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Black Holes and Holography: Insights and Applications University of California Santa Barbara Black Holes and Holography: Insights and Applications A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Jason Wien Committee in charge: Professor Donald Marolf, Chair Professor Gary Horowitz Professor David Stuart December 2017 The Dissertation of Jason Wien is approved. Professor Gary Horowitz Professor David Stuart Professor Donald Marolf, Committee Chair November 2017 Black Holes and Holography: Insights and Applications Copyright c 2017 by Jason Wien iii Acknowledgements I am fortunate to have been supported by a network of friends, family, peers, collab- orators, and mentors throughout my time in graduate school. This dissertation could not have been completed without the support of this strong community, for which I am incredibly thankful. First, I would like to thank my advisor, Don Marolf, for his tireless mentorship and guidance. He taught me the art of careful research guided by experience and intuition, and he patiently pointed me in the right direction any of the numerous times I found myself off course. The skills he helped me develop will help me through both my career and life in general. Thank you to the numerous professors who have be invaluable in my education. Thank you to Gary Horowitz, David Stuart, David Berenstein, Dave Morrison, Steve Giddings, Joe Polchinski, Nathaniel Craig, Mark Srednicki, and Andreas Ludwig for imparting physics wisdom both through numerous class lectures as well as informal dis- cussions. I had the pleasure of collaborating with Max Rota, Will Donnelly and Ben Michel for parts of this dissertation. Thank you for helping to broaden my research perspective. Additionally, thank you to Will Kelly, Sebastian Fischetti, Netta Englehardt, Gavin Hartnett, Eric Mefford, Henry Maxfield, Alex Miller, Jason Weinberg, Benson Way, and anyone else whose office I stumbled in to, deep in a calculation and looking for help. Thank you to my community of peers who helped me through late night problem sets and provided friendship and support during my time here. Thank you to Mickey, Ryan, Chai, Alex, Milind, Matt, Sky, Megan, Giulia, Brianna, Zicao, Seth, Alex, Gabriel, and Timothy. Thank you especially to Alex Miller for keeping the high energy theory journal club running strong. iv Thank you to my parents, Mike and Nannette Wien, for cultivating my curiosity from a young age, and to my brother Andrew for trying to distract me from doing physics all the time. Thank you to my brother Brian for teaching me the beauty of every moment. Thank you to my great aunt and uncle, Anita and Byron Wien, for being long supporters of and investors in my education and intellectual growth. I would especially like to thank my grandfather, Robert Earle Wien, for instilling in me a drive to wonder at the world. Thank you for that visit to Fermilab all those years ago. Finally, I would like to thank my life partner Elyse. You have been a source of emotional and intellectual support throughout this journey, keeping my brain sharp and my heart open. Truly we connect on every dimension, and no one understands me just as well as you do. I look forward to the rest of our lives together. This work was supported in part by the U.S. National Science Foundation under grant numbers PHY12-05500 and PHY15-04541, by the Simons foundation, and by funds from the University of California. v Curriculum Vitæ Jason Wien Education 2017 Ph.D. in Physics (Expected), University of California, Santa Bar- bara. 2016 M.A. in Physics, University of California, Santa Barbara. 2014 M.Sc. in Physics, University of Waterloo 2013 B.A. in Physics and Mathematics, Harvard Univeristy Publications 1. J. Wien, Numerical Methods for Handlebody Phases, [arXiv:1711.02711]. 2. D. Marolf and J. Wien, The Torus Operator in Holography, [arXiv:1708.03048]. 3. D. Marolf, M. Rota, and J. Wien, Handlebody phases and the polyhedrality of the holographic entropy cone, JHEP 1710 (2017) 069, [arXiv:1705.10736]. 4. W. Donnelly, D. Marolf, B. Michel, and J. Wien, Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers, JHEP 1704 (2017) 093, [arXiv:1611.05841]. 5. D. Marolf and J. Wien, Adiabatic corrections to holographic entanglement in ther- mofield doubles and confining ground states, JHEP 1609 (2016) 058, [arXiv:1605.02803]. 6. D. Marolf and J. Wien, Holographic confinement in inhomogeneous backgrounds , JHEP 1608 (2016) 015, [arXiv:1605.02804]. 7. J. Wien, A Holographic Approach to Spacetime Entanglement (Master's Thesis), [arXiv:1408.6005]. 