Scalar-Tensor Gravity in the Palatini Approach
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Formation of Structure in Dark Energy Cosmologies
HELSINKI INSTITUTE OF PHYSICS INTERNAL REPORT SERIES HIP-2006-08 Formation of Structure in Dark Energy Cosmologies Tomi Sebastian Koivisto Helsinki Institute of Physics, and Division of Theoretical Physics, Department of Physical Sciences Faculty of Science University of Helsinki P.O. Box 64, FIN-00014 University of Helsinki Finland ACADEMIC DISSERTATION To be presented for public criticism, with the permission of the Faculty of Science of the University of Helsinki, in Auditorium CK112 at Exactum, Gustaf H¨allstr¨omin katu 2, on November 17, 2006, at 2 p.m.. Helsinki 2006 ISBN 952-10-2360-9 (printed version) ISSN 1455-0563 Helsinki 2006 Yliopistopaino ISBN 952-10-2961-7 (pdf version) http://ethesis.helsinki.fi Helsinki 2006 Helsingin yliopiston verkkojulkaisut Contents Abstract vii Acknowledgements viii List of publications ix 1 Introduction 1 1.1Darkenergy:observationsandtheories..................... 1 1.2Structureandcontentsofthethesis...................... 6 2Gravity 8 2.1Generalrelativisticdescriptionoftheuniverse................. 8 2.2Extensionsofgeneralrelativity......................... 10 2.2.1 Conformalframes............................ 13 2.3ThePalatinivariation.............................. 15 2.3.1 Noethervariationoftheaction..................... 17 2.3.2 Conformalandgeodesicstructure.................... 18 3 Cosmology 21 3.1Thecontentsoftheuniverse........................... 21 3.1.1 Darkmatter............................... 22 3.1.2 Thecosmologicalconstant........................ 23 3.2Alternativeexplanations............................ -
General Relativity at the Turn of the Third Millennium
General Relativity at the turn of the third millennium Lorenzo Fatibene and Mauro Francaviglia Abstract. We shall review the development of the notion of SpaceTime and new challenges of theories of gravitation in the third millennium. We shall stress the fundations of GR in the geometry and physics of XIX Cen- tury and motivations leading to the Einstein paradigm. Then the math- ematical structure of standard GR is briefly reviewed and some possible generalizations are considered. In particular the so-called alternative the- ories of gravitation are considered. Their connection to dark matter/dark energy problem is presented. Finally we shall briefly review some open problems of gravitational theories which lead to motivations for quantum SpaceTimes and gravity. M.S.C. 2000: 83-XX, 83D05. Key words: General Relativity; structure of SpaceTime. 1 Introduction and motivation General Relativity is almost hundred years old. In spite of time passing by and in spite of the many further and important developments that Theoretical and Mathematical Physics had in XX Century (e.g., Quantum Mechanics, Quantum Field Theory, Gauge Theories, Strings and Superstrings) this beautiful theory developed by Einstein in 1915 to describe the gravitational ¯eld in terms of the curvature of SpaceTime is still fresh and alive and its paradigms still represent the basis for most attempts to \geometrize Physics". The keynote Lecture at DGDS 2009 upon which this review paper is grounded aimed to reviewing the mathematical and physical ideas that led Einstein to formulate General Relativity (GR) - starting from Newton's and Galilei Physics and from the Theory of Special Relativity (SR), that he did already develop in 1905 | together with the reasons why GR has been and it is still so important for the development of whole branches of mathematical research, from Di®erential and Riemannian Geometry to the theory of Ordinary and Partial Di®erential Equations (ODE's and PDE's), from the Calculus of Variations on Jet Bundles to the theory of topological invariants (characteristic classes, knots, . -
