Emergent Gravity: The Analogue Models Perspective Thesis submitted for the degree of “Doctor Philosophiæ” September 2009 Candidate Supervisor Lorenzo Sindoni Stefano Liberati SISSA International School for Advanced Studies Via Beirut 2-4, 34151 Trieste, Italy. E-mail: [email protected] To my family EMERGENT GRAVITY: The Analogue Models Perspective Lorenzo Sindoni — Ph.D. Thesis Supervisor: S. Liberati Abstract This thesis is devoted to the study of some aspects of emergent gravity scenarios, i.e. of non-gravitational systems which exhibit, under suitable conditions, the emergence of effective spacetime metrics and the associated gravitational dynamics. While this area of research is rather broad, we will assume a particular perspective: the starting point is the discussion of analogue models for gravity, which have so far provided precious insights on aspects of physics on curved spacetimes, on extensions of Riemannian geometry and on the possible role of high energy Lorentz symmetry violations in low energy physics. The first part is devoted to the study of the most relevant kinematical features of emergent spacetimes typical of condensed matter analogue models. In particular, it will be shown that a natural framework for the description of the geometrical properties of emergent spacetimes is Finsler geometry, which thus represent an interesting candidate for extensions of special (and eventually general) relativity. We will present the basic concepts of this particular generalization of Riemannian geometry, and hence we will pass to the careful analysis of the main issues to be solved and of the features that can be of major interest for physical applications, in particular the geometrical interpretation of modified dispersion relations and the fate of Lorentz symmetry at the Planck scale. Some emphasis will be given to the conditions under which pseudo-Riemannian geometries are selected among the larger class of Finsler geometries. The second part is devoted to the discussion of some features of the dynamics of the emergent spacetimes. In particular, we will discuss the lessons that can be learned from a Bose–Einstein condensate analogue model. It is shown that, in an appropriate regime, a modified non-relativistic gravitational theory does appear to be the effective description of quasiparticles and their interactions with the condensate. While this is by no means an attempt to realize a perfect analogue of a realistic gravitational theory, this particular example gives the possibility of discussing a number of interesting features (e.g. locality and the nature of the cosmological constant). These features could be used as an inspiration to understand some of the problems encountered in understanding gravity at both classical and quantum levels. Finally, a generalization to the relativistic case will give the possibility of discussing more refined questions, like the possibility of emerging diffeomorphism invariance and time out of a system defined in flat Euclidean space. Despite the particular perspective and the very specific form of the toy models considered, we believe that all the observations and hints that can be collected from these investigations, in particular the role of symmetries, can shed some light onto some of the most challenging problems in the formulation of a satisfactory theory of emergent gravity. List of papers The research presented in this thesis is part of the outcome of the scientific collaborations stated below, as well as of the author’s own work. The research activity has led to the following research papers published in refereed Journals. • Planck-scale modified dispersion relations and Finsler geometry. Florian Girelli, Stefano Liberati, Lorenzo Sindoni (SISSA, Trieste & INFN, Trieste) . Published in Phys.Rev.D75:064015, 2007. [e-Print: gr-qc/0611024] (Ref. [234], discussed in chapter 5) • Shell-mediated tunnelling between (anti-)de Sitter vacua. Stefano Ansoldi (MIT, LNS & MIT & ICRA, Rome), Lorenzo Sindoni (SISSA, Trieste & INFN, Trieste) . Published in Phys.Rev.D76:064020,2007. [e-Print: arXiv:0704.1073 [gr-qc]] • The Higgs mechanism in Finsler spacetimes. Lorenzo Sindoni (SISSA, Trieste & INFN, Trieste) . Published in Phys.Rev.D77:124009,2008. [e-Print: arXiv:0712.3518 [gr-qc]] (Ref. [241], discussed in chapter 5) • On the emergence of Lorentzian signature and scalar gravity. Florian Girelli, Stefano Liberati, Lorenzo Sindoni (SISSA, Trieste& INFN, Trieste) . Published in Phys.Rev.D79:044019,2009. [e-Print: arXiv:0806.4239 [gr-qc]] (Ref. [295], discussed in chapter 8) • Gravitational dynamics in Bose Einstein condensates. Florian Girelli, Stefano Liberati, Lorenzo Sindoni (SISSA, Trieste & INFN, Sezione di Trieste). Published in Phys.Rev.D78:084013,2008. [e-Print: arXiv:0807.4910 [gr-qc]] (Ref. [283], discussed in chapter 7) • Reconciling MOND and dark matter? Jean Philippe Bruneton, Stefano Liberati, Lorenzo Sindoni and Benoit Famaey Published in JCAP 0903:021,2009. [e-Print arXiv:0811.3143 [astro-ph]] (Ref. [315], mentioned in chapter 9) • Linking the trans-Planckian and the information loss problems in black hole physics. Stefano Liberati, Lorenzo Sindoni (SISSA, Trieste & INFN, Trieste) Sebastiano Sonego (Universit`a di Udine) e-Print: arXiv:0904.0815 [gr-qc] Submitted to Annals of Physics (Ref. [82], mentioned in chapter 2) Acknowledgments There are several people that I want to thank. First of all, my supervisor, a great teacher and a patient guide in these four years. His lessons about all the aspects of the research in physics are a precious treasure that I hope to be able to administrate wisely. Without his constant inspiration and deep insight, the work presented in this thesis would not have been possible. Not only: his human qualities have made all the endless conversations of these years very pleasant and special occasions, beyond the scientific aspects. I have to thank Sebastiano Sonego for his help and advice, for his good questions, for all the discussions about physics and for the trans-Planckian collaboration. He deserves a special mention for having shared endless hours of travels by train between Udine and Miramare. I want to warmly thank Florian Girelli, for having taught me a lot, for his patience, for his precious advice and for the great job done together. I had also the opportunity to work with Stefano Ansoldi and Jean-Philippe Bruneton: I have to thank them for the fruitful collaborations, for discussions and for their friendship. During the PhD I had also enjoyed several conversations with Matt Visser and Silke Weinfurtner: all the discussions with them have been a valuable source of help and inspiration, and for this I owe them a lot. I want to thank Christoph, Luca, Annibale, Sara, Stefano, Goffredo, Vincenzo, Gaurav, Lorena and all the several friends I have met in SISSA in these years. It has been wonderful to share with them the days of the PhD. I will miss them. Of course, all this would not have been possible without the constant support of my family: my mother, my brother Pietro and Gianni. They have been a constant presence in these years. I owe them everything, and for this I dedicate this thesis to them. August 2009 Lorenzo Sindoni Contents 1 Motivations and overview 1 I Kinematics 9 2 Preliminaries 10 2.1 Acousticspacetimes .............................. ...... 10 2.1.1 Horizons,blackholes. ..... 11 2.2 BECgeneralization............................... ...... 12 2.2.1 Second quantization and the mean field approximation . ............ 13 2.2.2 Gross–Pitaevski and its fluid interpretation . ............ 16 2.3 Phonons......................................... .. 17 2.3.1 Bogoliubovtransformations . ....... 17 2.3.2 Bogoliubovdispersionrelation . ........ 18 2.3.3 Thehealinglength .............................. ... 19 2.3.4 Thedepletionfactor . .. .. .. .. .. .. .. .. .. .. .. .... 20 2.3.5 TheBogoliubov–deGennesformalism . ....... 20 2.4 BEC-basedanaloguemodels. ....... 22 2.5 Multi-component systems: normal modes analysis . ............. 23 2.5.1 Birefringentcrystals . ...... 24 2.5.2 TwocomponentsBEC.............................. 26 2.6 Fromanaloguemodelsto Finslergeometry. ........... 27 3 Special Relativity beyond the Planck scale 29 3.1 Is Lorentz invariance a fundamental symmetry? . ............. 29 3.2 TestsofLorentzinvariance . ........ 31 3.2.1 Robertson–Mansouri–Sexlformalism . ......... 32 3.2.2 Lorentz Invariance violation in effective field theories.............. 32 3.2.3 Dispersionrelations . ..... 34 3.2.4 Modified dispersion relations and the effective field theoryapproach . 35 3.2.5 Quantumgravityphenomenology . ...... 37 3.3 Special Relativity revisited: axiomatic approach . ................. 38 i 3.4 Anisotropic relativity and Finsler geometry . .............. 40 3.4.1 Towards a generalization in 3 + 1 dimensions . ......... 44 3.4.2 Finsler geometry as a test theory for Minkowski spacetime........... 45 3.5 IntroducingthePlanckscale. ......... 46 3.6 Insummary....................................... .. 50 4 Finsler geometry: basics 52 4.1 Finslerspaces:definition. ......... 52 4.1.1 Basictheorems ................................. .. 53 4.1.2 TheCartantensor ............................... .. 55 4.1.3 Finslerspaces................................. ... 56 4.1.4 Theindicatrix ................................. .. 57 4.2 Examples ........................................ .. 58 4.2.1 Randersspaces ................................. .. 58 4.2.2 Berwald–Moor .................................. 59 4.2.3 (α, β)spaces .................................... 59 4.3