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Geometric Constructions for Repulsive Gravity and Quantization Geometric constructions for repulsive gravity and quantization vorgelegt von Manuel Hohmann geboren in Wolfsburg Hamburg 2010 Dissertation zur Erlangung des Doktorgrades des Fachbereichs Physik der Universit¨atHamburg ii Gutachter der Dissertation: Dr. Mattias N. R. Wohlfarth Prof. Dr. Klaus Fredenhagen Gutachter der Disputation: Dr. Mattias N. R. Wohlfarth Prof. Dr. Jan Louis Datum der Disputation: 29. Oktober 2010 Vorsitzender des Pr¨ufungsausschusses: Prof. Dr. G¨unter Sigl Vorsitzender des Promotionssausschusses: Prof. Dr. Jochen Bartels Leiterin des Fachbereichs Physik: Prof. Dr. Daniela Pfannkuche Dekan der MIN-Fakult¨at: Prof. Dr. Heinrich Graener iii Abstract In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathemat- ical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical po- sition expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity. iv v Zusammenfassung In dieser Arbeit stellen wir zwei geometrische Theorien vor, die entworfen wurden, um die allgemeine Relativit¨atstheoriezu erweitern. Es kann als eines der Ziele solcher Theorien angesehen werden, die beobachtete beschleunigte Expansion des Universums als Gravi- tationsph¨anomenzu beschreiben, or eine mathematische Struktur f¨urdie Formulierung von Quantenfeldtheorie auf gekr¨ummten Raumzeiten und Quantengravitation zu liefern. Diese Arbeit ist in zwei Teile aufgeteilt: Im ersten Teil betrachten wir multimetrische Gravitationstheorien mit N > 1 Stan- dardmodellkopien, die nur gravitativ wechselwirken und einander im Newtonschen Grenz- fall abstoßen. Die Dynamik jeder dieser Standardmodellkopien ist durch einen eige- nen metrischen Tensor bestimmt. Wir zeigen, dass der antisymmetrische Fall, in dem die gegenseitige Abstoßung zwischen den verschiedenen Materiesektoren von gleicher St¨arke ist wie die attraktive Gravitationskraft innerhalb jeden Sektors, durch ein No- Go-Theorem f¨ur N = 2 ausgeschlossen ist. Wir zeigen weiter, dass dieses Theorem f¨ur N > 2 nicht gilt, indem wir explizit eine antisymmetrische, multimetrische, repulsive Gravitationstheorie konstruieren. Wir untersuchen daraufhin einige Eigenschaften dieser Theorie. Wir konstruieren insbesondere ein kosmologisches Modell und zeigen, dass die beschleunigte Expansion des sp¨atenUniversums in der Tat durch die gegenseitige Abstoßung zwischen den verschiedenen Materiesorten erkl¨artwerden kann. Weiterhin stellen wir ein einfaches Modell f¨urStrukturbildung vor und zeigen, dass unser Modell zur Bildung von filamentartigen Strukturen und Voids f¨uhrt.Schließlich zeigen wir unter Anwendung des parametrisierten post-Newtonschen Formalismus, dass multimetrische, repulsive Gravitation mit Pr¨azisionsmessungenim Sonnensystem vertr¨aglich ist. Im zweiten Teil der Arbeit pr¨asentieren wir ein mathematisches Modell einer Quanten- Raumzeit in Form einer unendlichdimensionalen Mannigfaltigkeit, die hom¨oomorphzu einem geeigneten Schwartzraum ist. Dies erweitert und vereinigt sowohl die bekan- nte Funktionenraum-Konstruktion der Quantenmechanik als auch die differenzierbare Mannigfaltigkeitsstruktur der klassischen Raumzeit. In diesem Modell zeigen wir, dass die klassische Raumzeit in Form einer endlichdimensionalen Mannigfaltigkeit durch die topologische Identifikation aller Quantenpunkte mit identischem Ortserwartungswert entsteht. Wir spekulieren ¨uber die m¨ogliche Relevanz dieser Geometrie f¨urQuanten- feldtheorie und Gravitation. vi vii Acknowledgements First of all, I would like to thank Dr. Mattias Wohlfarth for accepting me as his PhD student, for countless valuable discussions, for supporting me with a constant flow of new ideas and helpful advice, and for teaching me how to properly write an article that has a good chance to be accepted. Further, I would like to thank