The D'alembert Paradox, 1900
1 The d’Alembert Paradox, 1900-1914: Levi-Civita and his Italian and French Followers Rossana Tazzioli Université de Lille 1, Sciences et Technologies UFR de Mathématiques, Laboratoire Paul Painlevé Cité Scientifique 59655 Villeneuve d’Ascq Cedex (France) Tel. 06 29 12 33 50 e-mail address: [email protected] Abstract Before the First World War, Tullio Levi-Civita (1873-1941) was already a well-known mathematician in ItalY and abroad, in particular in France. Professor at the UniversitY of Padua since 1898, he had published important contributions to tensor calculus, theory of relativity, differential GeometrY, hYdrodYnamics, and the three-bodY problem. In 1918, when he moved to the University of Rome, he engaged himself in creatinG an international school of mathematics. In this paper, we focus on d’Alembert’s paradox to which Levi- Civita and some of his Italian and French followers contributed remarkable solutions. This case-study is used to illustrate Levi-Civita’s approach to hYdrodYnamics and its influence in ItalY and France, especially in the period 1910-1914. Key-words Tullio Levi-Civita, Henri Villat, d’Alembert’s paradox, hYdrodYnamics 1. Introduction Since ancient times it had been known that a certain resistance is opposed to a bodY movinG in a fluid. However, in 1768 Jean le Rond d’Alembert published a treatise entitled “Paradoxe proposé aux Géomètres sur la résistance des fluides” in which for incompressible and inviscid potential flow he obtained the result of zero drag on head-tail sYmmetric two-dimensional bodies movinG with constant velocity relative to the fluid.1 The contradiction to experimental 1 See [1] Appendix A for a modern statement of the d’Alembert paradox. -
The Eyes of French Mathematicians on Tullio Levi-Civita - the Case of Hydrodynamics (1900-1930) Rossana Tazzioli
The eyes of French mathematicians on Tullio Levi-Civita - the case of hydrodynamics (1900-1930) Rossana Tazzioli To cite this version: Rossana Tazzioli. The eyes of French mathematicians on Tullio Levi-Civita - the case of hydrodynamics (1900-1930). Images of Italian Mathematics in France: The Latin Sisters, from Risorgimento to Fascism, 2016. hal-01436973 HAL Id: hal-01436973 https://hal.archives-ouvertes.fr/hal-01436973 Submitted on 22 Jan 2017 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. The eyes of French mathematicians on Tullio Levi-Civita – the case of hydrodynamics (1900-1930) Rossana Tazzioli Laboratoire Paul Painlevé Université de Lille 1, Sciences et Technologies 1. Introduction Tullio Levi-Civita (1871-1943), professor at the University of Rome from 1918 onwards, was a prominent Italian mathematician of the first part of the 20th century. He gave remarkable contributions to various mathematical fields, such as general relativity, the three-body problem, differential geometry, and hydrodynamics. French scholars generally appreciated Levi-Civita’s work – for instance, his work on differential geometry influenced Élie Cartan and his school, and his papers on the regularization of the three-body were essential for Jean Chazy’s research concerning the long-term behaviour of the solutions of the three-body problem. -
From Dark Energy and Dark Matter to Dark Metric
From Dark Energy & Dark Matter to Dark Metric S. Capozziello∗, M. De Laurentis♭, M. Francaviglia†,‡, S. Mercadante†, ∗ Dipartimento di Scienze Fisiche, Universit`adi Napoli ”Federico II”, and INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 - Napoli, Italy ♭ Dipartimento di Fisica, Politecnico di Torino and INFN Sez. di Torino, Corso Duca degli Abruzzi 24, I-10129 - Torino, Italy † Dipartimento di Matematica, Universit`adi Torino, and INFN Sez. di Torino, Via Carlo Alberto 10, I-10123 - Torino, Italy ‡ Laboratorio per la Comunicazione Scientifica dell’Universit`adella Calabria, Ponte Bucci, Cubo 30b, I-87036 - Arcavacata di