Undergraduate Lecture Notes in Physics Costas Christodoulides The Special Theory of Relativity Foundations, Theory, Verification, Applications Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuitableasabasisfor undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter summaries, andsuggestions for further reading. ULNP titles mustprovide at least oneof thefollowing: (cid:129) Anexceptionally clear andconcise treatment ofastandard undergraduate subject. (cid:129) Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject. (cid:129) Anovel perspective oranunusual approach toteaching asubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching at theundergraduate level. ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe reader’spreferred reference throughout theiracademic career. Series editors Neil Ashby ProfessorEmeritus, University of Colorado, Boulder, CO,USA William Brantley Professor, FurmanUniversity, Greenville, SC,USA MatthewDeady Professor, BardCollege Physics Program, Annandale-on-Hudson,NY, USA Michael Fowler Professor, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Professor, University of Oslo, Oslo,Norway Michael Inglis Professor, SUNY Suffolk CountyCommunity College, LongIsland, NY,USA Heinz Klose ProfessorEmeritus, Humboldt University Berlin, Germany HelmySherif Professor, University of Alberta, Edmonton, AB,Canada More information about this series at http://www.springer.com/series/8917 Costas Christodoulides The Special Theory of Relativity fi Foundations, Theory, Veri cation, Applications 123 Costas Christodoulides Department ofPhysics, Schoolof Applied MathematicalandPhysical Sciences National Technical University of Athens Athens Greece ThebookisarevisedversionoftheGreekedition.CopyrightforalllanguagesexceptGreek is owned by the author. Original Greek edition I eidiki theoria tis sxetikotitas kai oi efarmoges tis or Η EIΔIKH ΘΕΩΡΙΑ ΤΗΣ ΣΧΕΤΙΚΟΤΗΤΑΣ ΚΑΙ ΟΙ ΕΦΑΡΜΟΓΕΣ ΤΗΣ, ©Tziolas Publications, 2014. ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notesin Physics ISBN978-3-319-25272-8 ISBN978-3-319-25274-2 (eBook) DOI 10.1007/978-3-319-25274-2 LibraryofCongressControlNumber:2015951793 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface The book at hand has developed from the lecture notes used in the teaching of the SpecialTheoryofRelativityformanyyearsbytheauthor,originallytoengineering students and finally to students of Physics and Mathematics at the School of Mathematical and Physical Sciences of the National Technical University of Athens. About half the material contained in the book was covered in 25 hourly lectures during the second semester of the first year of studies. As a textbook, the book has some special characteristics: The proofs of the theorems are given in adequate detail, many figures are used, many examples are used for the comprehension of the theory and many problems are suggested for solution by the reader. The detailed solutions of all the problems are given at the end of the book. However, it must be understood that, for maximum benefit, the reader should really try to solve a problem before resorting to the solution given. Asassistanceinthisdirection,theanswerstoalltheproblemsaregiven,aswellas appropriate hints or suggestions on how to solve the problems. Special attention has been paid to presenting the historical approach of Physics to the Special Theory of Relativity, to its experimental foundation and to the experiments performed in order totest thevalidity ofthetheory.Forthis reason, a large number of experiments are described in detail. The Special Theory of Relativity found applications in a large number of problems,mainlyinPhysics,andanattemptismadetopresentthemostimportant of them, which every physicist should be familiar with. Many of the applications are developed in the Examples and Problems and the reader should pay particular attention to them. In order that the reader should take advantage of the ease of access to the scientificliteraturemadepossiblebytoday’stechnology,referencesaregiventothe original articles and review articles concerning each topic. ThebookgivesafairlycompletepresentationoftheSpecialTheoryofRelativity for a first approach to the subject at the undergraduate university level. A basic course on the Special Theory of Relativity could consist of Chaps. 2, 3, 6 and 9, supplemented by appropriate applications of the theory from Chaps. 4, 5 and 7. v vi Preface Additional,moredemandingtopicsmaybefoundintherestofthechaptersandin the appendices. The mathematics needed is that of the first year of a degree in Physics. Whenever necessary, a brief presentation of additional mathematics is given in the text. Itishopedthatthebookwillbeausefuladditiontotheexistingliteratureonthis fascinating subject. Athens Costas Christodoulides November 2015 Contents 1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Main Landmarks in the Development of the Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 ThePrincipleofRelativityofGalileo.Galileo’sInvariance Hypothesis.TheLawofInertia.InertialFramesofReference. . . 4 1.2.1 The Galilean Transformation . . . . . . . . . . . . . . . . . . 5 1.3 Rømer and the Speed of Light . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Newton’s Laws of Motion. Inertia and Inertial Frames of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 The Invariance of Newton’s Second Law of Motion Under the Galilean Transformation . . . . . . . . . . . . . . 11 1.5 The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1 The Discovery by Bradley of the Aberration of Stellar Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Arago’s Measurements Concerning the Constancy of the Speed of Light from Stars. . . . . . . . . . . . . . . . . . . . . . 18 1.7 Measurements of the Speed of Light in the Laboratory . . . . . . 21 1.7.1 The Possibility of Dependence of the Speed of Light in Vacuum on Its Frequency . . . . . . . . . . . . 