# SpecialRelativity

Undergraduate Lecture Notes in Physics Valerio Faraoni Special Relativity Undergraduate Lecture Notes in Physics For further volumes 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. 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Series Editors Neil Ashby Professor, Professor Emeritus, University of Colorado Boulder, Boulder, CO, USA William Brantley Professor, Furman University, Greenville, SC, USA Michael Fowler Professor, University of Virginia, Charlottesville, VA, USA Michael Inglis Professor, SUNY Suffolk County Community College, Selden, NY, USA Elena Sassi Professor, University of Naples Federico II, Naples, Italy Helmy Sherif Professor Emeritus, University of Alberta, Edmonton, AB, Canada Valerio Faraoni Special Relativity 123 Valerio Faraoni Physics Department Bishop’s University Sherbrooke, QC Canada ISSN 2192-4791ISSN 2192-4805electronic ISBN 978-3-319-01106-6ISBN 978-3-319-01107-3eBook DOI 10.1007/978-3-319-01107-3 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number 2013942663 Springer International Publishing Switzerland 2013 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are mpt from the relevant protective laws and regulations and therefore free for general use. While the advice and ination in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer ScienceBusiness Media To my parents Preface Special Relativity arises from the need for a textbook at intermediate level, especially in North-American universities. There are nowadays several techno- logical applications of Special Relativity in everyday life, including the GPS system, PET scanners, and other medical instruments. There are plenty of college- level, deliberately watered-down introductions to Special Relativity. These intro- ductions typically serve a Modern Physics course and devote two chapters to the subject. Such short introductions are inadequate for physicists or scientists requiring a modern university course. What is worse, and few teachers will deny that, fundamental aspects of Special Relativity are usually not understood by the students and often not presented by the teachers. The four-dimensional world view is usually introduced only in a very colloquial way while the mathematical for- malism of four-vectors and four-tensors, which is necessary to understand and alize the idea of spacetime, is ignored. Yet, the latter is not so complicated and is accessible to students who have taken a linear algebra course provided that room is made in the relativity course for it. For most students the Modern Physics course is the only introduction to Special Relativity that they receive in their entire curriculum and, as a result, physics and mathematics students the very ones who must know it miss this fundamental subject. This knowledge gap is very serious since Special Relativity is, without doubt, one of the great intellectual achieve- ments of mankind, exactly the kind of stuff that a student gets into physics for. From a teacher’s logistical point of view, it is also relatively simple in comparison with, e.g., General Relativity or particle physics. Yet, too often the science student is robbed of this part of his or her education by systems too bent on entertainment and student counts. An essential part of the physics curriculum is sacrifi ced for fear of mathematics. For students carrying on to a course on General Relativity, perhaps in graduate studies, this problem quickly becomes irrelevant but such students are a minority and physics departments in many small universities do not offer a General Rela- tivity course or offer it infrequently. This situation is too common and it is true that the mathematical background of second- or third-year undergraduates is limited and that the alism of tensors necessary for the study of Special Relativity needs to be taught in the course. One problem of this rationalization of the cur- riculum is the lack of a proper textbook on Special Relativity at a level higher than vii college physics or current Modern Physics courses, yet accessible to undergrad- uates who are not prepared to tackle General Relativity textbooks or Special Relativity classics such as Wolfgang Rindler’s Introduction to Special Relativity. Another complication is that, in an elementary course in Special Relativity, typical undergraduates have not yet taken a serious course in electromagnetism and do not yet master the Maxwell equations. Therefore, the usual textbook approach using Maxwell’s theory as a trampoline for Special Relativity ends up being ineffective for students at this level. In retrospect, although the historical line of thought and the traditional pedagogical approach was electromagnetism fi rst and then Special Relativity, and it is true that Maxwell’s theory receives its most elegant and revealing ulation in Special Relativity, the latter is not just electromagnetism pushed to the extreme this point was stressed early on by Pauli. Although electromagnetism was a useful trampoline to discover relativity, Einstein’s 1905 theory is about the unveiling of the four-dimensional nature of the world, spacetime and its Lorentzian geometry, the equivalence of mass and energy, the modifi ed mechanics, and the fundamental symmetries of this theory. The Maxwell fi eld is only one of the possible s of mass-energy that can live in Minkowski spacetime. This point of view is obvious after one takes a course in General Relativity