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 jΨ 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.