the current members of the Emmy Noether group, Dr. Claudio Dappiaggi and Christian Pfeifer, as well as the former members Martin von den Driesch, Niklas H¨ubel, J¨orgKulbartz, Christian Reichwagen, Felix Tennie and Lars von der Wense for many helpful ideas and numerous contributions to the discussions at our group meetings. I am also very grateful to my current and former officemates Thomas Danckaert, Christian Gross, Danny Mart´ınez-Pedrera, Martin Schasny, Bastiaan Spanjaard and Ha- gen Triendl for various discussions about general relativity, quantum mechanics, politics and many other interesting and funny topics. Moreover, I would like to thank all members of the II. Institute for theoretical physics and the DESY theory group for the nice atmosphere during both work times and lunch breaks. Finally, I am indepted to Dr. Raffaele Punzi, who used to be an irreplaceable source of inspiration and motivation for our group, and who was certainly one of the smartest and friendliest men I was pleased to meet. viii Contents Introduction of the thesis3 1 A brief history of gravity3 1.1 Newtonian gravity..............................3 1.2 Einstein gravity................................4 2 Foundations of general relativity7 2.1 Pseudo-Riemannian geometry........................7 2.1.1 Manifolds and tensors........................7 2.1.2 The metric tensor........................... 10 2.1.3 Parallel transport and Levi-Civita connection........... 12 2.1.4 Riemann curvature.......................... 14 2.2 Einstein-Hilbert action............................ 15 2.3 Matter and energy-momentum tensor.................... 16 3 Problems of general relativity 19 3.1 Astronomical observations.......................... 19 3.1.1 Peculiar velocities in galaxy clusters................. 20 3.1.2 Gravitational lensing......................... 20 3.1.3 Rotation curves of galaxies...................... 20 3.1.4 Structure formation.......................... 21 3.1.5 Accelerating expansion of the universe............... 21 3.1.6 Galaxies in the vicinity of voids................... 22 3.1.7 Pioneer anomaly........................... 23 3.2 Quantization................................. 23 3.2.1 Necessity of a quantum theory of gravity.............. 24 3.2.2 Problems of quantization....................... 24 3.2.3 Possible solutions........................... 25 ix x Contents 4 Outline of the thesis 27 I Multimetric repulsive gravity 31 5 Introduction 33 5.1 Repulsive extension of general relativity.................. 34 5.2 Repulsive gravity effects........................... 36 5.3 Experimental tests of repulsive gravity................... 38 6 No-go theorem for canonical bimetric repulsive gravity 41 6.1 Formulation of the theorem and its assumptions.............. 42 6.2 Proof of the no-go theorem.......................... 45 6.2.1 Field equations............................ 46 6.2.2 Gauge-invariant formalism...................... 47 6.2.3 Decoupling of modes......................... 49 6.2.4 Gauge-invariance and consistency.................. 51 6.2.5 Contradiction............................. 54 6.3 Possible ways around the theorem...................... 56 7 A simple multimetric repulsive gravity theory 59 7.1 Action..................................... 59 7.2 Derivation of field equations......................... 62 7.3 Particle content................................ 65 8 Multimetric cosmology 67 8.1 Simple cosmological model.......................... 67 8.2 Equations of motion............................. 69 8.3 Explicit solution................................ 70 9 Simulation of structure formation 75 9.1 Simple model for structure formation.................... 75 9.2 Implementation................................ 79 9.3 Results of the simulation........................... 80 Contents
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