Wolfgang Nolting Theoretical Physics 4 Special Theory of Relativity Theoretical Physics 4 Wolfgang Nolting Theoretical Physics 4 Special Theory of Relativity 123 WolfgangNolting Inst.Physik Humboldt-UniversitaRtzuBerlin Berlin,Germany ISBN978-3-319-44370-6 ISBN978-3-319-44371-3 (eBook) DOI10.1007/978-3-319-44371-3 LibraryofCongressControlNumber:2016954193 ©SpringerInternationalPublishingSwitzerland2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland General Preface The nine volumes of the series Basic Course: Theoretical Physics are thought to be textbookmaterials for the study of university-levelphysics. They are aimed to impart, in a compact form, the most importantskills of theoretical physics which can be used as basis for handling more sophisticated topics and problems in the advanced study of physics as well as in the subsequent physics research. The conceptualdesignofthepresentationisorganizedinsuchawaythat ClassicalMechanics(Vol.1) AnalyticalMechanics(Vol.2) Electrodynamics(Vol.3) SpecialTheoryofRelativity(Vol.4) Thermodynamics(Vol.5) are considered as the theory part of an integrated course of experimental and theoretical physics as is being offered at many universities starting from the first semester.Therefore,thepresentationisconsciouslychosentobeveryelaborateand self-contained,sometimessurelyatthe costofcertainelegance,sothatthecourse is suitableevenforself-study,at firstwithoutanyneedof secondaryliterature.At anystage,nomaterialisusedwhichhasnotbeendealtwithearlierinthetext.This holds in particular for the mathematical tools, which have been comprehensively developed starting from the school level, of course more or less in the form of recipes,suchthatrightfromthebeginningofthestudy,onecansolveproblemsin theoreticalphysics. The mathematicalinsertionsare always then pluggedin when theybecomeindispensabletoproceedfurtherintheprogramoftheoreticalphysics. It goes without saying that in such a context, not all the mathematical statements canbeprovedandderivedwithabsoluterigor.Instead,sometimesareferencemust be made to an appropriate course in mathematics or to an advanced textbook in mathematics. Nevertheless, I have tried for a reasonably balanced representation so that the mathematical tools are not only applicable but also appear at least “plausible.” v vi GeneralPreface Themathematicalinterludesareofcoursenecessaryonlyinthefirstvolumesof thisseries,whichincorporatemoreorlessthematerialofabachelorprogram.Inthe secondpartoftheserieswhichcomprisesthemodernaspectsoftheoreticalphysics, QuantumMechanics:Basics(Vol.6) QuantumMechanics:MethodsandApplications(Vol.7) StatisticalPhysics(Vol.8) Many-BodyTheory(Vol.9) mathematical insertions are no longer necessary. This is partly because, by the time one comes to this stage, the obligatory mathematics courses one has to take in order to study physics would have provided the required tools. The fact that training in theory has already started in the first semester itself permits inclusion ofpartsofquantummechanicsandstatisticalphysicsinthebachelorprogramitself. Itisclearthatthecontentofthelastthreevolumescannotbepartofanintegrated coursebutratherthesubjectmatterofpuretheorylectures.Thisholdsinparticular for many-body theory which is offered, sometimes, under different names, e.g., advancedquantummechanics,intheeighthorsosemesterofstudy.Inthispart,new methodsandconceptsbeyondbasicstudiesareintroducedanddiscussedwhichare developedinparticularforcorrelatedmany-particlesystemswhichinthemeantime havebecomeindispensablefora studentpursuingmaster’sora higherdegreeand forbeingabletoreadcurrentresearchliterature. In all the volumes of the series Theoretical Physics, numerous exercises are included to deepen the understanding and to help correctly apply the abstractly acquired knowledge. It is obligatory for a student to attempt on his own to adapt and apply the abstract concepts of theoretical physics to solve realistic problems. Detailedsolutionstotheexercisesaregivenattheendofeachvolume.Theideais tohelpastudenttoovercomeanydifficultyataparticularstepofthesolutionorto