or particle physics, but it is at odds with the discover-Special- Relativity-through-electromagnetism approach of many textbooks. An axiomatic approach to Special Relativity based on the Principle of Relativity and the con- stancy of the speed of light seems the best option herethe key physical concepts of the theory relativity of simultaneity, time dilation, and length contraction can be derived easily from these two postulates, and this is the path originally followed by Einstein. A constructive approach would need to be based on electromagnetism and the average undergraduate at this point still views the Maxwell equations as rather abstract or has not seen them at all. It is possible, however, to introduce the Lorentz transation as the transation relating inertial frames which leaves electromagnetism invariant without considering explicitly the Maxwell equations. An example showing a simple electromagnetic phenomenon will do, and this is the avenue taken in this book. The Lorentz invariance of the full Maxwell equations can be checked later when the student is familiar with them. The book begins at an elementary level exposing and discussing in Chap. 1 and Chap. 2 all the basic concepts normally contained in college-level expositions, including the Lorentz transation. Then, in Chap. 3, it introduces the student to the four-dimensional world view, making clear that this is implied by the Lorentz transations mixing time and space coordinates. This is as far as the best Modern Physics textbooks seem to get. In addition, we make use of spacetime diagrams already in this part of the book as well as, of course, in the following parts to visualize the relevant discussion. Following this introduction, in Chap. 4, is the part that is avoided in lower-level courses; the alism of tensors. It is my experience, gained by teaching Special Relativity courses in Canada, that once the student is persuaded that space and time do mix and is motivated by the need to understand the four-dimensional world, and once time is made during the course to explain this part, this chapter goes surprisingly easily and the fear of tensors proves viiiPreface unfounded. Chapter 5 then introduces the essential concept of causality missed in the Modern Physics courses and details the application of the general alism of tensors to Minkowski space. The following Chap. 6 discusses the relativistic mechanics of point particles, four-momentum and four-force, the equivalence between mass and energy, and some applications. Chapter 7 describes relativistic optics and, whenever possible, uses the similarities between the motion of mass- less and massive particles to facilitate understanding and memorization. An optional short Chap. 8 follows, in which measurements in Minkowski spacetime are discussed to dispel the widespread impression that physical observables are coordinate components of four-vectors or four-tensors and therefore, that all measurements are based on coordinate-dependent components. Matter in Minkowski spacetime is discussed in Chap. 9. Here the energy- momentum tensor of a continuous distribution of mass-energy and its covariant conservation are introduced, and various optional energy conditions are pre- sented. Angular momentum is discussed briefl y. This part is followed by a dis- cussion of the scalar fi eld presented fi rst, as this is the simplest fi eld in theoretical physics, of perfect fl uids, and of the Maxwell fi eld which is now presented as one of the possible energy distributions in Minkowski space, although as an important one describing one of only four fundamental interactions. For pedagogical reasons, it is easier to work in Cartesian coordinates and all the material introduced thus far with the exception of the tensor alism of Chap. 4, which is quite general is restricted to these coordinates. This restriction facilitates theintroductionofthe ideasandconceptsofSpecialRelativity,butistoorestrictive. At thispoint, Chap. 10 introduces general coordinates, covariantdifferentiation, and geodesics the emphasis is on computational skills rather than rigour or proof and reulates the previous material in arbitrary coordinate systems using covariant ulas. This chapter could be skipped if short of time during a course. The mathematics essential to study Special Relativity is relatively simple some calculus and linear algebra. The real diffi culties are conceptual, not mathematical. The beginning student is fi ghting his or her own physical sense derived by everyday low-speed intuition, which confl icts with the results of Special Rela- tivity. Mathematics is a tool which facilitates understanding and, by banning it from the course, current low-level oversimplifi ed courses preclude the under- standing of the basic concepts of the theory. It is much better to allow room for it in the course, although stripping it down to the essential, and then use it rather than paraphrasing it with obscure and wordy discussions which invariably fail to convey the essence of relativity. Every chapter is supplemented by a section containing practice problems. These rcises constitute an essential part of the textbook and the student is urged to try them. The solution to selected rcises, as well as the numerical answers to others, appears at the end of the book. For pedagogical purposes, in this book we retain explicitly the speed of light c in the ulas, i.e., we do not set c to unity except