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

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

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

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

Leonard Susskind, Primary Adviser

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

Patrick Hayden

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

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 in University 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 0 the perturbed state |Ψ i, 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 andv ˜ = 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 1 3.10 The size of the S 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 28 mechanically, we’d have to evolve forward a system of 1010 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

s β d t (x, y) = log N 2 + |x − y|. (1.1) ∗ 2π 2(d − 1)

In the first term, β is the inverse temperature of the quantum field theory, and N 2 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 β 2 as follows. The small perturbation has very little effect on any other system for a time 2π log N . (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 pd/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 α0 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 |ψi 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 [|ψihψ|], 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 |ψi 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 |ψi 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 |Ψi in the total Hilbert space:

1 X |Ψi = e−βEn/2 |ni |ni . (2.1) Z1/2 L R n Tracing over the R Hilbert space leaves a precisely thermal density matrix for the L system:

1 X ρ = e−βEn |nihn|. (2.2) L Z n

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 |Ψi describes a large eternal AdS Schwarzschild black hole with Hawking temperature TH = 1/β. In this context |Ψi 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 |ψi 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 |ψi. 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 log S 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 CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 6

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 |Ψi 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, |Ψ0i, 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 , or in the large N field theory context the coefficient of N 2, GN vanishes. There will continue to be a nonzero value of subleading strength. 3 In 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 and so runs forwards on the right CFT but backwards on the left. 4 The entanglement between L and R as a whole will remain unchanged. The entropy SL is invariant under any unitary operator, no matter how chaotic, applied to HL. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 7

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

E 2πtw/β horizon at t = 0 is proportional to M e , 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 β M when tw becomes of order t∗ ∼ 2π log E . Assuming E takes the smallest reasonable value, the energy in one quantum at the Hawking temperature E ∼ TH , the time t∗ is

β t ∼ log S (2.3) ∗ 2π 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

5 In particular [32] discussed the effect of a perturbation at large tw as a firewall candidate. [35] discussed, in the one sided black hole context, highly boosted horizon hugging branes. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 8

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: 10 X n (i) (i+1) (i) (i)o HL = σz σz − 1.05 σx + 0.5 σz . (2.4) i=1 The coefficients -1.05 and 0.5 are chosen, following [37], to ensure that the Hamiltonian is far from integrability.

2.5

2.0

1.5

1.0

0.5

0.0 0 2 4 6 8 10 tw

Figure 2.1: Mutual information (upper, blue) and spin-spin correlation function (lower, red) in the 0 perturbed state |Ψ i, 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.

Our procedure is to prepare the thermofield double state |Ψi, 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 (5,L) perturbation σz to the fifth qubit of the L system at a time tw in the past. In other words, we consider the perturbed state

0 −iHLtw (5,L) iHLtw |Ψ i = e σz e |Ψi. (2.5)

Notice that the applied operator acts trivially on the R system. In the state |Ψ0i, 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, CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 9

0 red) in the perturbed state |Ψ i, 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 |Ψi, 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 0 (1,L) (1,R) 0 function hΨ |σz σz |Ψ i, 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 |Ψ0i, 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

r2 − R2 `2 ds2 = − dt2 + dr2 + r2dφ2 (2.6) `2 r2 − R2 2π`2 φ ∼ φ + 2π R2 = 8G M`2 β = , (2.7) N R CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 10

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

t t

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

−4`2dudv + R2(1 − uv)2dφ2 ds2 = . (2.8) (1 + uv)2

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:

d cosh = T T 0 + T T 0 − X X0 − X X0 , (2.9) ` 1 1 2 2 1 1 2 2 where we’ve used the embedding coordinates

v + u 1 p Rt T = = r2 − R2 sinh 1 1 + uv R `2 1 − uv Rφ r Rφ T = cosh = cosh (2.10) 2 1 + uv ` R ` v − u 1 p Rt X = = r2 − R2 cosh 1 1 + uv R `2 1 − uv Rφ r Rφ X = sinh = sinh . 2 1 + uv ` R `

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. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 11

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

E` 2 E ∼ eRtw/` (2.11) p R 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 2 −Rtw/` of mass M to a solution of mass M + E across the null surface uw = e . 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 andu, ˜ v˜ to the left (future), so that the metric is always of the Kruskal form (3.7One shockequation.3.2.7). Because of the q ˜ M+E increase in mass, the radius is R to the right and R = M R to the left. We will fix a relative boost ambiguity in the relation between u, v andu, ˜ 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 2 −Rt˜ w/` coordinates asu ˜w = e . 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 1 − u˜ v˜ 1 − u v R˜ w = R w . (2.12) 1 +u ˜wv˜ 1 + uwv For small E/M, the solution is a simple shift

