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Emergent Gravity: the Bec Paradigm

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Emergent Gravity: the Bec Paradigm UNIVERSIDAD COMPLUTENSE DE MADRID FACULTAD DE CIENCIAS FISICAS Departamento de Física Teórica II EMERGENT GRAVITY: THE BEC PARADIGM MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR Gil Jannes Bajo la dirección de los doctores Carlos Barceló Serón Luis J. Garay Elizondo Madrid, 2009 ISBN: 978-84-692-8447-6 ©Gil Jannes,2009 Departamento de Física Teórica II Facultad de Ciencias Físicas Universidad Complutense de Madrid EMERGENT GRAVITY : THE BEC PARADIGM Gravedad emergente: el paradigma de CBE Gil Jannes1,2 Directores: Tesis presentada Dr. Carlos Barceló Serón1 para optar al grado de Dr. Luis J. Garay Elizondo3 Doctor en Ciencias Físicas Madrid, 2009 1 Inst. de Astrofísica de Andalucía, CSIC, Camino Bajo de Huétor 50, 18008 Granada, Spain. 2 Inst. de Estructura de la Materia, CSIC, C/ Serrano 121, 28006 Madrid, Spain. 3 Depto. de Física Teórica II, Universidad Complutense de Madrid, 28040 Madrid, Spain. Acknowledgements I first and foremost thank Carlos Barceló and Luis Garay, my research advisors. It has been a tremendous pleasure to work with two persons who are not only out- standing scientists, but moreover extraordinary personalities. And I immediately wish to extend this description to Guillermo Mena, even though we have not (yet) directly collaborated in terms of science. At all times, I felt like a colleague and friend more than (just) a PhD student, and that is the greatest compliment you could have paid me. I hope that this thesis will mark a starting point rather than an end- point. Many thanks to Stefano Liberati, Grisha Volovik and Matt Visser, for their hospi- tality and for many illuminating scientific discussions. Particular thanks to Matt for the extensive gastronomic and touristic tour of Wellington and surroundings. There is a long list of people with whom I had plenty of discussions that were totally irrelevant in any scientific way, but nonetheless very entertaining. Rather than attempting to give an exhaustive list and perhaps offending some by forgetting to mention them, I will only name one person explicitly: the one who has paid me the largest amount of beers. That honour goes without a doubt to José Antonio Sánchez Gil. Salud! Thanks to the few of my friends who have shown genuine interest in my work. Thanks also to the large majority who, totally understandably, have shown only a marginal interest, or none at all. After all, even if one finds great pleasure in work, being with friends who make you forget work is sometimes an even greater pleasure. Thanks to my father for being the most avid supporter of my scientific career, and for trying to convince the rest of the family, with variable degrees of success. Thanks to Wendy for enduring the absurdities and insecurity of a partner in academic life. And of course for being the most marvellous, smart, sweet, funny and sexy woman in the world, thus helping to make my life the wonderful adventure it is. 3 4 I would end with a word to Igor, but you wouldn’t care, and you are absolutely right. Everything that I have written in this thesis falls into utter meaninglessness compared to the simple pleasure of seeing you smile when I wake you up in the morning. So I will limit myself to apologise for having spent less time playing with you than I should while I was writing up this thesis. Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve. Karl R. Popper Contents Abstract 7 1 Introduction 9 1.1 Analogue gravity in condensed matter . 10 1.2 From analogue gravity to QG phenomenology . 11 1.3 From analogue gravity to emergent gravity . 12 1.4 A very brief history of analogue gravity . 14 1.5 Synopsis . 15 2 Preliminaries 21 2.1 General considerations . 21 2.2 Bose–Einstein condensates of dilute atomic gases . 23 2.3 Gross–Pitaevskii equation and quantum potential . 26 2.4 Effective acoustic metric . 28 2.5 Dispersion relation . 30 2.6 Validity of the hydrodynamic approximation . 32 2.7 Why Bose–Einstein condensates? . 34 3 Black hole configurations in BECs 37 3.1 A simple (1+1)-dimensional model . 37 3.2 Background configuration and numerical method . 38 4 Analysis of dynamical instabilities 43 4.1 Motivation . 43 4.2 Boundary conditions . 45 4.3 Case by case analysis and results . 47 5 6 CONTENTS 4.4 Discussion . 52 4.5 Summary and conclusions . 54 5 Quasinormal modes 57 5.1 Introduction . 57 5.2 Boundary conditions . 58 5.3 Results and discussion . 62 5.4 Summary and conclusions . 64 6 Lorentz violation and modified disp. relations 67 7 Hawking radiation in a collapse scenario 73 7.1 Introduction . 73 7.2 Classical geometry . 77 7.3 Standard Dispersion Relations . 81 7.4 Superluminally modified dispersion relations . 86 7.5 Graphical results and discussion . 96 7.6 Summary and discussion . 102 8 Emergent gravity 109 8.1 Introduction . 