in spacetime diagrams and we include several steps in the calculations to facilitate comprehension by the beginner. We avoid the obsolete notions of rest mass and dynamical mass, Prefaceix transversal and longitudinal mass which were popular in old textbooks. Sections marked with an asterisk are optional and can be omitted without jeopardizing the understanding of the material which follows. rcises referring to optional sec- tions are also marked with an asterisk. I hope that this book will help many students enjoy the beauty, elegance, and power of Einstein’s theory. Have fun Sherbrooke, Summer 2013Valerio Faraoni xPreface Acknowledgments It is a pleasure to thank all my students who provided feedback and comments over the years. Special thanks go to Andres Zambrano Moreno for typing most of the content of this book. Many thanks go also to Dr. Aldo Rampioni, Springer Editor for Theoretical and Mathematical Physics for his support and friendly assistance during the writing of this book. xi Contents 1Fundamentals of Special Relativity . . . . . . . . . . . . . . . . . . . . . . .1 1.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.2The Principle of Relativity. . . . . . . . . . . . . . . . . . . . . . . . . .2 1.3 HGroups The Galilei Group . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4Galileian Law of Addition of Velocities . . . . . . . . . . . . . . . .8 1.5The Lesson from Electromagnetism . . . . . . . . . . . . . . . . . . .8 1.5.1The Ether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 1.5.2The Michelson–Morley Experiment . . . . . . . . . . . . .11 1.6The Postulates of Special Relativity . . . . . . . . . . . . . . . . . . .13 1.6.1The Role of the Speed of Light . . . . . . . . . . . . . . . .14 1.7Consequences of the Postulates . . . . . . . . . . . . . . . . . . . . . .16 1.7.1Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . .16 1.7.2Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 1.7.3The Twin ‘‘Paradox’’ . . . . . . . . . . . . . . . . . . . . . . .23 1.7.4Length Contraction. . . . . . . . . . . . . . . . . . . . . . . . .23 1.8Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 2The Lorentz Transation. . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.2The Lorentz Transation . . . . . . . . . . . . . . . . . . . . . . . .29 2.3Derivation of the Lorentz Transation . . . . . . . . . . . . . . .31 2.4Mathematical Properties of the Lorentz Transation . . . . .33 2.5Absolute Speed Limit and Causality . . . . . . . . . . . . . . . . . . .36 2.6Length Contraction from the Lorentz Transation . . . . . . .39 2.7Time Dilation from the Lorentz Transation. . . . . . . . . . .40 2.8Transation of Velocities and Accelerations in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 2.8.1Relative Velocity of Two Particles . . . . . . . . . . . . . .43 2.8.2Relativistic Transation Law of Accelerations. . . .45 2.9Matrix Representation of the Lorentz Transation . . . . . . .46 xiii 2.10 HThe Lorentz Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.11The Lorentz Transation as a Rotation by an Imaginary Angle with Imaginary Time. . . . . . . . . . . . .51 2.12 HThe GPS System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.13Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 3The 4-Dimensional World View . . . . . . . . . . . . . . . . . . . . . . . . .59 3.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 3.2The 4-Dimensional World . . . . . . . . . . . . . . . . . . . . . . . . . .60 3.3Spacetime Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 3.4Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 4The alism of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 4.2Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 4.2.1Coordinate Transations. . . . . . . . . . . . . . . . . . .74 4.2.2Einstein Convention . . . . . . . . . . . . . . . . . . . . . . . .75 4.3Contravariant and Covariant Vectors. . . . . . . . . . . . . . . . . . .76 4.4Contravariant and Covariant Tensors . . . . . . . . . . . . . . . . . .79 4.4.1Tensor Symmetries. . . . . . . . . . . . . . . . . . . . . . . . .83 4.5Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 4.6Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 4.7 HIndex-Free Description of Tensors . . . . . . . . . . . . . . . . . . . 90 4.8The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 4.8.1Inverse Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 4.8.2Metric Determinant. . . . . . . . . . . . . . . . . . . . . . . . .101 4.9The Levi-Civita Symbol and Tensor Densities . . . . . . . . . . . .101 4.9.1Properties of the Levi-Civita Symbol in Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . .104 4.9.2Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . .106 4.10Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 5Tensors in Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . .111 5.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 5.2Vectors and Tensors in Minkowski Spacetime . . . . . . . . . . . .111 5.3The Minkowski Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 5.4Scalar Product and Length of a Vector in Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . .117 5.5Raising and Lowering Tensor Indices . . . . . . . . . . . . . . . . . .120 5.5.1Working with Tensors in Minkowski Spacetime. . . . .122 xivContents 5.6Causal Nature of 4-Vectors . . . . . . . . . . . . . . . . . . . . . . . . .123 5.7Hypersurfaces . . . . . . . . . . . . . . .