E 2 v˜ = v + α , α ≡ eRtw/` . (2.13) 4M

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 boundary and backwards on the left (see Fig. 2.2The Kruskal diagram (center) and Penrose diagram (right) for the BTZ geometryfigure.2.2). In particular, a perturbation released at time tw from the left boundary is in the past of the t = 0 slice if tw > 0. 7This metric corresponds to a boundary source adjusted to make all particles fall through the horizon at the same time. We comment further on this choice in the discussion. The use of a classical metric is simplest to justify if we consider a perturbation that corresponds to a large but fixed number of quanta in the small GN limit. However, we believe that our conclusions are also accurate for small numbers of quanta. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 12

which is relevant for the small but early perturbations we wish to consider, R˜ = R, and the metric can be written −4`2dudv + R2 [1 − u(v + αθ(u))]2 dφ2 ds2 = . (2.14) [1 + u(v + αθ(u))]2 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 andv ˜ = 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

−4`2dUdV + 4`2αδ(U)dU 2 + R2(1 − UV )2dφ2 ds2 = . (2.15) (1 + UV )2

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 andv ˜ = 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 α Tuu = δ(u), (2.16) 4πGN 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 CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 13

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:

d1 r 1 p 2 cosh = + r2 − R2 e−RtL/` (v + α) (2.17) ` R R d2 r 1 p 2 cosh = − r2 − R2 e−RtR/` v. (2.18) ` R R

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

d 2r h R α 2 i = 2 log + 2 log cosh (t − t ) + e−R(tL+tR)/2` . (2.19) ` R 2`2 R L 2

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, 2 if we fix tL = tR = 0 and take Rtw  ` , 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 2r Rφ = 2 log + 2 log sinh . (2.20) ` R 2`

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 CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 14

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

`  2r Rφ SA = SB = 2 log + 2 log sinh . (2.21) 4GN R 2`

Next, consider SA∪B. When φ < π, we have two possible choices of extremal surface. First, we (1) have the union of the two geodesics used to compute SA and SB. This gives SA∪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

  (2) ` 2r  α SA∪B = log + log 1 + . (2.22) GN R 2

Rφ (1) (2) For small regions with sinh 2` < 1, we have SA∪B < SA∪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

`  πφ`  Eβ  I(A; B) = log sinh − log 1 + e2πtw/β . (2.23) GN β 4S

This mutual information is a decreasing function of tw. For high temperature, I reaches zero when tw is equal to φ` β 2S t (φ) = + log . (2.24) ∗ 2 2π βE

8More precisely, this expression gives the contribution proportional to N 2 in the entropy. There may be numerically large but subleading terms, as well as finite λ corrections. The RT prescription also requires that the bulk surface must be homologous to Ω. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 15

When the string coupling gs (which is ∼ 1/N in terms of the large N gauge theory) is small, so S ∼ N 2 is large, and E assumes its smallest reasonable value E ∼ T = 1/β then

β t = log S. (2.25) ∗ 2π 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: hΨ|W †ϕ ϕ W |Ψi hϕ ϕ i ≡ L R , (2.26) L R W hΨ|W †W |Ψi 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

hϕ(x)ϕ(y)i ∼ 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 extremely difficult to interpret solely in terms of supergravity degrees of freedom outside the horizon. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 16

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

!2m` 1 hϕLϕRiW ∼ 2 . (2.28) E Rtw/` 1 + 8M e

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 ∼ 2 1 Rtw/` 12 ` e . When tw is large enough this energy can exceed string or even Planck scales . 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, possibly multiplied by curvatures. These would be multiplied by the appropriate powers of the string mass ms and the Planck 2 mass mp. In 3 dimensions these are related by ms ∼ gs mp where gs is the string coupling. Here GN ∼ 1/mp. The results of the previous section show that the high energy in the shock wave causes large derivatives with respect to proper time along the geodesic world line. These would seem to cause the 1/ms suppressed terms to become order one at times tw far smaller than t∗. Because there are no powers of gs involved, the time when this would happen CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 17

interested in the time tw when this quantity starts to differ substantially from the two point function hΨ|ϕLϕR|Ψi. The expectation value (2.26Correlation functionsequation.2.3.26) computes spacelike correlations in the state W |Ψi, 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 2 2 small when the dimensionless quantity  = gs sls is small. Putting in s = EpT , appropriate for a thermal quantum sourced by ϕR, one finds that  becomes of order one at a time t somewhat prior 14 to t∗, by an additive S-independent amount proportional to β log `/ls. So we find that flat-space stringy effects would not change the log S 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

p 2 bI = ls log sls (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 2 the phase shift is suppressed relative to the real part by (ls/b) . We now turn to scattering15 in the black hole, whose characteristic lengths are the horizon size would be much sooner, of order log ms/T rather than log mp/T ∼ log R/GN ∼ log S. The following remarks about flat-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 CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 18

