularity, when (t2 − t1) ≈ t∗.The presence of the timescale t∗ suggests that we interpret the “capture” of the first perturbation


in terms of scrambling. Indeed, the state W (t1)|TFD〉 is carefully tuned to produce an atypical

perturbation in the UV at time t1. If we additionally perturb this state by acting with W (t2) a

scrambling time before t1, this delicate tuning is upset, and the perturbation at t1 fails to materialize.

We can also think about this effect in terms of the square of the commutator

〈TFD|[W1(t1),W2(t2)]†[W1(t1),W2(t2)]|TFD〉. (3.17)

Expanding this out, we find two terms that each give a numerical contribution of one, minus two

terms involving the overlap of W1(t1)W2(t2)|TFD〉 and W2(t2)W1(t1)|TFD〉. According to the bulk

solution just described, the overlap of these states should be small if the time separation is greater

than t∗, indicating that (3.17Two shocksequation.3.2.17) becomes approximately equal to two once

|t1 − t2| ∼ t∗. This large commutator is a sharp diagnostic of chaos: perturbing one quantum

perturbs all quanta a scrambling time later [12].

3.2.3 Many shocks

A general geometry built from spherical shock waves can be analyzed in terms of a sequence of

two-shock collisions. This means that the matching conditions discussed above, together with the

recursive procedure for adding a W perturbation, allow us to construct the dual to arbitrary states

of the form

Wn(tn)...W1(t1)|TFD〉. (3.18)

By varying the times t1, ..., tn, one finds a very wide array of possible metrics. We will focus on a

particular slice through the space of these states, in which all even-numbered times are equal to tw,

and all odd-numbered times are equal to −tw.

We will also assume that the asymptotic energy of each shock, E, is very small compared to

the unperturbed mass M . The large-N limit in the gauge theory allows us to take E/M → 0 and

tw →∞, with

α =E

4Me2πtw/β (3.19)

held fixed. In this limit, the iterative construction process described above becomes rather straight-

forward: we alternately add shocks traveling backwards in time from the top left corner, and forwards

in time from the bottom left. The associated null shifts, which alternate in the u and v directions,

have the effect of extending the wormhole to the left, as illustrated in Fig. 3.4The thermofield double

and the first six multi-W states are drawn. In each case, the next geometry is obtained from the

previous by adding a shock either from the top left or bottom left corner. The gray regions are

sensitive to the details of a collision, but the white regions are not. Using the time-folded bulk of

[64], these states can be combined as different sheets of an “accordion” geometryfigure.3.4.

Because of the null shifts, all but one of the shock waves run from singularity to singularity. Still,


Figure 3.4: The thermofield double and the first six multi-W states are drawn. In each case, thenext geometry is obtained from the previous by adding a shock either from the top left or bottomleft corner. The gray regions are sensitive to the details of a collision, but the white regions are not.Using the time-folded bulk of [64], these states can be combined as different sheets of an “accordion”geometry.

the leftmost one touches the boundary at time ±tw,7 making this time locally distinguished in the

CFT. One can also consider bulk solutions with the property that all shocks run from singularity to

singularity, leaving no locally distinguished time. At the level of the bulk theory, there is nothing

wrong with these geometries. However, unlike the multi-W states described in this paper, we are

not sure how or whether they can be constructed in the CFT.

Our assumption that the ti are equal in magnitude and alternating in sign means that the

interior region of the resulting wormhole has a discrete translation symmetry. We can understand

this as follows: after step k in the iterative procedure, the geometry to the left of all shocks will

be unperturbed AdS-Schwarzschild. The geometry that gets built in that region during subsequent

steps is therefore independent of k.8

Using this translation invariance, we can understand the full geometry of the wormhole by

studying a “unit cell,” for which the geometry depends on α but not n. Let us begin by computing

the length of the wormhole, i.e. the regularized length of the shortest geodesic that passes from

the left boundary to the right. Up to an n-independent deficit, this is simply n times the length

across the central layer of a unit cell. The portion of the geodesic that passes through this unit cell

(see Fig. 3.5A geodesic passes across a portion of the wormhole. It intersects the null boundaries of

the central regions halfway across their widthfigure.3.5) is a geodesic in the BTZ geometry passing

from Kruskal coordinates (u = 0, v = α/2) to (u = α/2, v = 0). The length of such a geodesic is

7Here, we are backing off the limit tw →∞.8Notice that at finite E, this symmetry would be broken by a smoothly varying mass profile in the wormhole,

increasing from right to left. If we relax the assumption of equal times, this translation invariance would also bebroken by the fact that different W operators source shocks of varying strength.