109 8.2 Quantum gravity and dark energy . 111 8.3 Diffeomorphism invariance: kinematical vs. dynamical . 113 8.4 Diffeomorphism invariance at the kinematical level . 115 8.5 Diffeomorphism invariance at the dynamical level . 120 8.6 Summary . 125 9 General summary and outlook 127 10 Resumen en español 135 10.1 Introducción . 138 10.2 Objetivos y contenido . 140 10.3 Resumen general y perspectivas futuras . 145 References 153 Abstract We study selected aspects of quantum gravity phenomenology inspired by the grav- itational analogy in Bose–Einstein condensates (BECs). We first review the basic ideas and formalism of analogue gravity in BECs, with particular emphasis on the possibility of simulating black holes. We stress that the dispersion relation in a BEC has a relativistic form in the hydrodynamic limit, and acquires non-relativistic, ‘su- perluminal’ modifications at high frequencies. This makes it a particularly interest- ing model for many scenarios of quantum gravity phenomenology which consider a possible violation of local Lorentz invariance at high energies. In particular, it al- lows the study of kinematical corrections that such quantum gravity scenarios could impose on general relativity. We present a simple model for a (1+1)-dimensional acoustic black hole configu- ration in a BEC, and study its dynamical stability. We find that black hole horizons in a BEC are stable under quite general boundary conditions, inspired by similar stud- ies of the stability of black holes in general relativity. This allows us to study the stable dynamical modes or quasinormal modes of such configurations, as well as to confirm the long-expected possibility of (at least in principle) simulating Hawking radiation in a BEC. We find that there indeed exist short-lived quasinormal modes in our model, in spite of its (1+1)-dimensionality, and moreover that their spectrum covers a contin- uous region of the complex frequency plane. This is in sharp contrast to general relativistic black holes and to acoustic black holes in the hydrodynamic limit, where no quasinormal modes exist in 1+1 dimensions, whereas in higher dimensions the spectrum is discrete. The quasinormal modes that we find are characteristic of a relaxation at the microscopic level, and we trace their presence back to the superlu- minality of the dispersion relation, and the associated permeability of the horizon. We argue that a similar effect could show up for astrophysical black holes if quan- tum gravity turns out also to lead to superluminality at high frequencies. We then study the impact of superluminal dispersion relations on the Hawking 7 8 CONTENTS radiation for a collapsing geometry in which a black hole is created. A crucial aspect of modified dispersion relations is that the horizon becomes a frequency-dependent concept. In particular, with superluminal dispersion relations, at every moment of the collapse process, there is a critical frequency above which no horizon is experi- enced. If moreover there is an overall saturation level to the collapse (as is certainly the case in BECs due to the interatomic separation, and as is expected in the gravi- tational case due to quantum gravity resolving the general relativistic singularity), then there is also a global critical frequency, such that higher frequencies never see a horizon and hence do not participate in the thermal radiation process. This leads to several important differences with the standard Hawking radiation. Generally speaking, a finite critical frequency implies that the intensity of the late-time radia- tion is weaker than in the standard (Lorentz invariant) case. The Planckian form of the thermal spectrum is largely preserved (although with a decreased intensity) unless the Lorentz symmetry violation scale (the scale at which non-relativistic deviations in the dispersion relation become important) lies below the critical frequency. In the latter case, high frequencies (above the Lorentz symmetry violation scale) can ac- quire extremely large surface gravities, leading to important ultraviolet contributions in the radiation spectrum. Finally, the radiation originating from the collapse dies out on a relatively short time-scale. In the final part of this thesis, we discuss some questions related to the possibil- ity of constructing a serious toy model for Planck-scale gravity based on the con- densed matter analogy, namely a model of emergent gravity. The basic idea is that gravity (including its dynamical aspects, i.e., the Einstein equations) might emerge as an effective low-energy macroscopic description of the collective behaviour of the microscopic constituents of a system. A crucial consequence if this turns out to be a correct approach is that there would not be any direct quantisation procedure leading from the macroscopic to the microscopic degrees of freedom.
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