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 p 2 covers the horizon bI /R times which is roughly log(/gs ). Interaction effects should be at most p 2 16  log (/gs ), which give a log log correction to t∗, which we ignore. 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 log S.

For tw > t, several effects come into play. First, there is the increasing effect of the Einstein β gravity scattering, which becomes of order one at the time t∗ = 2π log S. 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 2 2 ls/` , 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 D−3 √ rough estimate suggests this occurs when the Schwarzschild radius RS, RS ∼ 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. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 19

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 hϕLϕRiW 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 β dependence of t∗ is universal, t∗ = 2π log S. 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

18 The perturbation at tw can also affect conserved quantities, such as the energy. This will give rise to small but non-decaying terms in the correlation function. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 20

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 hbbi can be viewed as arising from the CFT correlation hΨ|ϕLϕR|Ψi characteristic of the thermofield double state |Ψi. One might worry that the smallness † ˜ of hΨ|W ϕLϕRW |Ψi implies de-correlation of b and b and a firewall in the state W |Ψi. 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- 20 planation. 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 |Ψi can be made [12]. If we assume (i) 21 ˜ that the perturbation at sufficiently early tw acts like a random unitary, (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 iθN right horizons. To make this argument, one considers the operator e b , where Nb is the number operator. This rotates the phase of hb˜bi, but leaves the ensemble generated by random unitaries invariant, so we conclude that the ensemble average of hb˜bi 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. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 21

c

φL ~ b ~ φL b b φR φR b

Figure 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.

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 waves on both horizons. At least for spherical shock waves this is not the case. The future horizons in the resulting geometry are well beyond the location of the collision and do not coincide with the shock waves [34]. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 22

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

−itwH itwH In the main text of the paper, we’ve considered the effect of an operator OL(tw) = e OLe on the the entanglements between local subsystems A ⊂ L and B ⊂ R in a thermofield double state

|ΨiLR. 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 |L| 1 X |Ψ0i = U |mi |ni . (2.30) |L|1/2 nm L R n,m=1 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 = kρAB − ρA ⊗ ρBk1, where the 1-norm of a matrix M is defined as kMk1 = 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 d2 ≤ I(A, B) ≤ d log |A| − d log d . (2.31) 2 1 1 1 1

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 kMk ≤ √ 1 p † rank MkMk2, where the 2-norm is defined as kMk2 = 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 |Ψi is maximally mixed, ρA(U) = ρB(U) = 1/|A|, we compute

2 2 2 kρAB(U) − ρA(U) ⊗ ρB(U)k2 = tr[(ρAB(U) − 1/|A| ) ] (2.32) 2 1 = tr[ρ (U)2] − tr[ρ ] + tr[1] (2.33) AB |A|2 AB |A|4 1 = tr[ρ (U)2] − . (2.34) AB |A|2

23This approach has the benefit of emphasizing that Haar randomness is not essential to the calculation, and that a 2-design would lead to identical results. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 23

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 the m and n indices of Umn into m → (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

2 1 X ∗ ∗ tr[ρ (U) ] = U U 0 0 U 0 0 0 0 U 0 0 . (2.35) AB |L|2 iI jJ i I j J i I j J iI jJ iIjJi0I0j0J 0

We can now take the expectation value using Z ∗ ∗ 1   0 0 0 0 0 0 0 0 dU Ui1j1 Ui2j2 Ui0 j0 Ui0 j0 = δi1i δi2i δj1j δj2j + δi1i δi2i δj1j δj2j (2.36) 1 1 2 2 |L|2 − 1 1 2 1 2 2 1 2 1 1   − δi i0 δi i0 δj j0 δj j0 + δi i0 δi i0 δj j0 δj j0 . |L|(|L|2 − 1) 1 1 2 2 1 2 2 1 1 2 2 1 1 1 2 2

The terms on the bottom line are subleading and we will drop them, along with the “1” in 2 the first line. Summing as in Eq. (3.30Discussionequation.3.4.30), we find that tr[ρAB(U) ] = |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