α

α/2

Figure 3.5: A geodesic passes across a portion of the wormhole. It intersects the null boundaries ofthe central regions halfway across their width.

` cosh−1(1 + α2/2). Thus the regularized length across the entire wormhole is

L

`= n cosh−1

(1 +

α2

2

)+O(n0). (3.20)

This function interpolates between nα for small α and 2n logα for large α. We can make this length

large, and in particular greater than S, by making α and/or n large. Such wormhole geometries

therefore describe CFT states with very weak local correlation ∼ e−(const.)L between the two sides.

Note, however, that if we make L ∼ S by fixing α and taking n ∼ S, then the mass of the left black

hole will be larger than that of the right by an amount δM ∼ SE ∼M . Instead, we could fix n and

take the time differences to be of order S. In this case, the energies of the shocks are extremely high

∼ eS , and the geometrical computation of the correlator is completely out of control. We interpret

the geodesic estimate as an upper bound on the true correlator.

Having computed the length, we would like to understand the qualitative shape of the unit cell as

a function of α. First, let us consider the case in which α is large compared to one. The construction

of the geometry is very simple in this limit, because the post-collision regions are pushed near the

singularities, and almost none of the geometry is affected by the details of the collisions. This should

be clear from the large-α four-W geometry shown in Fig. 3.6The large-α four-W geometry is shown.

Notice that the post-collision regions are small and isolated near the singularities. The Kruskal

diagram at the bottom emphasizes the kinkiness of the geometryfigure.3.6.

For intermediate values of α . 1, we have geometries similar to those in Fig. 3.6The large-α

four-W geometry is shown. Notice that the post-collision regions are small and isolated near the

singularities. The Kruskal diagram at the bottom emphasizes the kinkiness of the geometryfigure.3.6.

The central white region is unaffected by details of the collisions, but the Mandelstam s invariant in

each collision is of order α2M2, and the shaded regions will be sensitive to string and Planck scale

physics.

For small values of α (with αn fixed) it is natural to guess that the large kinks of size α in

Fig. 3.6The large-α four-W geometry is shown. Notice that the post-collision regions are small and

isolated near the singularities. The Kruskal diagram at the bottom emphasizes the kinkiness of


α

Figure 3.6: The large-α four-W geometry is shown. Notice that the post-collision regions are smalland isolated near the singularities. The Kruskal diagram at the bottom emphasizes the kinkiness ofthe geometry.

the geometryfigure.3.6 will be smoothed out,9 allowing an analysis in terms of an averaged stress

tensor. For most values of α 1, inelastic stringy effects, proportional to GNα2M2`2s/`

D−2 [75],

will be important in determining the form of this stress energy. As an example, though, we will

work out the geometry appropriate for the case in which α is small enough that we can ignore these

effects10 Thus, we look for a solution to Einstein’s equations with radial null matter moving in both

directions, and with translation symmetry plus spherical symmetry.11

Specifically, we make an ansatz

ds2 = −`2dτ2 + h(τ)2dx2 + g(τ)2dφ2 (3.21)

and compute the stress tensor implied Einstein’s equations. In order for Tφ,φ to be pure cosmological

constant, h(τ) must be proportional to cos τ . In order for Tτ,τ and Tx,x to be pure cosmological

constant plus traceless matter, we find an equation for g. By requiring that the solution be differ-

entiable at τ = 0, we find that the metric is uniquely determined (up to the scales ` and R, which

we now restore) as:

ds2 = −`2dτ2 + `2 cos2 τdx2 + g(τ)2dφ2 (3.22)

g(τ)

R= 1− sin τ log

1 + sin τ

cos τ.

9We are grateful to Raphael Bousso for making this suggestion.10We need α small enough that the probability of oscillator excitation per collision, GNα

2M2`2s/`D−2, times the

number of collisions, 1/α, is small. Roughly, we support the wormhole with a large number of relatively soft quanta,with boost factor e2πtw/β of order `2/`2s. The mild boost means that doubling the mass of the left black hole onlyleads to a wormhole of length `3/`2s.

