UvA-DARE (Digital Academic Repository) Emergent gravity in a holographic universe Visser, M.R. Link to publication Citation for published version (APA): Visser, M. R. (2019). Emergent gravity in a holographic universe. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl) Download date: 23 Jul 2019 Emergent Gravity in a Holographic Universe This research was performed at the Institute for Theoretical Physics Amsterdam (ITFA) of the University of Amsterdam (UvA) and was supported by the European Research Council (ERC) and by the Spinoza grant, which is funded by the Netherlands Organisa- tion for Scientific Research (NWO). ISBN 978-94-6323-653-9 c Manus Visser, 2019 Cover design: Charlotte Kalshoven All rights reserved. Without limiting the rights under copyright reserved above, no part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the written permission of both the copyright owner and the author of the book. Emergent Gravity in a Holographic Universe Academisch Proefschrift Ter verkrijging van de graad van doctor aan de Univer- siteit van Amsterdam op gezag van de rector magnificus prof. dr. ir. K. I. J. Maex ten overstaan van een door het College voor Promoties ingestelde commissie, in het open- baar te verdedigen in de Aula der Universiteit op vrijdag 14 juni 2019 , te 11.00 uur door Manus Renze Visser geboren te Edam-Volendam Promotiecommissie Promotor prof. dr. E. P. Verlinde Universiteit van Amsterdam Co-Promotor dr. D. M. Hofman Universiteit van Amsterdam Overige leden prof. dr. J. de Boer Universiteit van Amsterdam dr. A. Castro Anich Universiteit van Amsterdam prof. dr. J. A. E. F. van Dongen Universiteit van Amsterdam prof. dr. T. A. Jacobson University of Maryland dr. J. P. van der Schaar Universiteit van Amsterdam Faculteit der Natuurwetenschappen, Wiskunde en Informatica Publications This thesis is based on the following publications: (Note that the authors are sorted alphabetically in theoretical high-energy physics.) [1] Ted Jacobson and Manus Visser, “Gravitational Thermodynamics of Causal Diamonds in (A)dS,” Submitted to SciPost Physics, arXiv:1812.01596 [hep-th]. Presented in Chapter 2. MV derived the Smarr formula and first law of causal diamonds in (A)dS space. Further, he established the connection between entan- glement equilibrium and stationarity of free energy, computed the York time for diamonds in (A)dS and worked out various limiting cases of the first law. He wrote the majority of Sections 2.2, 2.3, 2.4.1, 2.4.4, 2.6 and Appendices A.4–A.6. [2] Pablo Bueno, Vincent Min, Antony Speranza and Manus Visser, “Entanglement equilibrium for higher order gravity,” Physical Review D 95 (2017) 4, 046003, arXiv:1612.04374 [hep-th]. Presented in Chapter 3. MV extended the first law of causal diamonds to higher curvature gravity. He mainly contributed to Sections 3.2, 3.3.2 and Appendix A.8. [3] Sam van Leuven, Erik Verlinde and Manus Visser, “Towards non-AdS Holography via the Long String Phenomenon,” Journal of High Energy Physics 1806 (2018) 097, arXiv:1801.02589 [hep-th]. Presented in Chapter 4. MV worked out the definitions of the three holographic quantities and connected them to standard quantities in 2d CFT. In particular, he wrote large parts of Sections 4.2, 4.3.3, 4.5.2 and Appendix A.10. v Other publications by the author: [4] Pablo Bueno, Pablo Cano, Vincent Min and Manus Visser, “Aspects of general higher-order gravities,” Physical Review D 95 (2017) no. 4, 044010, arXiv:1612.03034 [hep-th]. [5] Margot Brouwer, Manus Visser et al., “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing measurements,” Monthly Notices of the Royal Astronomical Society 466 (2017) no. 3, arXiv:1612.03034 [astro-ph.CO]. [6] Niels Linnemann and Manus Visser, “Hints towards the Emergent Nature of Gravity,” Studies in History and Philosophy of Modern Physics B 64 (2018), arXiv:1711.10503 [physics.hist-ph]. [7] Juan Pedraza, Watse Sybesma and Manus Visser, “Hyperscaling violating black holes with spherical and hyperbolic horizons,” Classical and Quantum Gravity 36 (2019) no. 5, arXiv:1807.09770 [hep-th]. [8] Sebastian de Haro, Jeroen van Dongen, Manus Visser and Jeremy Butterfield, “Conceptual Analysis of Black Hole Entropy in