Z |A| dUkρ (U) − ρ (U) ⊗ ρ (U)k ≤ . (2.37) AB A B 1 |Ac|

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 Z |A| dU I(A, B) ≤ log |Ac|. (2.38) |Ac| 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 the R 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. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 24

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

2 2 −1 2 2 2 ds = −f(r)dt + f (r)dr + r dΩD−2. (2.39)

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

4f(r) 0 ds2 = − e−f (R)r∗(r)dudv + r2dΩ2 (2.40) f 0(R)2 D−2 0 0 uv = −ef (R)r∗(r) u/v = −e−f (R)t, (2.41)

−1 with dr∗ = f dr 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 coordinatesu, ˜ v˜ to the left of the perturbation, and continue to use u, v to the right. The shell propagates on the surface

f˜0(R˜) f0(R) (˜r∗(∞)−tw) (r∗(∞)−tw) u˜w = e 2 uw = e 2 , (2.42) and the matching condition relatesv ˜ to v via

0 0 f˜ (R˜)˜r∗(r) f (R)r∗(r) u˜wv˜ = −e uwv = −e . (2.43)

We would like to use these equations to find the shiftv ˜ = v + α, to linear order in E and at large tw. Small E allows us to approximateu ˜w = uw. Large tw pushes us to a limit where r approaches 0 0 f (R)r∗(r) R, so we can expand f(r) = f (R)(r − R) + .... Evaluating r∗, we find e = (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

E d h i E dR α = (R − r)C(r, R) = C(R,R). (2.44) uw dM r=R uw dM

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

24 The main point, that the coefficient of the logarithm in t∗ is dimension-independent, should already be clear from the discussion in § 2.4String and Planck scale effectssection.2.4. CHAPTER 2. BLACK HOLES AND THE BUTTERFLY EFFECT 25

dR/dM. Also using f 0(R) = 4π/β, we find that α becomes equal to one at time25

 D−3  β (D − 2)ΩD−2 T R t∗ = r∗(∞) + log . (2.45) 2π 4C(R,R) E GN

−2 β 2 Fixing E,R,T and taking GN ∝ N to zero, we have t∗ ∼ 2π log N 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 ansatzv ˜ = v + h(Ω). Evaluating the Ricci tensor of the patched metric (see, e.g. appendix A of [39]) and plugging into the Einstein D−2 equations with Tuu ∝ δ(u)δ (Ω), we find that h must satisfy an equation

  2 D − 2 0 D−2 ∇ D−2 − Rf (R) h(Ω) ∝ δ (Ω). (2.46) S 2

For a large AdS black hole with R  `, we have f 0(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 q − (D−1)(D−2) Rθ/` pole as h ∝ e 2 . Comparing this to the rate at which the perturbation grows as we 0 f (R)tw/2 push tw earlier, e , we find that as we increase tw, the level sets of h expand outward with a “speed of propagation” s D − 1 v = , (2.47) D 2(D − 2) where D is the spacetime dimension of the AdS space.

25 β Note that the combination r∗(∞) − 2π log C(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

1 X |TFDi = e−βEn/2|ni |ni . (3.1) Z1/2 L R n

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

β t = log S (3.2) ∗ 2π 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 hϕL(t)ϕR(t)i, 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 hϕL(t)ϕR(t)i 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]. CHAPTER 3. MULTIPLE SHOCKS 28

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 |TFDi 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)|TFDi. (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 a W 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

2 Here, we mean time evolution with HL + HR. CHAPTER 3. MULTIPLE SHOCKS 29

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)|TFDi, (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, 2 while M ∼ 1/GN , which is proportional to N in the large-N gauge theory. CHAPTER 3. MULTIPLE SHOCKS 30

M+E

M+E t 1 M M α

M+E t1

Figure 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.