11In a realistic setting, the shocks won’t be exactly spherically symmetric. Suppose we build each shell as a sum ofparticles localized on the S1. After a collision, these can be deflected by an angle ∼ α [75]. Each experiences ∼ 1/αcollisions before hitting the singularity, but if the initial inhomogeneity is small, deflections will tend to cancel, andthe total effect will remain small.


In order to check that this metric actually corresponds to the small α limit of the dense network

BTZ

Vaidya

VaidyaVaidya

BTZ

Vaidya

ds2=-ℓ2dτ2+h(τ)2dx2+g(τ)2dϕ2

Figure 3.7: The wormhole created from a large number of weak shocks (top) becomes a smoothgeometry in the α→ 0 limit (bottom).

of shock waves, we write down recursion relations for the patched-together geometry in Appendix

3.5Appendix A: Recursion relations for many shock wavessection.3.5. By taking α = 0.01, solving

the recursion relations numerically, and computing the size of the S1 as a function of proper time in

the direction orthogonal to the symmetry axis, we find excellent agreement with the function g(τ).

The metric (3.22Many shocksequation.3.2.22) gives us the translationally invariant part in the

interior of the wormhole. To complete the geometry, we need to understand how to patch it together

with the BTZ exteriors. Here, we go back to the shock wave construction sketched in Fig. 3.7The

wormhole created from a large number of weak shocks (top) becomes a smooth geometry in the

α→ 0 limit (bottom)figure.3.7, and notice that the intersecting network of shocks in the interior of

the wormhole is matched to the empty exteriors across a region in which the shock waves are moving

in only one direction. These regions are therefore a piece of the BTZ-Vaidya spacetime, with mass

profile determined in Appendix 3.6Appendix B: Vaidya matching conditionssection.3.6.

3.3 Ensembles

In the previous section we have discussed a family of geometries with long wormholes, describing

weak correlation between the left and right CFTs. In particular, by taking a large number of shocks

or large time separations, the wormhole length can exceed S, consistent with a two point correlator

of order e−S , the value in a typical state found by Marolf and Polchinski [69]. However, as we will

emphasize in the discussion section, the states constructed in this manner are not typical in the

two-CFT Hilbert space.

In this section, we will put the W states aside and address the question of whether truly typical


states could be described by smooth geometries. First let us define “typical state” more carefully.

This concept is straightforward in classical statistical mechanics. The standard phase space measure

on an energy shell in phase space determines the probability for finding a phase space region. Typical

regions are those with typical probability in this measure. For an ergodic system time evolution

reproduces this probability. The fraction of time such a system spends in a region is equal to the

measure of the region. So typical states can also be defined as ones that occur typically in the time

evolution of the system.

Quantum mechanics is different. If a state |ψ〉 =∑s cs|Es〉 then

|ψ(t)〉 =∑s

cse−iEst|Es〉. (3.23)

Time evolution does not change the magnitude of the coefficient of an eigenvector, only its phase.

But there are natural notions of a distribution for the magnitudes. For example, in a Hilbert space

of dimension D, there is unique distribution that is invariant under U(D) transformations. This is

given by acting on a reference state with a Haar random unitary.12 For large D, the probability is

proportional to

P (|ψ〉) ∼ exp(−D∑s=1

|cs|2/2f2) (3.24)

where f is chosen so that the state normalization condition 〈ψ|ψ〉 = 1 is satisfied (up to small

fluctuations), 2f2 = 1/D. This measure gives a natural notion of a typical state. In a less completely

random situation we expect the probabilities in an ensemble to depend on the energy of states. A

natural generalization of (3.24Ensemblesequation.3.3.24) to this case is

P (|ψ〉) ∼ exp(−D∑s=1

|cs|2/2f2(Es)) (3.25)

where f is smooth over the spread in energies of the system being sampled, and satisfies the normal-

ization condition∑s 2f(Es)

2 = 1. The ensemble (3.25Ensemblesequation.3.3.25) provides a natural,

but not unique, notion of a typical state. Note that this ensemble is invariant under time evolution,

which just changes the phases of the cs.

