Undergraduate Lecture Notes in Physics Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions Second Edition Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts coveringtopicsthroughoutpureandappliedphysics.Eachtitleintheseriesissuit- ableasabasisforundergraduateinstruction,typicallycontainingpracticeproblems, workedexamples,chaptersummaries,andsuggestionsforfurtherreading. ULNPtitlesmustprovideatleastoneofthefollowing: • Anexceptionallyclearandconcisetreatmentofastandardundergraduatesubject. • Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standard subject. • Anovelperspectiveoranunusualapproachtoteachingasubject. ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysics teachingattheundergraduatelevel. The purpose of ULNP is toprovide intriguing, absorbing books that willcontinue tobethereader’spreferredreferencethroughouttheiracademiccareer. SeriesEditors NeilAshby UniversityofColorado,Boulder,CO,USA WilliamBrantley DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA MatthewDeady PhysicsProgram,BardCollege,Annandale-on-Hudson,NY,USA MichaelFowler DepartmentofPhysics,UniversityofVirginia,Charlottesville,VA,USA MortenHjorth-Jensen DepartmentofPhysics,UniversityofOslo,Oslo,Norway MichaelInglis Department of Physical Sciences, SUNY Suffolk County Community College, Selden,NY,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/8917 Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions Second Edition MichaelTsamparlis FacultyofPhysics DepartmentofAstronomy– Astrophysics–Mechanics UniversityofAthens Athens,Greece ISSN2192-4791 ISSN2192-4805 (electronic) UndergraduateLectureNotesinPhysics ISBN978-3-030-27346-0 ISBN978-3-030-27347-7 (eBook) https://doi.org/10.1007/978-3-030-27347-7 1stedition:©Springer-VerlagBerlinHeidelberg2010 2ndedition:©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Omniameamecumfero WhateverIpossess,Icarrywithme Preface to the First Edition WritinganewbookontheclassicsubjectofSpecialRelativity,onwhichnumerous important physicists have contributed and many books have already been written, canbelikeaddinganotherepicycletothePtolemaiccosmology.Furthermore,itis our belief that if a book has no new elements, but simply repeats what is written in the existing literature, perhaps with a different style, then this is not enough to justify its publication. However, after having spent a number of years, both in classandresearchwithRelativity,Ihavecometotheconclusionthatthereexistsa placeforanewbook.Sinceitappearsthatsomewherealongtheway,mathematics may have obscured and prevailed to the degree that we tend to teach Relativity (andIbelieve,TheoreticalPhysics)simplyusing‘heavier’mathematicswithoutthe inspiration and the mastery of the classic physicists of the last century. Moreover current trends encourage the application of techniques in producing quick results and not tedious conceptual approaches resulting in long lasting reasoning. On the otherhand,Physicscannotbedoneálacartestrippedfromphilosophy,or,toputit inasimplebutdramaticcontext: Abuildingisnotanaccumulationofstones! As a result of the above, a major aim in the writing of this book has been the distinction between the mathematics of Minkowski space and the Physics of Relativity. This is necessary for one to understand the Physics of the theory and not stay with the geometry, which by itself is a very elegant and attractive tool. Thereforeinthefirstchapterwedevelopthemathematicsneededforthestatement and development of the theory. The approach is limited and concise but sufficient for the purposes it is supposed to serve. Having finished with the mathematical conceptswecontinuewiththefoundationofthephysicaltheory.Chapter2setsthe frameworkonthescopeandthestructureofatheoryofPhysics.Weintroducethe PrincipleofRelativityandtheCovariancePrinciple,bothPrinciplesbeingkeystones ineverytheoryofPhysics.Subsequentlyweapplythescenariofirstlytoformulate NewtonianPhysics(Chap.3)andthenSpecialRelativity(Chap.4).Theformulation of Newtonian Physics is done in a relativistic way, in order to prepare the ground foraproperunderstandingoftheparallelformulationofSpecialRelativity. vii viii PrefacetotheFirstEdition Having founded the theory we continue with its application. The approach is systematic in the sense that we develop the theory by means of a stepwise introductionofnewphysicalquantities.SpecialRelativitybeingaKinematictheory forces us to consider as the fundamental quantity the position four-vector. This is doneinChap.5wherewedefinetherelativisticmeasurementofthepositionfour- vectorbymeansoftheprocessofchronometry.TorelatethetheorywithNewtonian reality,weintroducerules,whichidentifyNewtonianspaceandNewtoniantimein SpecialRelativity. In Chaps.6 and 7 we introduce the remaining elements of kinematics, that is, the four-velocity and the four-acceleration. We discuss the well-known relativistic composition law for the 3-velocities and show that it is equivalent to the Einstein Relativity Principle, that is, the Lorentz transformation. In the chapter of four- acceleration we introduce the concept of synchronization which is a key concept in the relativistic description of motion. Finally, we discuss the phenomenon of acceleration redshift which together with some other applications of four- acceleration show that here the limits of Special Relativity are reached and one mustgoovertoGeneralRelativity. After the presentation of kinematics, in Chap.8 we discuss various paradoxes, whichplayanimportantroleinthephysicalunderstandingofthetheory.Wechoose to present paradoxes which are not well known, as for example, it is the twin paradox. InChap.9weintroducethe(relativistic)massandthefour-momentumbymeans ofwhichwedistinguishtheparticlesinmassiveparticlesandluxons(photons). Chapter10isthemostusefulchapterofthisbook,becauseitconcernsrelativistic reactions,wheretheuseofSpecialRelativityisindispensible.Thischaptercontains manyexamplesinordertofamiliarizethestudentwithatool,thatwillbenecessary