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Emergence of Spacetime: from Entanglement to Einstein

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Emergence of Spacetime: from Entanglement to Einstein Emergence of Spacetime: From Entanglement to Einstein Andrew Sveskoa aDepartment of Physics, Arizona State University, Tempe, Arizona 85287, USA E-mail: [email protected] Abstract: Here I develop the connection between thermodynamics, entanglement, and grav- ity. I begin by showing that the classical null energy condition (NEC) can arise as a con- sequence of the second law of thermodynamics applied to local holographic screens. This is accomplished by essentially reversing the steps of Hawking's area theorem, leading to the Ricci convergence condition as an input, from which an application of Einstein's equations yields the NEC. Using the same argument, I show logarithmic quantum corrections to the Bekenstein-Hawking entropy formula do not alter the form of the Ricci convergence condi- tion, but obscure its connection to the NEC. Then, by attributing thermodynamics to the stretched horizon of future lightcones { a timelike hypersurface generated by a collection of radially accelerating observers with constant and uniform proper acceleration { I derive Einstein's equations from the Clausius relation T ∆Srev = Q, where ∆Srev is the reversible en- tropy change. Based on this derivation I uncover a local first law of gravity, ∆E = T ∆S −W , connecting gravitational entropy S to matter energy E and work W . I then provide an entan- glement interpretation of stretched lightcone thermodynamics by extending the entanglement equilibrium proposal. Specifically I show that the condition of fixed volume can be under- stood as subtracting the irreversible contribution to the thermodynamic entropy. Using the AdS3=CFT2 correspondence, I then provide a microscopic explanation of the `thermodynamic volume' { the conjugate variable to the pressure in extended black hole thermodynamics { and reveal the super-entropicity of AdS3 black holes is due to the gravitational entropy over- counting the number of available dual CFT2 states. Finally, I conclude by providing a recent generlization of the extended first law of entanglement, and study its non-trivial 2 + 1- and arXiv:2006.13106v2 [hep-th] 8 Dec 2020 1 + 1-dimensional limits. This thesis is self-contained and pedagogical by including useful background content relevant to emergent gravity. Contents 1 OVERVIEW4 2 FOUR ROADS TO EMERGENT GRAVITY 10 2.1 Induced Gravity 10 2.2 Spacetime Thermodynamics 13 2.3 Entropic Gravity 16 2.4 Spacetime Entanglement 18 3 THERMODYNAMIC ORIGIN OF THE NULL ENERGY CONDITION 23 3.1 From the Second Law to the NEC 25 3.2 Time Derivatives of Entropy 25 3.3 Thermodynamics of Spacetime 27 3.4 Quantum Corrections to Entropy and the NEC 29 4 GRAVITY FROM EQUILIBRIUM THERMODYNAMICS 34 4.1 Einstein's Equations from the Stretched Future Lightcone 36 4.1.1 Geometry of Stretched Lightcones 36 4.2 Stretched Lightcone Thermodynamics 39 4.2.1 Generalized Equations of Gravity 43 4.3 A Local First Law of Gravity 47 4.4 Gravity from Causal Diamond Thermodynamics 52 4.4.1 Geometry of Causal Diamonds 53 4.4.2 Causal Diamond Thermodynamics 55 4.4.3 Gravity from Thermodynamics 59 5 GRAVITY FROM ENTANGLEMENT EQUILIBRIUM 63 5.1 Vacuum Entropy and Gravity 63 5.2 Entanglement of Stretched Lightcones 67 5.3 Gravity from Entanglement of Stretched Lightcones 70 6 MICROSCOPIC HERALDS OF THERMODYNAMIC VOLUME 77 6.1 CFT and Standard BTZ 80 6.2 Charged BTZ Black Holes 82 6.3 Super{Entropicity and Instability 83 6.4 Generalized Exotic BTZ Black Holes 84 { i { 7 THE EXTENDED FIRST LAW OF ENTANGLEMENT IN ARBITRARY DIMENSIONS 87 7.1 Killing Horizons in Pure AdS and Extended (Bulk) First Law 89 7.2 Mapping to Boundary CFT 93 7.3 Extended First Law of Entanglement 95 7.4 Shifting Conformal Frames 96 7.5 Killing Horizons in Pure Two-Dimensional AdS 100 7.5.1 Einstein-Dilaton Theories 102 7.5.2 Jackiw-Teitelboim gravity 104 7.6 Beyond Pure AdS in Three Dimensional Gravity 106 7.6.1 Extended Thermodynamics and Volume 108 8 FINAL REMARKS 116 A FUNDAMENTALS OF SPACETIME THERMODYNAMICS 118 A.1 Geodesic Congruences 118 A.2 Black Hole Thermodynamics 121 A.3 Extended Black Hole Thermodynamics 127 A.4 The Einstein Equation of State: A Review 132 A.5 Horizon Entropy, Noether Charge, and Beyond 136 B FUNDAMENTALS OF SPACETIME ENTANGLEMENT 147 B.1 Gibbs and the Thermofield Double 147 B.2 Rindler Entanglement 148 B.3 The CHM Map 154 B.4 The First Law of Entanglement and its Extension 158 C FAILURE OF KILLING'S IDENTITY 163 C.1 Stretched Lightcones 163 C.2 Causal Diamonds 176 D CAUSAL DIAMONDS AND ENTANGLEMENT EQUILIBRIUM 184 D.1 First Law of Causal Diamond Mechanics 184 D.2 Entanglement Equilibrium 186 E IYER-WALD FORMALISM FOR STRETCHED LIGHTCONES 189 E.1 Iyer-Wald Formalism 189 E.2 Generalized Area of Stretched Lightcones 191 F D ! 