We now turn to the question of how time evolution can approximate this ensemble. Assuming

that the Hamiltonian of the system H is sufficiently chaotic, and that the initial state is typical with

respect to this distribution, then time evolution eventually brings this state to within a distance of

12Random matrix techniques show that the eigenstates of a random Hamiltonian are distributed in the same wayas states obtained by acting a random unitary on a reference basis.


order one of nearly all states in the ensemble. To see this, we compute∫d|ψ〉d|χ〉P (|ψ〉)P (|χ〉) max

t|〈χ|e−iHt|ψ〉| (3.26)

=1

N 2

∫ ∏s

(d2csd

2c′se−(|cs|2+|c′s|

2)/2f2(Es))

maxt

∣∣∑r

c∗rc′re−iErt

∣∣ (3.27)

≈∑s

(1

2πf2(Es)

∫d2cse

−|cs|2/2f2(Es)|cs|)2

(3.28)

=∑s

π

2f2(Es) =

π

4. (3.29)

In the second equality, we have used the assumption that all energy levels are incommensurate,

so we can find a time t such that c∗sc′se−iEst = |cs||c′s| for nearly all s (this time will typically be

double-exponential in the entropy S). The factor N normalizes the probability distribution. In the

final equality, we used the normalization condition for f .

In our specific situation we will imagine following [69] and adding a weak “wire” between the left

and right sides that lets the system as a whole thermalize. We can imagine the wire allowing the

exchange of one quantum with thermal energy between the left and right sides every large number

of thermal times. Denote this wire by an operator Ω which is a smeared product of local operators

in the left and right systems and the total Hamiltonian H = H0 + Ω where H0 = HL + HR.

Now thermalize by evolving |TFD〉 forward with U(t). By choosing a random time t, we form

an ensemble of states that is invariant under time translation. How similar is this ensemble to

(3.25Ensemblesequation.3.3.25)? We expect the expansion of |TFD〉 in eigenstates of H to have

coefficients |cs| that are typical of the distribution (3.25Ensemblesequation.3.3.25) for an appropriate

f(Es). Therefore, after some time the state comes within an overlap of π/4 of any typical state in

that ensemble.13 This overlap is enough to ensure that the states cannot be distinguished, with an

optimal measurement of a linear operator, with probability better than roughly 80%.

The ensemble generated by the wire raises a question of time scales: how much evolution is

required to produce a state that we may treat as typical? As a lower bound, it seems reasonable

to allow at least a time S, so that all quanta can equilibrate across the wire. An (extreme?)

upper bound is provided by the quantum recurrence time, schematically ∼ eeS . Another potentially

interesting time scale is the time ∼ eS , after which point states can be written as a superposition of

naively orthogonal states at earlier times. These recurrence timescales, if relevant, would be vastly

longer than those over which the geometrical constructions of the previous sections are reliable.

Having defined these ensembles, we will now use their time-translation invariance to derive a

constraint. Suppose that a typical state |ψ〉 is described by a smooth geometry with a long wormhole.

Then U(−t)|ψ〉 is also typical, and hence by assumption also described by a smooth geometry with

13To improve upon the π/4, we could take our initial state and evolve it with two different chaotic Hamiltonians(“wires”) for various lengths of time in various orders. To be safe one should use order D different time evolutionintervals.


a long wormhole. Roughly, the two geometries are related by a boost. This is dangerous: imagine

that part of the matter supporting the |ψ〉 wormhole is a light ray behind the horizon. If Bob starts

Bob

light

ray

II

I

III

IV

Figure 3.8: Bob falls in from the boundary at tB = 0 and experiences a mild interaction with thestress energy supporting the solution. If he jumped in at a much earlier time tB ∼ −t∗, he wouldexperience a dramatic interaction.

falling into the |ψ〉 black hole at time tB = 0, he might experience a mild collision. But consider the

geometry associated with U(−t)|ψ〉. If Bob falls into this geometry at time tB = 0 his experience

will be the same as falling into |ψ〉 at time tB = −t. If t ∼ t∗, Bob will experience a violent collision.

It typical states are dual to smooth geometries, avoiding this boosting effect would require all

three regions I, II, III on the figure to be essentially the same as the empty eternal black hole. This

is a powerful constraint on the form of such geometries. These empty regions would have to be joined

in some way onto a long wormhole. The joining locus on the Penrose diagram (Fig. 3.9A candidate

for the geometrical dual to a typical state?figure.3.9) would have to be a surface containing timelike

curves of infinite length, quite different from the intuitive notion of a long thin wormhole. If we

Boblig

ht ra

yII

I

III

IV???

