On the phenomenology of quantum gravity : stability properties of Hawking radiation in the presence of ultraviolet violation of local Lorentz invariance Antonin Coutant To cite this version: Antonin Coutant. On the phenomenology of quantum gravity : stability properties of Hawking ra- diation in the presence of ultraviolet violation of local Lorentz invariance. Other [cond-mat.other]. Université Paris Sud - Paris XI, 2012. English. NNT : 2012PA112213. tel-00747874 HAL Id: tel-00747874 https://tel.archives-ouvertes.fr/tel-00747874 Submitted on 21 Dec 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universite´ Paris-Sud 11 LPT Orsay On the phenomenology of quantum gravity: Stability properties of Hawking radiation in the presence of ultraviolet violation of local Lorentz invariance PhD thesis by Antonin Coutant Defended on October 1, 2012, in front of the jury Pr. Vitor Cardoso Referee Pr. Ted Jacobson Referee Pr. Vincent Rivasseau Jury president Pr. Roberto Balbinot Jury member Pr. Stefano Liberati Jury member Pr. Renaud Parentani PhD advisor Abstract (EN): In this thesis, we study several features of Hawking radiation in the presence of ultraviolet Lorentz violations. These violations are implemented by a modified dispersion relation that becomes nonlinear at short wavelengths. The motivations of this work arise on the one hand from the developing field of analog gravity, where we aim at measuring the Hawking effect in fluid flows that mimic black hole spacetimes, and on the other hand from the possibility that quantum gravity effects might be approximately modeled by a modified dispersion relation. We develop several studies on various aspects of the prob lem. First we obtain precise characterizations about the deviations from the Hawking result of black hole radiation, which are induced by dispersion. Second, we study the emergence, both in white hole flows or for massive fields, of a macroscopic standing wave, spontaneously produced from the Hawking effect, and known as ‘undulation’. Third, we describe in detail an instability named black hole laser, which arises in the presence of two horizons, where Hawking radiation is selfamplified and induces an exponentially growing in time emitted flux. Tags: Hawking radiation, Analog gravity, Lorentz violation, Instabilities, Undulations R´esum´e(FR): Dans cette th`ese,nous ´etudionsplusieurs aspects de la radiation de Hawking en pr´esence de violations de l’invariance locale de Lorentz. Ces violations sont introduites par une modification de la relation de dispersion, devenant nonlin´eaireaux courtes longueurs d’onde. Les principales motivations de ces travaux ont une double origine. Il y a d’une part le d´eveloppement en mati`erecondens´eede trous noirs analogues, ou l’´ecoulement d’un fluide est per¸cucomme une m´etriqued’espacetemps pour les ondes de perturba tions et ou la radiation de Hawking pourrait ˆetre d´etect´eeexp´erimentalement. D’autre part, il se pourrait que des effets de gravit´equantique puissent ˆetremod´elis´espar une modification de la relation de dispersion. En premier lieu, nous avons obtenu des car act´erisationspr´ecises des conditions n´ecessairesau maintien de l’effet Hawking en pr´esence de violation de l’invariance de Lorentz. De plus, nous avons ´etudi´el’apparition d’une onde macroscopique de fr´equencenulle, dans des ´ecoulements de type trous blancs et ´egalement pour des champs massifs. Une autre partie de ce travail a consist´e`aanalyser une insta bilit´eengendr´eepar les effets dispersifs, ou la radiation de Hawking est autoamplifi´ee, g´en´erant ainsi un flux sortant exponentiellement croissant dans le temps. Mots-cl´es: Radiation de Hawking, Gravit´eanalogue, Violation de Lorentz, Instabilit´es, Undulations Th`esepr´epar´eedans le cadre de l’Ecole Doctorale 107 au Laboratoire de Physique Th´eoriqued’Orsay (UMR 8627) Bˆat. 210, Universit´eParisSud 11, 91405 Orsay Cedex. i ii Acknowledgment First of all, I would like to thank my PhD advisor, Renaud Parentani. Working under his supervision was not only fruitful, it was also a great pleasure. I can’t thank him enough for the countless conversations we had about various topics of physics, and for all the knowledge he shared with me. I particularly enjoyed our discussions on the interpre tation of quantum mechanics or the origin of the second principle. I am sincerely grateful to Vitor Cardoso and Ted Jacobson, who accepted to be my PhD referees. The