checkone’sowneffort.Importantly,thesesolutionsshouldnotseducethestudentto followtheeasywayoutasasubstituteforhisowneffort.Attheendofeachbigger chapter,Ihaveaddedself-examinationquestionswhichshallserveasaself-testand maybeusefulwhilepreparingforexaminations. I should not forget to thank all the people who have contributed one way or another to the success of the book series. The single volumes arose mainly from lectures which I gave at the universities of Münster, Würzburg, Osnabrück, and Berlin (Germany),Valladolid (Spain), and Warangal(India). The interest and constructive criticism of the students provided me the decisive motivation for preparing the rather extensive manuscripts. After the publication of the German version,Ireceivedalotofsuggestionsfromnumerouscolleaguesforimprovement, and this helped to further develop and enhance the concept and the performance of the series. In particular, I appreciate very much the support from Prof. Dr. A. Ramakanth,a long-standingscientific partnerand friend,who helpedme in many respects,e.g.,whatconcernsthecheckingofthetranslationoftheGermantextinto thepresentEnglishversion. GeneralPreface vii SpecialthanksareduetotheSpringercompany,inparticulartoDr.Th.Schneider and his team. I remember many useful motivations and stimulations. I have the feelingthatmybooksarewelltakencareof. Berlin,Germany WolfgangNolting May2016 Preface to Volume 4 ThemaingoalofVol.4(SpecialTheoryofRelativity)remainsexactlythesameas thatofthetotalcourseontheoreticalphysics.Itisthoughttobeanaccompanying textbook material for the study of university-levelphysics. It aims to impart, in a compactform,themostimportantskillsoftheoreticalphysicswhichcanbeusedas basisforhandlingmoresophisticatedtopicsandproblemsintheadvancedstudyof physicsaswellasinthesubsequentphysicsresearch.Itispresentedinsuchaway thatitenablesself-studywithouttheneedforademandingandlaboriousreference to secondaryliterature. For the understandingof the text, it is only presumedthat thereaderhasa goodgraspofwhathasbeenelaboratedin theprecedingVols. 1– 3.Mathematicalinterludesarealwayspresentedinacompactandfunctionalform andarepracticedwhentheyappearindispensableforthefurtherdevelopmentofthe theory.Suchmathematicalinsertions,though,areofcoursebecomingdecreasingly necessarywithincreasingvolumenumber.Forthewholetext,itistruethatIhadto focusontheessentials,presentingtheminadetailedandelaborateform,sometimes consciouslysacrificingcertainelegance.Itgoeswithoutsayingthatafterthebasic course, secondaryliterature is needed to deepenthe understandingof physicsand mathematics. The special theory of relativity belongs to the classical theories and is thus advisably taught immediately after classical mechanics (Vols. 1 and 2) and elec- trodynamics (Vol. 3). Due to this reason, the special theory of relativity, with its relativistic extension of classical mechanics and especially of electrodynamics, is presented as Vol. 4 in the underlying textbook series. The mathematically demanding nature of electrodynamics and therewith also of special theory of relativity makes practicing the application of concepts and methods especially mandatory.Inthiscontext,theexerciseswhichareofferedtoeachofthesubsections playanindispensableroleforeffectivelearning.Theelaboratesolutionsofexercises attheendofthebookshouldnotkeepthelearnerfromattemptinganindependent treatmentoftheproblemsbutshouldonlyserveasacheckupofone’sownefforts. The special theory of relativity discussed in Vol. 4 deals with the dependence of physical statements on the reference system of the observer. Important in this connectionaretheinertialsystemsforwhichNewton’slawofinertiaisvalidwithout ix x PrefacetoVolume4 anycontributionofpseudoforces.AccordingtoEinstein’sprincipleofequivalence, inertialsystems basicallyare allon anequalfooting.However,theyare nolonger transformed into each other by the Galilean transformation known from nonrela- tivistic mechanicsbut rather by the yet to be derivedLorentz transformation.The most fundamental consequence of Lorentz transformation consists