Max-Planck-Institut f¨urGravitationsphysik Dr. Frederic P. Schuller Institut f¨urPhysik und Astronomie Prof. Dr. Martin Wilkens Tensorial spacetime geometries carrying predictive, interpretable and quantizable matter dynamics Dissertation zur Erlangung des akademischen Grades “doctor rerum naturalium” (Dr. rer. nat.) in der Wissenschaftsdisziplin “Mathematische Gravitationsphysik” eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atPotsdam von Sergio Rivera Hern´andez Potsdam, den 15. Februar 2012 Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2012/6186/ URN urn:nbn:de:kobv:517-opus-61869 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61869 A mis padres, con mucho amor y admiraci´on. Contents Acknowledgements 1 Chapter 1. Introduction to spacetime geometries 3 1.1. The conceptual path to general relativity 3 1.2. Observations 8 1.3. New tensorial geometries 11 Chapter 2. Condition I: Hyperbolicity 15 2.1. Initial value problem and hyperbolicity 15 2.2. Hyperbolic polynomials 17 Chapter 3. Condition II: Time-orientability 21 3.1. The short wave approximation and the massless dispersion relation 21 3.2. Solution to the eikonal equation 23 3.3. Massless duality theory: the Gauss map and action for massless particles 25 3.4. Existence and computability of the dual polynomial 28 3.5. Time-orientability 31 Chapter 4. Condition III: Energy-distinguishability 35 4.1. Energy-distinguishing condition 35 4.2. The massive dispersion relation 36 4.3. Massive duality theory: the Legendre map and action for massive particles 38 4.4. Lorentzian Finsler geometry and freely falling observers 41 Chapter 5. General properties of tensorial spacetimes 45 5.1. Observer frames and observer transformations 45 5.2. Temporal-spatial split of modified dispersion relations 47 5.3. `Superluminal' propagation of matter and vacuum Cherenkov process 48 Chapter 6. Concrete tensorial spacetime geometries 53 6.1. Lorentzian geometry 53 6.2. Area metric geometry 57 6.3. Testing modified dispersion relations 70 Chapter 7. Free QED on area metric spacetimes 73 7.1. Hamiltonian formulation and gauge fixing for area metric electrodynamics 73 v vi CONTENTS 7.2. Quantization 77 7.3. Casimir effect in a birefringent linear optical medium 81 Chapter 8. Coupling currents 85 8.1. Covariant propagator 85 8.2. Quantum point charges 87 8.3. General structure of lower order field equations 92 8.4. Examples of second and first order field equations 96 Chapter 9. Towards gravity 101 9.1. Deformation algebra of hypersurfaces. 101 9.2. Dynamical evolution of the geometry. 103 Chapter 10. Conclusions 105 Bibliography 109 Acknowledgements Dear Frederic, this work would not have been possible without your tremendous support in science and in all other aspects, and I have not enough space here to thank you for everything. Thank you for having supervised my doctoral research, I have learnt a lot of physics (but also about other things) from you. In particular, I thank you for conveying the aim to really under- stand and improve things. I really appreciate that you were always open to discuss about any subject and ideas. I thank you for the opportunity to come to this beautiful country, for your trust, for your advices, for your patience, for your help, and, of course, for the coffees. Next, I would like to thank my PhD-colleagues Dennis R¨atzeland Christof Witte. I enjoyed a lot the time we spent working together. Dear Dennis, it was a great time working in our first paper together with Frederic, and also in the quantization of matter fields. Dear Christof, I thank you for the insightful discussions, and it was a pleassure to have worked with you at the \Deutsche Sch¨ulerakademie". I would also like to thank Prof. Dr. Martin Wilkens for acting as my supervisor at the university of Potsdam and Prof. Daniel Sudarsky for his understanding about finishing my PhD in Germany. I thank my wife Vicky and my son Sergito for their love and support over the last years. It is wonderful to share my life with both of you. I also thank my parents, Fernando and Alicia, for their love and support in all circumstances, and my brother Fernando for all experiences we have had together. I would also like to thank Fritz Henke, Frank Michalet, Natividad Batista, Wolfgang and Ingrid Christoph, and my friends Odir Rodriguez and Ivette Cruz for helping me (or my family) in several ways. Finally, I would like to thank the Albert Einstein Institute for its hospitality and support where this doctoral work has been done, and the finacial support of the German Academic Ex- change Service (DAAD). 