Rende (CS), Italy. Abstract: It is nowadays clear that General Relativity cannot be the definitive theory of Gravitation due to several shortcomings that come out both from theoretical and experimental viewpoints. At large scales (astrophysical and cosmological) the attempts to match it with the latest observational data lead to invoke Dark Energy and Dark Matter as the bulk components of the cosmic fluid. Since no final evidence, at fundamental level, exists for such ingredients, it is clear that General Relativity presents shortcomings at infrared scales. On the other hand, the attempts to formulate more general theories than Einstein’s one give rise to mathematical difficulties that need workarounds that, in turn, generate problems from the interpretative viewpoint. We present here a completely new approach to the mathematical objects in terms of which a theory of Gravitation may be written in a first-order (`ala Palatini) formalism, and introduce the concept of Dark Metric which could completely bypass the introduction of disturbing concepts as Dark Energy and Dark Matter. -
Fermions in Curved Spacetimes
fermions in curved spacetimes Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von M. Sc. Stefan Lippoldt geboren am 31.07.1990 in Jena Gutachter 1. Prof. Dr. Holger Gies (Friedrich-Schiller-Universität Jena) 2. Prof. Dr. Martin Reuter (Johannes Gutenberg-Universität Mainz) 3. Prof. Dr. Roberto Percacci (SISSA, Trieste) Tag der Disputation: 21. Januar 2016 Zusammenfassung In dieser Promotionsschrift untersuchen wir eine Formulierung von Dirac-Fermionen in gekrüm- mten Raumzeiten, die sowohl eine allgemeine Koordinateninvarianz, sowie eine lokale Spinbasen- Invarianz besitzt. Wir beleuchten die Vorteile der Spinbasen-invarianten Formulierung aus konzeptioneller und praktischer Sicht. Dies legt nahe, dass die lokale Spinbasen-Invarianz auf die Liste der (effektiven) Eigenschaften der (Quanten-) Gravitation gehört. Wir finden Argumente für diesen Standpunkt durch die Konstruktion einer globalen Realisierung der Clifford-Algebra auf der 2-Sphäre, eine Struktur die es im (nicht Spinbasen-invarianten) Vielbein-Formalismus nicht gibt. Die natürlichen Variablen für diese Formulierung sind die raumzeitabhängigen Dirac-Matrizen, welche die Clifford-Algebra erfüllen müssen. Insbesondere benötigt man kein “Achsenkreuz”, d. h. ein Vielbein. Wir decken die versteckte Spinbasen-Invarianz des Vielbein- Formalismus auf. Explizite Formeln für den Spin-Zusammenhang in Abhängigkeit der Dirac- Matrizen werden angegeben. Dieser Zusammenhang besteht aus einem kanonischen Teil, der vollständig durch die Dirac-Matrizen fixiert ist, und einem freien Teil, der als Spin-Torsion in- terpretiert werden kann. Die übliche Lorentz-symmetrische Eichung für das Vielbein wird auf die Dirac-Matrizen verallgemeinert, auch für Metriken die nicht linear verbunden sind. Nach bestimmten Kriterien bildet sie die einfachste Eichung – das zeigt warum diese Eichung so nützlich ist. -
Tullio Levi Civita
Tullio Levi Civita Tullio Levi Civita (Source: http://en.wikipedia.org/wiki/Tullio_Levi -Civita ) Born: 29 March 1873, Padua, Italy Died: 29 December 1941 (aged 68) Rome, Italy Tullio Levi-Civita was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics. Biography Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Mechanics where he met and, in 1914, married Libera Trevisani, one of his pupils. He remained in his position at Padua until 1918, when he was appointed to the Chair of Higher Analysis at the University of Rome; in another two years he was appointed to the Chair of Mechanics there. In 1900 he and Ricci-Curbastro published the theory of tensors in Méthodes de calcul différentiel absolu et leurs applications, which Albert Einstein used as a resource to master the tensor calculus, a critical tool in the development of the theory of general relativity.