String Theory,” Submitted to Studies in History and Philosophy of Modern Physics, arXiv:1904.03232 [physics.hist-ph]. [9] Jeroen van Dongen, Sebastian de Haro, Manus Visser and Jeremy Butterfield, “Emergence and Correspondence for String Theory Black Holes,” Submitted to Studies in History and Philosophy of Modern Physics, arXiv:1904.03234 [physics.hist-ph]. [10] Ted Jacobson and Manus Visser, “Spacetime Equilibrium at Negative Temperature and the Attraction of Gravity,” Honorable mention in Gravity Research Foundation Essay Competition 2019, arXiv:1904.04843 [gr-qc]. vi Contents Preface and Outline xiii 1 Introduction 1 1.1 Emergent gravity and holography . .2 1.1.1 Definition of emergent gravity . .2 1.1.2 Black hole thermodynamics . .8 1.1.3 The holographic principle . 10 1.2 Thermodynamics of de Sitter space . 13 1.2.1 Classical geometry of de Sitter space . 13 1.2.2 De Sitter entropy and temperature . 16 1.3 Einstein equation from entanglement equilibrium . 23 1.3.1 Stationarity of generalized entropy . 24 1.3.2 Area and volume variations in Einstein gravity . 25 1.3.3 Entanglement entropy of a causal diamond . 30 1.4 The AdS3/CFT2 correspondence . 36 1.4.1 The 3d/2d holographic dictionary . 36 1.4.2 Black hole entropy from the Cardy formula . 38 2 Gravitational Thermodynamics of Causal Diamonds 41 2.1 Introduction . 41 2.2 Maximally symmetric causal diamonds . 45 2.3 Mechanics of causal diamonds in (A)dS . 49 2.3.1 Smarr formula for causal diamonds . 49 2.3.2 First law of causal diamonds . 53 2.4 Thermodynamics of causal diamonds in (A)dS . 60 2.4.1 Negative temperature . 60 2.4.2 Quantum corrections . 62 2.4.3 Entanglement equilibrium . 65 2.4.4 Free conformal energy . 67 2.5 Further remarks on the first law . 70 2.5.1 Role of maximal volume . 70 vii Contents 2.5.2 Fixed volume and fixed area variations . 71 2.5.3 Varying the cosmological constant . 72 2.5.4 Gravitational field Hamiltonian and York time . 73 2.6 Limiting cases: small and large diamonds . 74 2.6.1 De Sitter static patch . 75 2.6.2 Small diamonds and Minkowski space . 76 2.6.3 Rindler space . 78 2.6.4 AdS-Rindler space . 79 2.6.5 Wheeler-DeWitt patch in AdS . 82 2.7 Discussion . 82 3 Higher Curvature Gravity from Entanglement Equilibrium 85 3.1 Introduction . 85 3.2 First law of causal diamond mechanics . 89 3.2.1 Iyer-Wald formalism . 90 3.2.2 Geometric setup . 91 3.2.3 Local geometric expressions . 93 3.2.4 Variation at fixed W ...................... 96 3.3 Entanglement Equilibrium . 97 3.3.1 Subleading entanglement entropy divergences . 98 3.3.2 Equilibrium condition as gravitational constraints . 100 3.4 Field equations from equilibrium condition . 101 3.5 Comparison to other “gravitational dynamics from entanglement” approaches . 103 3.6 Discussion . 105 4 Towards non-AdS Holography 107 4.1 Introduction . 107 4.2 Lessons from the AdS/CFT correspondence . 109 4.2.1 General features of a microscopic holographic theory . 110 4.2.2 An example: AdS3/CFT2 ................... 112 4.2.3 Geometric definition and generalization to sub-AdS . 113 4.3 Towards holography for non-AdS spacetimes . 117 4.3.1 A conjecture on the microscopics of conformally related space- times . 117 p−2 ∼ d−2 4.3.2 An example: AdSd × S = AdSp × S ......... 120 4.3.3 Towards holography for sub-AdS, Minkowski and de Sitter space . 122 4.4 A long string interpretation . 125 4.4.1 The long string phenomenon . 127 4.4.2 Sub-AdS scales . 130 viii Contents 4.4.3 Minkowski space . 134 4.4.4 De Sitter space . 135 4.4.5 Super-AdS scales revisited . 138 4.5 Physical implications . 140 4.5.1 Black hole entropy and negative specific heat . 140 4.5.2 Vacuum energy of (A)dS . 142 4.6 Discussion . 144 Summary and Outlook 149 A Appendices 153 A.1 Acceleration and velocity of the conformal Killing flow . 153 A.2 Conformal isometry of causal diamonds in (A)dS . 155 A.3 Conformal Killing time and mean curvature . 156 A.4 Zeroth law for bifurcate conformal Killing horizons . 159 A.5 Conformal group from two-time embedding formalism . 161 A.5.1 Conformal Killing vectors of de Sitter space . 162 A.5.2 Conformal Killing vectors of Anti-de Sitter space . 163 A.6 Conformal transformation from Rindler space to causal diamond .