Figure 3.9: A candidate for the geometrical dual to a typical state?

imagine this curve to be boost invariant, the configuration in quadrant IV resembles the dual of a

cut off CFT. This suggests that there are other quantum states present than the standard ones at

the UV boundary of quadrant II.14

Of course another possibility is that typical states do not have smooth geometries outside of

region II [33]. An observer falling through the horizon immediately encounters a firewall [11].

14The “mirror operators” of [76] might be candidates for these. This possibility arose in a discussion with JuanMaldacena and Edward Witten.


3.4 Discussion

In the context of 2+1 dimensional Einstein gravity, we have identified a large class of two-sided

AdS black hole geometries with long wormholes. These geometries are dual to perturbations of the

thermofield double state of two CFTs,

Wn(tn)...W1(t1)|TFD〉, (3.30)

and they provide constructible examples of highly entangled states with two-sided correlators that

are small at all times. The key geometrical effect is boost enhancement of the GN -suppressed

backreaction associated to each perturbation [1]. If the time between perturbations is sufficiently

large, their shock wave backreaction must be included, lengthening the wormhole.

The scrambling time t∗ emerges as an important dynamical timescale in the construction of the

metrics. For example, perturbations at widely separated times, ∆t ∼ 2t∗, create kinked geometries

with high energy shocks, while large numbers of perturbations at smaller time separation lead to

smoother wormholes. As a second example, even though a multi-W state includes operators local

at n different times, if the separations |ti+1 − ti| are greater than t∗, our bulk analysis indicated

that the CFT state (3.30Discussionequation.3.4.30) has a locally detectable disturbance only at the

“outermost” time tn. Roughly, the action of Wn(tn) disturbs the delicate tuning required for a local

perturbation to appear at time tn−1; in bulk language, the Wn−1 shock is captured by a tiny increase

in size of the horizon due to the Wn shock.

Although these states display the very small correlation between L and R characteristic of typical

states, they are atypical in important ways. They have a distinguished time, tn, at which a shock

wave approaches the boundary. Also, the W operators increase the energy without increasing the

two-sided entanglement. In a typical ensemble, the distribution of entanglement is very sharply

peaked, and deficits are highly suppressed in the measure [77]. Another feature of these states is

that boosting them gives a high energy shock wave on the horizon. If typical states are dual to

smooth geometries, they would have to be of the kind discussed in §3.3Ensemblessection.3.3.

One could attempt to build a typical state out of a basis consisting of the multi W states, each

described by a geometry. It might seem unlikely that a superposition of distinct geometries could

again be represented as a geometry, but this is difficult to exclude: in expectation values, the large

number of off diagonal terms will dominate, rendering semiclassical reasoning invalid.

By estimating correlators using geodesic distance, we have ignored the backreaction of the field

sourced by the correlated operators. Although this should provide an upper bound on the correla-

tion, an interesting possibility is that nonlinear effects might make it possible for relatively short

wormholes with high energy shocks running between the singularities to represent states with ∼ e−S

local correlation between the two sides.

Using the methods discussed in this paper it is straightforward to construct states containing a


few particles behind the horizon. Constructing actual field operators in this region is an open and

interesting problem.

3.5 Appendix A: Recursion relations for many shock waves

In this appendix, we will write the recursion relations for the translationally-invariant network of

intersecting shock waves. By solving these relations numerically in the α → 0 limit, one finds

agreement with the smooth metric given in Eq. (3.22Many shocksequation.3.2.22).

Exploiting the discrete translational invariance of the arrangement of shock waves, we can rep-

resent the metric in terms of the radii of the collisions, rn, and the BTZ R parameters of the

geometries between collisions, Rn (see Fig. 3.10The size of the S1 at the vertices is labeled rn, and

the R parameter of the BTZ geometry forming each plaquette is labeled Rnfigure.3.10). We would

like to check the function g(τ) in the case ` = R = 1. In order to do so, we will write recursion

relations for rn and Rn, and then compute the geodesic distance “straight up” from the first collision

to the n’th. Identifying this with the interval in τ , we will then be able to confirm that the radius

of the S1 (determined by rn) depends on τ as g(τ).

r1

r2

r3

r4

R1

R2

R3

R4

R2

r3R3

R1

R4r4

r2

r1

R2

R4

R3R3

R1 R1

r2

r1

r3

r4

r2

r4

Figure 3.10: The size of the S1 at the vertices is labeled rn, and the R parameter of the BTZgeometry forming each plaquette is labeled Rn.

