Special Relativity: An Introduction with 200 Problems and Solutions PDF
Preview Special Relativity: An Introduction with 200 Problems and Solutions
Special Relativity Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions 123 Dr.MichaelTsamparlis DepartmentofAstrophysics,AstronomyandMechanics UniversityofAthens Panepistimiopolis GR15784ZOGRAFOS Athens Greece [email protected] Additionalmaterialtothisbookcanbedownloadedfromhttp://extra.springer.com. Password:978-3-642-03836-5 ISBN978-3-642-03836-5 e-ISBN978-3-642-03837-2 DOI10.1007/978-3-642-03837-2 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2009940408 (cid:2)c Springer-VerlagBerlinHeidelberg2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:eStudioCalamarS.L. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Omniameamecumfero WhateverIpossessIbearwithme Preface 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 place for a new book. Since it appears that somewhere along the way, mathemat- ics 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,physicscannotbedonea´ lacartestrippedfromphilosophy,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 rel- ativity. This is necessary for one to understand the physics of the theory and not staywiththegeometry,whichbyitselfisaveryelegantandattractivetool.There- fore in the first chapter we develop the mathematics needed for the statement and developmentofthetheory.Theapproachislimitedandconcisebutsufficientforthe purposesitissupposedtoserve.Havingfinishedwiththemathematicalconceptswe continue with the foundation of the physical theory. Chapter 2 sets the framework on the scope and the structure of a theory of physics. We introduce the principle ofrelativityandthecovariance principle,bothprinciplesbeingkeystonesinevery theoryofphysics.SubsequentlyweapplythescenariofirsttoformulateNewtonian Physics(Chap.3)andthenSpecialRelativity(Chap.4).TheformulationofNewto- nianPhysicsisdoneinarelativisticway,inordertopreparethegroundforaproper understandingoftheparallelformulationofSpecialRelativity. Havingfoundedthetheorywecontinuewithitsapplication.Theapproachissys- tematicinthesensethatwedevelopthetheorybymeansofastepwiseintroduction vii viii Preface of new physical quantities. Special Relativity being a kinematic theory forces us to consider as the fundamental quantity the position four-vector. This is done in Chap.5wherewedefinetherelativisticmeasurementofthepositionfour-vectorby means of the process of chronometry. To relate the theory with Newtonian reality, weintroducerules,whichidentifyNewtonianspaceandNewtoniantimeinSpecial Relativity. 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 three-velocities and show that it is equivalent to the Ein- steinrelativityprinciple,thatis,theLorentztransformation.Inthechapteroffour- accelerationweintroducetheconceptofsynchronizationwhichisakeyconceptin therelativisticdescriptionofmotion.Finally,wediscussthephenomenonofaccel- eration redshift which together with some other applications of four-acceleration showsthatherethelimitsofSpecialRelativityarereachedandonemustgooverto GeneralRelativity. After the presentation of kinematics, in Chap. 8 we discuss various paradoxes, whichplayanimportantroleinthephysicalunderstandingofthetheory.Wechoose topresentparadoxeswhicharenotwellknown,asforexample,itisthetwinpara- dox. InChap.9weintroducethe(relativistic)massandthefour-momentumbymeans ofwhichwedistinguishtheparticlesinmassiveparticlesandluxons(photons). Chapter10isthemostusefulchapterofthisbook,becauseitconcernsrelativistic reactions,wheretheuseofSpecialRelativityisindispensible.Thischaptercontains manyexamplesinordertofamiliarizethestudentwithatool,thatwillbenecessary toothermajorcoursessuchasparticlephysicsandhighenergyphysics. InChap.11wecommencethedynamicsofSpecialRelativitybytheintroduction of the four-force. We discuss many practical problems and use the tetrahedron of Frenet–Serrettocomputethegenericformofthefour-force.Weshowhowthewell- knownfour-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 under- stand 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 thoughitisapowerfulandnaturaltoolbothforthetheoryandtheapplications. Chapter13isthenextpillarofSpecialRelativity,thatis,electromagnetism.We presentinaconcisewaythestandardvectorformofelectromagnetismandsubse- quentlyweareledtothefourformalismformulationasanaturalconsequence.After discussingthestandardmaterialonthesubject(four-potential,electromagneticfield tensor,etc.)wecontinuewithlesserknownmaterial,suchasthetensorformulation ofOhm’slawandthe1+3decompositionofMaxwell’sequations.Thereasonwhy weintroducethesemoreadvancedtopicsisthatwewishtopreparethestudentfor coursesonimportantsubjectssuchasrelativisticmagnetohydrodynamics(RMHD). Therestofthebookconcernstopicswhich,toourknowledge,cannotbefound intheexistingbooksonSpecialRelativityyet.InChap.14wediscusstheconcept Preface ix of spin as a natural result of the generalization of the angular momentum tensor in Special Relativity. We follow a formal mathematical procedure, which reveals what“thespinis”withouttheuseofthequantumfieldtheory.Asanapplication,we discussthemotionofachargedparticlewithspininahomogeneouselectromagnetic fieldandrecoverthewell-knownresultsintheliterature. Chapter15dealswiththecovariantLorentztransformation,aformwhichisnot widelyknown.AllfourtypesofLorentztransformationsareproducedincovariant form and the results are applied to applications involving the geometry of three- velocityspace,thecompositionofLorentztransformations,etc. Finally,inChap.16westudythereaction A+B −→C+Dinafullycovariant form.Theresultsaregenericandcanbeusedtodevelopsoftwarewhichwillsolve suchreactionsdirectly,providedoneintroducestherightdata. The book includes numerous exercises and solved problems, plenty of which supplementthetheoryandcanbeusefultothereaderonmanyoccasions.Inaddi- tion, a large number of problems, carefully classified in all topics accompany the book. The above does not cover all topics we would like to consider. One such topic isrelativisticwaves,whichleadstotheintroductionofDeBrogliewavesandsub- sequently to the foundation of quantum mechanics. A second topic is relativistic hydrodynamicsanditsextensiontoRMHD.However,onehastodrawalinesome- whereandleavethefuturetotakecareofthingstobedone. 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 undertakentheproject.However,Ifeelthatitwouldbeunfairtoallthestudentsand colleagues,whoformorethat30yearshavehelpedmetounderstandanddevelop therelativisticideas,tofindandsolveproblems,andingeneraltokeepmyinterest alive.Thereforethepresentbookisacollectiveworkandmyrolehasbeensimply to compile these experiences. I do not mention specific names – the list would be toolong,andIwillcertainlyforgetquiteafew–buttheyknowandIknow,andthat isenough. I close this preface, with an apology to my family for the long working hours; thatIwaskeptawayfromthemforwritingthisbookandIwouldliketothankthem fortheircontinuoussupportandunderstanding. Athens,Greece MichaelTsamparlis October2009
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