toothermajorcoursessuchasparticlePhysics,HighEnergyPhysicsetc. InChap.11wecommencethedynamicsofSpecialRelativitybytheintroduction of the four-force. We discuss many practical problems and use the tetrahedron of Frenet – Serret to compute the generic form of the four-force. We show how the wellknownfour-forcescomplywiththegenericform. In Chap.12 we introduce the concept of covariant decomposition of a tensor along a vector and give the basic results concerning the 1 + 3 decomposition in Minkowski space. The mathematics of this chapter is necessary in order to understand properly the relativistic Physics. It is used extensively in General Relativity but up to now we have not seen its explicit appearance in Special Relativity, even though it is a powerful and natural tool both for the theory and theapplications. Chapter13isthenextpillarofSpecialRelativity,thatis,electromagnetism.We presentinaconcisewaythestandardvectorformofelectromagnetismandsubse- quently are led to the four formalism formulation as a natural consequence. After discussingthestandardmaterialonthesubject(four-potential,electromagneticfield tensoretc)wecontinuewithlesserknownmaterial,suchasthetensorformulation of Ohm’s law and the 1 + 3 decomposition of Maxwell’s equations. The reason we introduce these more advanced topics, is that we wish to prepare the student PrefacetotheFirstEdition ix for courses on important subjects such as, Relativistic Magnetohydrodynamics (RMHD). Therestofthebookconcernstopicswhich,toourknowledge,cannotbefound intheexistingbooksonSpecialRelativityyet.InChap.14wediscusstheconcept ofspinasanaturalresultofthegeneralizationoftheangularmomentumtensorin SpecialRelativity.Wefollowaformalmathematicalprocedure,whichrevealswhat ‘the spin is’ without the use of the Quantum Field Theory. As an application, we discussthemotionofachargedparticlewithspininahomogeneouselectromagnetic fieldandrecoverthewellknownresultsintheliterature. Chapter15dealswiththecovariantLorentztransformation,aformwhichisnot widely known. All four types of Lorentz transformation are produced in covariant formandtheresultsareappliedtoapplicationsinvolvingthegeometryof3-velocity space,thecompositionofLorentztransformationsetc. Finally,inChap.16westudythereactionA+B −→C+Dinafullycovariant form.Theresultsaregenericandcanbeusedtodevelopsoftwarewhichwillsolve suchreactionsdirectly,providedoneintroducestherightdata. The book includes numerous exercises and solved problems, plenty of which supplement the theory and can be useful to the reader on many occasions. In addition,alargenumberofproblems,carefullyclassifiedinalltopicscovered,are includedintheCDaccompanyingthebook. The above does not cover all topics we would like to consider. One such topic is relativistic waves, which leads to the introduction of De Broglie waves and subsequentlytothefoundationofQuantumMechanics.Asecondtopicisrelativistic hydrodynamics and its extension to RMHD. However, one has to draw a line somewhereandleavethefuturetotakecareofthingstobedone. Looking back at the long hours over the many years which were necessary for the preparation of this book, I cannot help feeling that, perhaps, I should not have undertaken the project. However, I feel that would be unfair to all the students and colleagues, who for more than thirty years have helped me to understand and developtherelativisticideas,tofindandsolveproblemsandingeneraltokeepmy interestalive.Thereforethepresentbookisacollectiveworkandmyrolehasbeen simplytocompiletheseexperiences.Idonotmentionspecificnames-thelistwould betoolong,andIwillcertainlyforgetquiteafew-buttheyknowandIknow,and thatisenough. Iclosethispreface,withanapologytomyfamilyforthelongworkinghoursI was kept away from them in writing this book and I would like to thank them for theircontinuoussupportandunderstanding. Athens,Greece MichaelTsamparlis 4July2009 Preface to the Second Edition Elevenyearshavepassedbysincethefirsteditionofthisbook.Duringthisperiodof time,manythingshappenedwhichhavechangedthewayweapproachknowledge and books. My generation had a sensory contact with the book. We could see it, touchit,thumbthroughitspages,andcertainlykeepitinourlibraryformanyyears. Thanks to technology, however, the book today is a file stored in a digital device. Thewin-wincaseappearstobeabusinesssituationonly. Allthistechnologicaldevelopmenthasn’ttouchedtotheleastthemaintenetsof thetheoryofrelativitywhichbynowaremorethanacenturyold.Thesetenetsare theobjectofthepresentbook. Anewchapter,number16,hasbeenincluded,whichdealswiththeuseofnull triads in the definition of tetrads and the proper Lorentz transformation. The aim ofthischapteristoshowanotherapproachtothecovariantLorentztransformation basedonnullratherthanontimelikevectors.Furthermore,theuseofthistypeof tetradshasnumerousapplicationsintherelativisticwaves,inRMHD,andinother topics. Intheprefaceofthefirstedition,therewasaremarkabouttheexclusionofthe topicsofrelativisticwavesandrelativisticfluidsfromthebook.Itwasemphasized that these topics are necessary for the reader to fully comprehend the Schrödinger equation (hence Quantum Mechanics) and the gravitational field equations and, consequently,GeneralRelativityandCosmology. The present edition covers these two key topics in Chaps.18, 19, 20, 21, and 22, respectively. The need for writing extensively on relativistic fluids is that they are indispensable in General Relativity, their understanding assumed but rarely provided. In our approach, the theory of relativistic fluids follows as a natural extensionthetheoryofNewtonianfluidsdevelopedingeometriclanguage. As it has been the approach in the first edition of the book, a new area of Special Relativity requires the introduction of a new relativistic physical quantity andconsequentlytheintroductionofanewLorentztensor.Forexample,inthecase of waves, the tensor quantity is the four vector of frequency; in the case of the relativisticfluids,itisasecond-ordersymmetrictensorfield,theenergymomentum tensor. xi