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Alberto Vecchiato from Newton to Einstein and Beyond Undergraduate Lecture Notes in Physics Alberto Vecchiato Variational Approach to Gravity Field Theories From Newton to Einstein and Beyond Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: • An exceptionally clear and concise treatment of a standard undergraduate subject. • A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject. • A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career. Series editors Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Humboldt University, Oldenburg, Niedersachsen, Germany More information about this series at http://www.springer.com/series/8917 Alberto Vecchiato Variational Approach to Gravity Field Theories From Newton to Einstein and Beyond 123 Alberto Vecchiato Astrophysical Observatory of Torino INAF - National Institute of Astrophysics Pino Torinese Italy ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-319-51209-9 ISBN 978-3-319-51211-2 (eBook) DOI 10.1007/978-3-319-51211-2 Library of Congress Control Number: 2016962044 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Gabriella Preface The variational approach is probably the most powerful technique and unifying concept of theoretical physics, but in the same way it is not common to explain it to its full extent to undergraduate students, which later results in a splitting between the method and the languages of theorists and those of “the rest of the world,” at least for what concerns physicists. I think it is important to try to fill in this gap by introducing them as early as possible, thus keeping a common understanding in this discipline. With the notable exception of Lagrangian and Hamiltonian mechanics, there is a consolidated teaching tradition in undergraduate physics courses in which there is little room for a complete exposition of variational techniques. This is indeed for good reason. Classical physics can be explicated in an extremely efficient way without resorting to these relatively advanced methods, which on the contrary can appear too abstract in such an elementary context. However, there is a price to pay with this approach. At some point the student comes to more advanced subjects such as quantum or relativistic physics, where such techniques are extremely useful, if not necessary. The risk, then, is that these are perceived as completely detached from the familiar background, and accepted without a real understanding. In this way the nontheoretical physicist will quickly forget this “anomalous event.” This problem might be avoided if, after the regular exposition of classical physics, the same concepts were revised from the point of view of the variational approach. This book undertakes the problem from the point of view of the gravity field theories, trying to introduce the variational approach by stressing its continuity from the classical to the relativistic realm. Such a job can be accomplished only by treating in parallel the evolution of the dynamics along the same theoretical path. Despite its great power, however, the variational approach is ultimately a technique, whereas the essential physical meaning of theories lies on their funda- mental principles. It is for this reason that the exposition tries to highlight as clearly as possible the connection between theories and principles at the very basic mathematical level. vii viii Preface The book is organized in four parts, which follow the basic ideas underlined above. The first one tries to give a gentle introduction to variational principles using Newtonian dynamics and gravity as a case study. In the next chapters, the link between classical physics and Euclidean geometry is analyzed from the point of view of the founding principles. This part concludes with the discovery of the internal inconsistency between electrodynamics and these principles in their clas- sical formulation. The second part starts from the failure of classical physics with respect to the principle of relativity to arrive at an alternative formulation of such a principle. Building upon it to get to special relativity in its Minkowskian formulation, this part ends by observing the infeasibility of a special relativistic theory of gravitation. By analyzing this problem in more detail, the third part deals with general relativity, some of its applications, and how and why this theory could be extended or modified. In this book, I tried to gauge the exposition with a particular emphasis on the physical concepts, which sometimes required momentarily delaying some mathe- matical details. For this reason the appendices contained in the fourth part are not to be considered just as supplementary material. Rather, they carry essential infor- mation needed for the complete understanding of the text, and should be tackled in parallel to the respective chapters, where appropriate references can be found. Finally, this book contains 42 exercises. It is important to solve all of them entirely, because in some cases they include other mathematical details referred to in the main text. Furthermore, this number might not be a mere coincidence, if it has to be the “Answer to the Ultimate Question of Life, the Universe, and Everything,” as somebody argued. I wish to thank my family for their support during the writing of this book, and I am deeply grateful to my wife. Without her continuous encouragement, steady support, and unshakeable patience this book would have never been completed. Torino, Italy Alberto Vecchiato November 2016 Contents Part I Introduction and Classical Physics 1 A Short Introduction to Field Theories and Variational Approach............................................... 3 1.1 What Is a Field Theory: A Naive View ................... 3 1.2 Equations of Motion ................................. 6 1.2.1 Lagrangian Formalism ......................... 6 1.2.2 The Variational Approach....................... 9 1.3 Field Equations and Variational Approach ................. 13 1.3.1 The Euler–Lagrange Equations for the Fields ........ 13 1.3.2 Poisson Equations from Variational Principles ....... 16 1.4 Exercises .......................................... 18 2 The Geometrical Character of Physics Theories................ 23 2.1 A Scientific Theory as a Model ......................... 24 2.2 Geometry and Physics: Tools for Modeling the Reality ....... 26 2.3 When Should a Scientific Theory Be Changed?............. 28 3 Fundamental Principles of Classical Physics................... 29 3.1 Principle of Covariance ............................... 29 3.1.1 Euclidean Space .............................. 30 3.1.2 Euclidean Space and Time ...................... 34 3.1.3 Covariance Revisited .......................... 35 3.1.4 Rotational Covariance.......................... 38 3.2 Principle of (Galilean) Relativity ........................ 43 3.2.1 Principle of Relativity as a “Kinematic” Covariance Principle .................................... 44 3.3 Equivalence Principle................................. 47 3.3.1 Equivalence Principle as a “Dynamic” Covariance Principle .................................... 49 3.4 Exercises .......................................... 51 ix x Contents 4 Classical Physics, Fundamental Principles, and Lagrangian Approach..............................................
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