same gratitude goes to the other members of my jury, Roberto Bal binot, Stefano Liberati and Vincent Rivasseau. They agreed to give some of their time to examine my work, and that was an honor for me. I also wish to thank my other collab orators Stefano Finazzi, Alessandro Fabbri, and Paul R. Anderson, and I hope our work together was as pleasant for them as it was for me. These three years in the LPT Orsay have been a real enjoyment. Working among the Cosmo group has been wonderful, and has only confirmed my willing to pursue my career as a physicist. I particularly thank my fellows from the SinJe, as much for the passionate scientific debates as for the human experience. A special thank goes to Xavier Bush for his careful reading of my manuscript. I am also very grateful to Patricia DuboisViolette, Philippe Molle, and the rest of the administrative staff, whose efficiency and patience has been a precious help. I am specially thankful to my friends and colleagues Marc Geiller, Sylvain Carrozza and Yannis Bardoux. The many conversations we had about physics has been, and con tinue to be, a great inspiration for me. I can only hope we will have many occasions to work together in the future. I can’t say how much I owe to my parents Isabelle and Bernard, my brother Balt hazar and sister B´er´enice,as well as my friends Gr´egoireand Ludovic, who has always been there for me. Their constant support has always been inflexible, even though they probably still wonder what my job really is. Finally, my last words go to my dear AnneSophie. These last years has been con stantly enlightened by her presence, support and love. iii iv Contents Introduction 1 The problem of quantum gravity . 1 The background field approximation . 3 What can we learn from black holes ? . 4 1 Geometry of space-time and black holes 7 1.1 Fundamental principles of relativity . 7 1.2 Elements of geometry . 9 1.2.1 Smooth manifolds . 9 1.2.2 Tangent bundle . 11 1.2.3 Cotangent bundle and tensor fields . 16 1.2.4 Spacetime as a Lorentzian manifold . 19 1.3 General relativity . 23 1.3.1 Einstein’s equations . 23 1.3.2 Black hole spacetimes . 24 1.4 Spherically symmetric Black holes . 26 1.4.1 Geodesic flow . 26 1.4.2 Near horizon region . 29 1.4.3 Field propagation around a black hole . 31 1.5 Hints of black hole thermodynamics . 32 2 Quantum field theory in curved space-time 35 2.1 Quantum field in flat space . 35 2.1.1 Field quantization . 36 2.1.2 Green functions . 39 2.2 The UnruhDeWitt particle detector . 41 2.2.1 Particle detector model . 41 2.2.2 Inertial detector . 43 2.2.3 Unruh effect . 44 2.3 Hawking radiation . 47 2.3.1 A collapsing model . 47 2.3.2 Radiation of eternal black holes . 56 2.3.3 Observables . 60 v Contents 3 Breaking Lorentz invariance 65 3.1 Dispersive field theories . 65 3.1.1 The transplanckian question . 65 3.1.2 Analog models . 67 3.1.3 General relativity with ultraviolet Lorentz violations . 70 3.2 Hawking radiation in the presence of dispersion . 74 3.2.1 Modifications at the eikonal approximation . 75 3.2.2 Hawking radiation as a scattering problem . 78 3.2.3 Mode analysis . 81 3.2.4 Globally defined modes in the near horizon region . 86 3.2.5 The various modes in the near horizon region . 87 3.2.6 Connection matrix and onshell Bogoliubov transformation . 91 3.2.7 Validity of the connection formula . 93 3.2.8 General superluminal dispersion . 95 3.2.9 Relating subluminal dispersion relations to superluminal ones . 96 3.3 Main conclusions of this study . 97 4 Infrared instabilities 99 4.1 Infrared divergences in Hawking radiation . 99 4.2 The BoseEinstein condensate context . 102 4.3 White hole undulations . 105 4.3.1 The black holewhite hole correspondence . 105 4.3.2 Undulation in a white hole . 106 4.4 Massive fields . 109 4.4.1 Settings, mode mixing, structure of the Smatrix . 109 4.4.2 Exactly solvable models . 117 4.5 Massive undulations in black holes . 126 4.6 Massive dispersive undulations in white holes . 127 4.6.1 The scattering . 128 4.6.2 The undulation . 130 4.7 Double horizon undulation . 131 4.7.1 Warpdrive analogy . 131 4.7.2 Propagation on a double horizon metric . 132 4.8 Conclusions about undulations . 138 5 Dynamical instability 141 5.1 Presentation of the black hole laser effect . 142 5.2 The settings . 143 5.3 The set of asymptotically bounded modes . 145 5.3.1 Main results . 145 5.3.2 Asymptotic behavior and roots kλ . 146 5.3.3 The continuous spectrum .