in an intimate entanglement of space and time coordinates, from which a series of spectacular phenomenaevolve. Some of them seem to even contradict the so-called common sense. Termssuch as space, time, simultaneity, etc., mustbe thoroughlyreconsid- ered.Einstein’ssecondpostulatestatesthatthevelocityoflightinthevacuumatall spacepointsonceandforallisconstantandinparticulariscompletelyindependent ofthekindofmotionperformedbythelightsource.Fromthispostulate,thespecial formofthedecisiveLorentztransformationmatrixcanbederived.Thekeyissueof thespecialtheoryofrelativityconsistsinverifyingthephysicallawsandthecogent conclusions of mechanics and electrodynamics regarding their compatibility with respect to the Lorentz transformation between inertial systems. Deviations of the relativisticallycorrectmechanicsfromthe“familiar”Newtonianmechanicsbecome evident above all when the relative velocities of physical systems are comparable to the velocity of light. The special theory of relativity leads, therefore, in this sense,tosomethinglikeasuperordinatemechanicsthatcontainsthenonrelativistic formulationasthelimitingcaseforsmallrelativevelocities. ThisvolumeonthespecialtheoryofrelativityarosefromlecturesIgaveatthe Germanuniversitiesin Münsterand Berlin.The animatinginterestof thestudents inmylecturenoteshasinducedmetopreparethetextwithspecialcare.Thepresent oneaswellastheothervolumesisthoughttobethetextbookmaterialforthestudy ofbasicphysics,primarilyintendedforstudentsratherthanforteachers. I am thankful to the Springer company, especially to Dr. Th. Schneider, for acceptingandsupportingtheconceptofmyproposal.Thecollaborationwasalways delightfulandveryprofessional.Adecisivecontributiontothebookwasprovided by Prof. Dr. A. Ramakanth from the Kakatiya Universityof Warangal(India).He deservesalotofthanks! Berlin,Germany WolfgangNolting May2016 Contents 1 BasicPhysicalPrinciples.................................................... 1 1.1 InertialSystems......................................................... 3 1.2 Michelson-MorleyExperiment......................................... 5 1.3 Einstein’sPostulates .................................................... 9 1.4 LorentzTransformation................................................. 11 1.4.1 MatrixofLorentzTransformation............................. 11 1.4.2 RelativityofSimultaneity...................................... 16 1.4.3 TimeDilatation ................................................. 17 1.4.4 ContractionofLength.......................................... 19 1.4.5 AdditionTheoremofVelocities................................ 20 1.5 LightCone,MinkowskiDiagram ...................................... 23 1.6 Exercises................................................................. 27 1.7 Self-ExaminationQuestions............................................ 30 2 CovariantFour-DimensionalRepresentations............................ 33 2.1 CovariantandContravariantTensors................................... 33 2.1.1 Definitions ...................................................... 33 2.1.2 CalculationRules............................................... 38 2.1.3 DifferentialOperators .......................................... 41 2.2 CovariantFormulationofClassicalMechanics........................ 42 2.2.1 ProperTime,World-Velocity .................................. 42 2.2.2 Force,Momentum,Energy..................................... 43 2.2.3 ElasticCollision ................................................ 50 2.3 CovarianceofElectrodynamics ........................................ 61 2.3.1 ContinuityEquation ............................................ 62 2.3.2 ElectromagneticPotentials..................................... 63 2.3.3 Field-StrengthTensors.......................................... 65 2.3.4 MaxwellEquations............................................. 67 2.3.5 TransformationoftheElectromagneticFields ................ 71 2.3.6 LorentzForce ................................................... 78 2.3.7 FormulaeofRelativisticElectrodynamics..................... 81 xi