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

(t=0) 2πt1/β Ep ∼ Ee , (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

E α = e2πt1/β ∼ e2π(t1−t∗)/β. (3.6) 4M

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

−4`2dudv + R2(1 − uv)2dφ2 ds2 = . (3.7) (1 + uv)2

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 will consider in this paper, it will be permissible to treat all matter in this way. CHAPTER 3. MULTIPLE SHOCKS 31

3.2.2 Two shocks

Next, we consider a state of the form

W (t2)W (t1)|TFDi. (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)|Φi, (3.9) assuming that we already know the geometry for |Φi. In general, the prescription is as follows: we start with the geometry for |Φi 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)|TFDi, 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.

t1 Mt

M M+2E M M+2E M+E M M+E M+E t2

Figure 3.2: The 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 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 CHAPTER 3. MULTIPLE SHOCKS 32

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 2 2 −1 2 2 2 2 2 ds = −fdt +f dr +r dΩ . Explicitly, for the 2+1 dimensional BTZ case, f(r) = r −8GN M` , where M is the mass of the black hole and ` is the AdS length. The DTR condition becomes

 2 2 2 2  2 2 2 2 r − 8GN Mt` r − 8GN (M + E)` = r − 8GN (M + 2E)` r − 8GN M` . (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 1 − uv = , (3.12) R 1 + uv

2 2 where the radius of the horizon, R, is determined by R = 8GN M` , 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:

2 2 u = e−Rt/` , v = −eRt/` . (3.13)

In particular, in the Kruskal system of the bottom quadrant, the v coordinate of the left-moving 2 2 shock is −eRbt1/` , while the u coordinate of the right-moving shock is e−Rbt2/` .6 This determines the r value of their collision as 2 r 1 + eRb(t1−t2)/` = . (3.14) R (t −t )/`2 Rb 1 − e b 1 2

6 2 2 Rb is the BTZ radius in the lower quadrant, defined by Rb = 8GN (M + E)` . CHAPTER 3. MULTIPLE SHOCKS 33

Plugging this value of r into Eq. (3.11Two shocksequation.3.2.11), we find

E2 R (t − t ) M = M + E + sinh2 b 2 1 . (3.15) t M + E 2`2

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.

t1 t1

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 shifts later, 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 2 quadrant. If v is negative, then the shell hits the boundary at time eRt/` = −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 2 2 the u coordinate in the left Kruskal system e−Rlt2/` ≈ e−Rt2/` . We find

2 E 2 v ≈ −eRt1/` + eRt2/` . (3.16) 4M

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 CHAPTER 3. MULTIPLE SHOCKS 34

in terms of scrambling. Indeed, the state W (t1)|TFDi 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

† hTFD|[W1(t1),W2(t2)] [W1(t1),W2(t2)]|TFDi. (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)|TFDi and W2(t2)W1(t1)|TFDi. 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)|TFDi. (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 E α = e2πtw/β (3.19) 4M 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, CHAPTER 3. MULTIPLE SHOCKS 35

Figure 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.

7 the leftmost one touches the boundary at time ±tw, 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

7 Here, 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 be broken by the fact that different W operators source shocks of varying strength. CHAPTER 3. MULTIPLE SHOCKS 36

α α/2

Figure 3.5: A geodesic passes across a portion of the wormhole. It intersects the null boundaries of the central regions halfway across their width.

` cosh−1(1 + α2/2). Thus the regularized length across the entire wormhole is

L  α2  = n cosh−1 1 + + O(n0). (3.20) ` 2

This function interpolates between nα for small α and 2n log α 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 α2M 2, 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 CHAPTER 3. MULTIPLE SHOCKS 37

α

Figure 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.

the geometryfigure.3.6 will be smoothed out,9 allowing an analysis in terms of an averaged stress 2 2 2 D−2 tensor. For most values of α  1, inelastic stringy effects, proportional to GN α M `s/` [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 = −`2dτ 2 + 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 = −`2dτ 2 + `2 cos2 τdx2 + g(τ)2dφ2 (3.22) g(τ) 1 + sin τ = 1 − sin τ log . R cos τ 9We are grateful to Raphael Bousso for making this suggestion. 10 2 2 2 D−2 We need α small enough that the probability of oscillator excitation per collision, GN α M `s/` , times the number of collisions, 1/α, is small. Roughly, we support the wormhole with a large number of relatively soft quanta, 2πtw/β 2 2 with boost factor e of order ` /`s. The mild boost means that doubling the mass of the left black hole only 3 2 leads to a wormhole of length ` /`s. 11In a realistic setting, the shocks won’t be exactly spherically symmetric. Suppose we build each shell as a sum of particles localized on the S1. After a collision, these can be deflected by an angle ∼ α [75]. Each experiences ∼ 1/α collisions before hitting the singularity, but if the initial inhomogeneity is small, deflections will tend to cancel, and the total effect will remain small. CHAPTER 3. MULTIPLE SHOCKS 38

In order to check that this metric actually corresponds to the small α limit of the dense network