2 LIMIT OF THE EXTENDED BULK FIRST LAW 193 { 1 { G EXTENDED FIRST LAW OF ENTANGLEMENT AND JT GRAVITY 195 G.1 JT Gravity and Wald Entropy 195 G.2 Extended First Law of Entanglement 196 { 2 { Acknowledgements and Dedication Physics is a practice that cannot and should not be done by oneself. I have had the good fortune of learning from and working with some inspiring individuals who have my gratitude. I would like to therefore briefly acknowledge all of those who have aided me along the way in reaching this goal, starting with my thesis committee. I would like to thank Professor Cindy Keeler for taking me on for multiple projects not even discussed in this dissertation. You have pushed me and given me a unique perspective of the field. Professor Damien Easson, who was my first research rotation advisor, thank you for giving me a first glimpse at what research is like and giving me the freedom to pursue my own project. Professor Tanmay Vachaspati, thank you for asking me honest questions forcing me to think about the `big picture'. And lastly my thesis advisor, Professor Maulik Parikh, who I was told normally does not take on first year students. You were one of the big reasons why I decided to come to ASU. Not only have you provided great insight into science, you taught me how to be self-reliant and given me tremendous freedom. I couldn't have asked for a better advisor. There are other professors of physics, both at ASU and other institutions, who deserve my thanks. In particular Professor Ted Jacobson at University of Maryland, who has greatly influenced the ideas that are in this thesis and given me encouragement, and Professor Clifford Johnson of University of Southern California, who has influenced my mode of thinking and lent me support and hospitality { each of you inspire me to be a better physicist. I must also thank Dr. Darya Dolenko of ASU for showing me how to be a better instructor and granting me the opportunity to try different styles of teaching. Individuals who are not (yet) professors of physics have my gratitude as well. Felipe Rosso and Batoul Banihashemi, I thank them for their insight, kindness, and hospitality. I must also express my thanks to the entire cosmology graduate student office. Each of you have made my time at ASU a joy. Even though I might get more work done at home, I look forward to coming into the office everyday for the lively conversation. I must also thank the group of postdoctoral researchers in the cosmology office, two of whom I should single out: George Zahariade and Victoria Martin. Both of you have inspired me to be a better physicist, but, most of all, I am fortunate to say that you are both colleagues and dear friends. Lastly, there are numerous people outside of the world of physics who have my deepest appreciation. These include the friends/family I didn't expect to find in Arizona, as well as my friends and family back home in Oregon. Listing you all would take up too much time, but I must highlight a few individuals, starting with my older sister Kacie Svesko. She showed me what it takes to be a good student, and has harbored in me a healthy spirit of competitiveness. I also thank my parents, Mike and Lorie Svesko, who have encouraged me every step of the way, in more ways than a few, including buying my very first textbook on physics. And, of course, my wonderful wife Abby Mccoy, who has shown me love and tremendous patience from the time we were in high school through now as a PhD graduate. { 3 { 1 OVERVIEW The discovery that black holes carry entropy [1,2], A S = H ; (1.1) BH 4G provides the two following realizations: (i) A world with gravity is holographic [3], and (ii) spacetime is emergent [4]. The former of these comes from the observation that the thermodynamic entropy of a black hole (1.1) goes as the area of its horizon AH, and the latter from noting that black holes are spacetime solutions to Einstein's equations. In fact, black holes are not the only spacetime solutions which carry entropy; any solution which has a horizon, e.g., Rindler space and the de Sitter universe, also possess a thermodynamic entropy proportional to the area of their respective horizons. The fact that Rindler space carries an entropy is particularly striking as there the notion of horizon is observer dependent. This leads to the proposal that an arbitrary spacetime { which may appear locally as Rindler space { is equipped with an entropy proportional to the area of a local Rindler horizon, and that thermodynamic relationships, e.g., the Clausius relation T ∆S = Q, have geometric meaning.
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