8. M. Headrick, R. Myers, and J. Wien, Holographic Holes and Differential Entropy, JHEP 1410 (2014) 149, [arXiv:1408.4770]. vi Abstract Black Holes and Holography: Insights and Applications by Jason Wien This dissertation focuses on the role classical black hole spacetimes play in the AdS/CFT correspondence. We begin by introducing some of the puzzles surrounding black holes, and we review their connection to strongly correlated CFT states through holography. Additionally, we detail numerical methods for constructing black hole states of non-trivial topology in three dimensions and evaluating their actions. In part I we focus on using black hole spacetimes to derive insights into holography and quantum gravity. Using numerical methods, we study a class of non-local operators in the CFT, defined via a path integral over a torus with two punctures. In particular, we are interested in determining the spectrum of such operators at various points in moduli space. In the dual gravitational theory, such an operator might be used to construct black hole spacetimes with arbitrarily high topology behind the horizon. We present evidence suggesting this fails, and along the way encounter a puzzle related to the positivity of these operators. The resolution of this puzzle lies in developing technology to better catalogue the relevant gravitational phases. Additionally, we use multi-boundary wormhole spacetimes to investigate the con- straints on the subregion entanglement entropies of holographic states. We find tension with previously claimed properties of these constraints, namely that they define a poly- hedral cone in the space of entanglement entropies. These results either suggest the possible existence of further unknown constraints, or the need for a more complicated vii construction procedure to realize the extremal states. In part II we focus on the holographic description of CFT states via black hole spacetimes, focusing on spacetimes perturbatively constructed from the planar AdS- Schwarzschild metric. First, we consider corrections to properties of confining ground states of holographic CFTs as we introduce spatial curvature. Next, we compute shifts in vacuum entanglement entropy in a thermal state with a locally varying temperature as well as similar shifts in the confining ground states with spatial curvature. viii Contents Curriculum Vitae vi Abstract vii 1 Introduction 1 1.1 Preamble: From black holes to thermal states . .7 1.2 Permissions and Attributions . 13 2 Numerical Methods 15 2.1 Finite Element Methods . 16 2.2 Handlebody Phases . 23 2.3 A Mathematica package . 37 2.4 An example . 39 Part I Insights 44 3 The Torus Operator and Holography 45 3.1 Introduction . 45 3.2 Gaps, ground states, and traces . 48 3.3 Handlebody Phases . 53 3.4 Computing Tr(T n).............................. 63 3.5 Single Boundary States . 75 3.6 Discussion . 83 3.A Phase Space of T j0i ............................. 88 3.B Estimation of Numerical Error . 90 4 The Holographic Entropy Cone 99 4.1 Introduction . 99 4.2 Entropy cones . 102 4.3 Constructing holographic geometries for the extremal rays . 107 4.4 Discussion . 118 ix 4.A Bulk action computation . 120 Part II Applications 124 5 Confinement in Inhomogenous Backgrounds 125 5.1 Introduction . 125 5.2 Adiabatically Varying Confining Vacua . 127 5.3 Gauge Theory Implications . 137 5.A 2+1 Dimensional Bulk . 143 6 Adiabatic Corrections to Holographic Entanglement 145 6.1 Introduction . 145 6.2 Preliminaries . 147 6.3 Adiabatic Thermofield Doubles . 151 6.4 States of Confining Theories . 169 6.5 Discussion . 175 6.A Adiabatic Thermofield Doubles in 1+1 Dimensions . 177 6.B Estimation of Numerical Uncertainty . 181 7 Outlook 184 Bibliography 186 x Chapter 1 Introduction Black holes are one of the most enchanting aspects of the known universe, providing inspiration for countless science fiction stories while remaining a puzzle to physicists as to their exact nature. These mysterious objects follow directly from Einstein's theory of General Relativity [1], and in fact they are as essential to the theory as a mass on a spring is to Newtonian mechanics. General Relativity describes how massive objects warp and stretch the fabric of space and time itself, and how objects travel through such a warped universe. These two notions are linked through Einstein's equations, which relate the curvature of a spacetime encoded in the tensor Gµν with the matter and energy content encoded in the stress tensor Tµν: 8πG G = N T : (1.1) µν c4 µν These equations were elegantly summarized by John Wheeler as, \Matter tells space how to curve; space tells matter how to move." A black hole is a region of space that is warped so strongly that nothing can escape it, not even light. Anyone or anything that enters a black hole is ultimately doomed to 1 Introduction Chapter 1 be torn apart by the extreme gravitational forces within. These forces become infinitely large inside of the black hole, where the solution has a singularity, and so some theory other than General Relativity must become relevant in the interior. The prototypical black hole solution of Einstein's equations, the Schwarzschild solution [2], was derived just one year after Einstein developed his theory, although physicists did not understand it to be a black hole until decades later.
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