23 1.8 Attempts to Measure the Dragging of Aether by Moving Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.8.1 The Experiment of Fizeau . . . . . . . . . . . . . . . . . . . . 24 1.8.2 The Experiment of Hoek . . . . . . . . . . . . . . . . . . . . . 27 1.9 Maxwell’s Equations and the Wave Equation. . . . . . . . . . . . . 28 1.10 The Experiment of Michelson and Morley. . . . . . . . . . . . . . . 30 1.11 The Lorentz-FitzGerald Contraction Hypothesis . . . . . . . . . . . 37 1.12 The Increase of the Mass of the Electron with Speed. . . . . . . . 38 1.13 The Invariance of Maxwell’s Equations and the Lorentz Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.14 The Formulation of the Special Theory of Relativity. . . . . . . . 43 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 vii viii Contents 2 Prolegomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.1 Inertial Frames of Reference. . . . . . . . . . . . . . . . . . . . . . . . . 47 2.1.1 The Earth, the Sun and the Galaxy as Inertial Frames of Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2 The Calibration of a Frame of Reference and the Synchronization of Its Clocks. . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 The Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . . . . 51 2.4 The Relativity of Time and Length. . . . . . . . . . . . . . . . . . . . 53 2.4.1 The Invariance of the Dimensions of a Body Which Are Perpendicular to Its Velocity . . . . . . . . . . 53 2.4.2 The Relativity of Time . . . . . . . . . . . . . . . . . . . . . . 54 2.4.3 The Relativity of Length . . . . . . . . . . . . . . . . . . . . . 56 2.5 The Inevitability of the Special Theory of Relativity . . . . . . . . 58 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 The Lorentz Transformation for the Coordinates of an Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.1 The Contraction of Length. . . . . . . . . . . . . . . . . . . . 69 3.1.2 The Dilation of Time. . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 The Transformation of Velocity . . . . . . . . . . . . . . . . . . . . . . 83 3.2.1 The Transformation of the Components of Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.2 The Transformation of the Magnitude of Velocity. . . . 84 3.2.3 The Transformation of the Lorentz Factor, γ. . . . . . . . 86 3.2.4 Speed c as an Upper Limit for Speeds. . . . . . . . . . . . 87 3.3 The Transformation of Acceleration . . . . . . . . . . . . . . . . . . . 91 3.3.1 Proper Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Applications of Relativistic Kinematics. . . . . . . . . . . . . . . . . . . . . 97 4.1 The ‘Meson’ Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1.1 Experimental Verification of the Dilation of Time with Muon Experiments at CERN. . . . . . . . . 102 4.2 The Apparent Focusing of Fast Charged Particle Beams Due to the Dilation of Time. . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3 The Sagnac Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4 Clocks Moving Around the Earth . . . . . . . . . . . . . . . . . . . . . 109 4.5 The Experiment of Hafele and Keating . . . . . . . . . . . . . . . . . 116 4.6 Einstein’s Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.7 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.8 Motion with a Constant Proper Acceleration. Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.8.1 The Dilation of Time and Journeys in Space . . . . . . . 126 Contents ix 4.9 Two Successive Lorentz Transformations. The Wigner Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.9.1 The Thomas Precession. . . . . . . . . . . . . . . . . . . . . . 133 4.9.2 The Thomas Precession in the Cases of a Planetary or a Stellar System. . . . . . . . . . . . . . . 135 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Optical Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1 The Aberration of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.2 Fizeau’s Experiment. Fresnel’s Aether Dragging Theory . . . . . 141 5.3 The Doppler Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.1 The Longitudinal Doppler Effect. . . . . . . . . . . . . . . . 143 5.3.2 The General Doppler Effect . . . . . . . . . . . . . . . . . . . 146 5.3.3 The Experimental Verification of the Relativistic Terms of the Doppler Effect. . . . . . . . . . . . . . . . . . . 152 5.4 Relativistic Beaming or the Headlight Effect . . . . . . . . . . . . . 155 5.4.1 Synchrotron Radiation or Magnetic Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.5 The Forces Exerted by Light . . . . . . . . . . . . . . . . . . . . . . . . 161 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1 The Definition of Relativistic Momentum. Relativistic Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2 Relativistic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.3 The Relationship Between Momentum and Energy . . . . . . . . . 170 6.3.1 Units of Energy, Mass and Momentum . . . . . . . . . . . 170 6.4 Classical Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.5 Particles with Zero Rest Mass. . . . . . . . . . . . . . . . . . . . . . . . 175 6.6 The Conservation of Momentum and of Energy . . . . . . . . . . . 176 6.7 The Equivalence of Mass and Energy . . . . . . . . . . . . . . . . . . 178 6.7.1 The Validity of the Conservation of Momentum and Energy During the Transmutation of Nuclei and the Annihilation and Creation of Particles . . . . . . 179 6.8 The Transformation of Momentum and Energy. . . . . . . . . . . . 186 6.9 The Zero-Momentum Frame of Reference . . . . . . . . . . . . . . . 189 6.10 The Transformation of the Total Momentum and the Total Energy of a System of Particles . . . . . . . . . . . . 193 6.10.1 The Invariance of the Quantity E2(cid:1)c2P2 for a System of Particles . . . . . . . . . . . . . . . . . . . . . 194 6.11 The Collision of Two Identical Particles . . . . . . . . . . . . . . . . 197 6.11.1 Experimental Test of the Special Theory of Relativity with Colliding Electrons . . . . . . . . . . . . 200 6.12 The Transformation of Force . . . . . . . . . . . . . . . . . . . . . . . . 200