1 CHAPTER 1 Introduction to spacetime geometries In this doctoral work, we will determine the requirements that a pair (M; G) of a finite- dimensional smooth manifold M and an a priori arbitrary tensor field G must satisfy in order to provide a classical spacetime geometry. This introductory chapter will be devoted to motivate our investigation, to make precise which problem we will solve, and to describe the logical path we will follow. In the first section, we present a review on general relativity based on an idealized historical logic for its construction, which indeed mirrors how we will determine the conditions on (M; G). In the second section, we shortly review the main observations interpreted in the framework of general relativity and the standard model of particle physics. We will argue that these suggest new tensorial geometries to replace Lorentzian metrics as an appropriate mathe- matical structure modelling the physical spacetime structure. Finally, in the third section, we state key conditions that any pair (M; G) must satisfy in order to provide a classical spacetime geometry. The enterprise undertaken in this thesis is to make these conditions mathematically precise, derive rigorous conclusions and apply them to physics. 1.1. The conceptual path to general relativity Historically, Einstein did not directly postulate that spacetime is a pair (M; g) of a finite- dimensional smooth manifold M equipped with a Lorentzian metric g and then made physical constructions on it, as sometimes it is the standard textbook presentation of general relativity. It was rather the study of a matter field, namely the electromagnetic field, that ultimately led Einstein to conclude that the geometry of spacetime could be encoded in a Lorentzian metric [1]. Here it will be precisely this logical path followed by Einstein, namely starting from a study of matter fields, that allows for the identification of the pairs (M; G) that can provide a classical spacetime structure. So let us review, using modern mathematical language, how Einstein constructed general relativity by studying Maxwell electrodynamics. We begin by considering a one-form field A, to be identified with the electromagnetic poten- tial, on a finite-dimensional smooth manifold M. If the manifold M on which the field A lives is equipped with a metric g, a priori not necessarily of Lorentzian signature, we can stipulate the well-known Maxwell action Z 1 4 p ab cd S[A; g] = − d x j det(g)j g g FacF (1.1) 4 bd to provide dynamics for the field A, where F = dA is defined as the field strength, and the ab components of g and Fab are chosen with respect to some coordinate system. Variation of this action with respect to the electromagnetic degrees of freedom A yields a consistent set of partial 3 4 1. INTRODUCTION TO SPACETIME GEOMETRIES differential equations, a priori for any signature of the metric. But if we want to do physics, we want to predict \future" values of the field A, i.e., we want the field equations for the field A to be predictive. More precisely, given initial values of the field A and its first derivatives on an \adequate" hypersurface Σ0 of M, which is all we have access to measure, we want the dynamics arising from the action above to be able to reconstruct in a unique way the values of the field A at a \later" hypersurface Σt. It is a well-known result from the theory of partial differential equations [2, 3] that predictivity in this sense can only be possible if the field equations arising from the considered action are of hyperbolic type. This already amounts to the requirement that the inverse metric g−1 in the action, and therefore the metric g itself, be of Lorentzian signature (and for definiteness, we choose the mainly minus signature (+−· · · −) in the following). Thus the predictivity requirement on the dynamics of the matter field A already restricts a metric manifold (M; g) to be Lorentzian. The adequate hypersurfaces Σ on which initial data can be provided are then recognized as those ∗ whose co-normals q 2 Tx M at each point x of the hypersurface Σ (which are determined up to a non-zero scale) are contained in one of the convex cones C or −C, which are defined as the set of covectors q that are positive with respect to g−1, e.g. the cones defined by −1 gx (q; q) > 0 ; (1.2) see figure 1.1. The physical interpretation of the covectors k on the boundary of the cones C and −C, i.e., those satisfying g−1(k; k) = 0, is obtained by studying the geometric optical limit of the field equations arising from the considered action. This study proceeds by considering wave-like approximate solutions of the form eiS(x)/λ for the field equations obeyed by A, where S is a scalar real function on M, and studying these for arbitrary small wavelength λ. These approximate solutions then correspond to high-frequency propagating waves whenever the wave- length is negligible in comparison to other relevant length scales.