Vaidya Vaidya

BTZ ds2=-ℓ2dτ2+h(τ)2dx2+g(τ)2dϕ2 BTZ

Vaidya Vaidya

Figure 3.7: The wormhole created from a large number of weak shocks (top) becomes a smooth geometry 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 CHAPTER 3. MULTIPLE SHOCKS 39

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. P Quantum mechanics is different. If a state |ψi = s cs|Esi then

X −iEst |ψ(t)i = cse |Esi. (3.23) s

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 D X 2 2 P (|ψi) ∼ exp(− |cs| /2f ) (3.24) s=1 where f is chosen so that the state normalization condition hψ|ψi = 1 is satisfied (up to small fluctuations), 2f 2 = 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

D X 2 2 P (|ψi) ∼ exp(− |cs| /2f (Es)) (3.25) s=1 where f is smooth over the spread in energies of the system being sampled, and satisfies the normal- P 2 ization condition s 2f(Es) = 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 way as states obtained by acting a random unitary on a reference basis. CHAPTER 3. MULTIPLE SHOCKS 40

order one of nearly all states in the ensemble. To see this, we compute Z d|ψid|χiP (|ψi)P (|χi) max |hχ|e−iHt|ψi| (3.26) t

1 Z  2 0 2 2  Y 2 2 0 −(|cs| +|cs| )/2f (Es) X ∗ 0 −iEr t = d csd cse max crcre (3.27) N 2 t s r  Z 2 X 1 2 2 ≈ d2c e−|cs| /2f (Es)|c | (3.28) 2πf 2(E ) s s s s X π π = f 2(E ) = . (3.29) 2 s 4 s

In the second equality, we have used the assumption that all energy levels are incommensurate,

∗ 0 −iEst 0 so we can find a time t such that cscse = |cs||cs| 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 |TFDi 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 |TFDi 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?) S upper bound is provided by the quantum recurrence time, schematically ∼ ee . 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 |ψi is described by a smooth geometry with a long wormhole. Then U(−t)|ψi 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 evolution intervals. CHAPTER 3. MULTIPLE SHOCKS 41

a long wormhole. Roughly, the two geometries are related by a boost. This is dangerous: imagine that part of the matter supporting the |ψi wormhole is a light ray behind the horizon. If Bob starts

I

IV II Bob light ray III

Figure 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.

falling into the |ψi black hole at time tB = 0, he might experience a mild collision. But consider the geometry associated with U(−t)|ψi. If Bob falls into this geometry at time tB = 0 his experience will be the same as falling into |ψi 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

I

??? IV II light ray III Bob

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 Juan Maldacena and Edward Witten. CHAPTER 3. MULTIPLE SHOCKS 42

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)|TFDi, (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 CHAPTER 3. MULTIPLE SHOCKS 43

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 1 geometries between collisions, {Rn} (see Fig. 3.10The size of the S 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 1 of the S (determined by rn) depends on τ as g(τ).

R R4 R4 4 r4 r4 r4 r4

R3 R3 R3 R3 r3 r3 r3

R2 R2 R2 r2 r2 r2 r2

R1 R1 R1 R1 r1 r1 r1

1 Figure 3.10: The size of the S at the vertices is labeled rn, and the R parameter of the BTZ geometry 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 2 2 f(r) = r − Rn. This gives 2 2 2 2 2 (Rn − rn) Rn+1 = rn + 2 2 . (3.31) Rn−1 − rn 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 CHAPTER 3. MULTIPLE SHOCKS 44

shocksequation.3.2.12) we must have 2 rn−1 1 − ub = 2 . (3.32) Rn 1 + ub

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

2 rn+1 1 − (ub + ∆) = 2 . (3.33) Rn 1 + (ub + ∆)

Eliminating ub and ∆, we find the recursion relation

2 2 2 2rnRn − rn−1Rn − rn−1rn rn+1 = 2 2 . (3.34) Rn + rn − 2rnrn−1

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:

1 − α2 p r = ,R = 1 + 4α2. (3.35) 2 1 + α2 2

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

s s R + r R − r R − r 2 tan−1 n n−1 n − 2 tan−1 n n−1 . (3.36) Rn − rn−1 R + rn Rn + rn−1

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) CHAPTER 3. MULTIPLE SHOCKS 45

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`g0(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 g0(τ)2 = ρ2(V )−r2. Rearranging these equations, we determine ρ(V ) as follows. First, find V (τ) along the matching surface via

Z τ dτ V (τ) = −2` . (3.38) g0(τ)

Next, invert this to find τ(V ), and fix ρ(V ) using

ρ(V )2 = g(τ(V ))2 + g0(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.

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