We need two recursion relations, one each for rn and Rn. One of these equations is given

simply by applying the DTR relation Eq. (3.10Two shocksequation.3.2.10) at a given vertex, with

f(r) = r2 −R2n. This gives

R2n+1 = r2

n +(R2

n − r2n)2

R2n−1 − r2

n

. (3.31)

To get the other equation, we proceed as follows. We focus on a given plaquette, with BTZ parameter

Rn, and assume that we know the radii rn, rn−1 of the side and bottom vertices. Let us choose a

Kruskal frame for this patch in which u = v = ub at the bottom vertex. Then using Eq. (3.12Two


shocksequation.3.2.12) we must havern−1

Rn=

1− u2b

1 + u2b

. (3.32)

Now, holding v = ub fixed, we solve for ∆, the change in u that is necessary to reach the radius of

the side vertex, rn. The radius of the top vertex is then determined by

rn+1

Rn=

1− (ub + ∆)2

1 + (ub + ∆)2. (3.33)

Eliminating ub and ∆, we find the recursion relation

rn+1 =2rnR

2n − rn−1R

2n − rn−1r

2n

R2n + r2

n − 2rnrn−1. (3.34)

For a wormhole that connects BTZ regions with R = 1, the initial conditions are R1 = r1 = 1.

Since the recursion relations are second order, we also need to determine R2 and r2. These can be

found using the two-shock solution:

r2 =1− α2

1 + α2, R2 =

√1 + 4α2. (3.35)

The equations (3.31Appendix A: Recursion relations for many shock wavesequation.3.5.31) and

(3.34Appendix A: Recursion relations for many shock wavesequation.3.5.34), together with these

initial conditions, completely determine the geometry. In order to compare with the smooth worm-

hole, we also need to compute the geodesic distance “straight upwards.” Using ub and ∆ derived

above, along with the Kruskal metric Eq. (3.7One shockequation.3.2.7), one can check that the

timelike distance from the bottom vertex to the top vertex of the n’th plaquette is

2 tan−1

√Rn + rn−1

Rn − rn−1

R− rnR+ rn

− 2 tan−1

√Rn − rn−1

Rn + rn−1. (3.36)

Taking α = 0.01, numerically solving the recursion relations, and plotting rn as a function of the

total geodesic distance from the initial slice, one finds excellent agreement with g(τ).

3.6 Appendix B: Vaidya matching conditions

We will work out the matching condition in detail for the top left Vaidya region in the lower panel of

Fig. 3.7The wormhole created from a large number of weak shocks (top) becomes a smooth geometry

in the α→ 0 limit (bottom)figure.3.7. This is a portion of the geometry

ds2 = (ρ(V )2 − r2)dV 2 + 2`drdV + r2dφ2. (3.37)


The V coordinate is −∞ on the horizon, and it increases in the inward null direction (i.e. up and

to the right). The function ρ(V ) is determined by matching onto the metric in Eq. (3.22Many

shocksequation.3.2.22) across a null slice. In particular, we require that the metric should be C1

across the matching surface.15 Continuity of the S1 implies that r = g(τ) along the join. By taking

the derivative along the patching surface, we can relate the normalization between the inward-

pointing null vectors in the two coordinate systems. In this way, one finds that 2`g′(τ)dτ = (r2 −ρ2(V ))dV along the surface. The C1 property of the metric relates the normalization of the outward-

pointing null vectors, by matching the derivative of the size of the S1. Requiring the inner product of

these vectors to be continuous across the matching surface, we find g′(τ)2 = ρ2(V )−r2. Rearranging

these equations, we determine ρ(V ) as follows. First, find V (τ) along the matching surface via

V (τ) = −2`

∫ τ dτ

g′(τ). (3.38)

Next, invert this to find τ(V ), and fix ρ(V ) using

ρ(V )2 = g(τ(V ))2 + g′(τ(V ))2. (3.39)

For our specific g(τ), we were not able to compute ρ(V ) exactly.16 However, it is clear that these

conditions completely fix the geometry, up to the undetermined overall length of the central region

of the wormhole.

15Requiring continuity alone would allow a δ-function stress tensor traveling along the null surface.16A surprisingly good approximation to the metric is g(τ) ≈ R−Rτ2, from which one finds ρ(V ) ≈ R+Re2RV/`.

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