Special Relativity (Princeton University )

SpecialRelativityDavidW.HoggSchoolofNatur[e-mail]1December1997Contents1Principlesofrelativity11.1Whatisaprincipleofrelativity?.......................................11.2Einstein'sprincipleofrelativity........................................21.3TheMichelson-Morleyexperiment......................................31.4The\specialnessofspecialrelativity....................................52Timedilationandlengthcontraction72.1Timedilation..................................................72.2Observingtimedilation............................................82.3Lengthcontraction...............................................92.4Magnitudeoftheeects............................................102.5Experimentalconrmation...........................................103Thegeometryofspacetime133.1Spacetimediagrams..............................................133.2Boosting:changingreferenceframes.....................................133.3The\ladderandbarnparadox........................................153.4Relativityofsimultaneity...........................................163.5Theboosttransformation...........................................163.6Transformingspaceandtimeaxes......................................174TheLorentztransformation194.1Propertimeandtheinvariantinterval....................................194.2DerivationoftheLorentztransformation...................................204.3TheLorentztransformation..........................................204.4Velocityaddition................................................214.5Thetwinparadox................................................225Causalityandtheinterval255.1Theladderandbarnrevisited.........................................255.2Causality....................................................265.3Nothingcantravelfasterthanthespeedoflight..............................266Relativisticmechanics296.1Scalars......................................................296.24-vectors.....................................................296.34-velocity....................................................306.44-momentum,restmassandconservationlaws...............................306.5Collisions....................................................316.6PhotonsandComptonscattering.......................................326.7Masstransportbyphotons..........................................336.8Particleproductionanddecay.........................................346.9Velocityaddition(revisited)andtheDopplershift.............................346.104-force......................................................34i7Opticsandapparenteects:specialrelativityappliedtoastronomy377.1Dopplershift(revisited)............................................377.2StellarAberration...............................................387.3Superluminalmotion..............................................387.4Relativisticbeaming..............................................397.5Theappearanceofpassingobjects......................................407.6Asimplemindedcosmology..........................................40References43Index45iiPrefaceForme,thewonderofspecialrelativityliesinitssuccess-fulpredictionofinterestingandverynonintuitivephe-nomenafromsimpleargumentswithsimplepremises.Thesenoteshavethree(perhapsambitious)aims:(a)tointroduceundergraduatestospecialrelativityfromitsfoundingprincipletoitsvariedconsequences,(b)toserveasareferenceforthoseofuswhoneedtousespe-cialrelativityregularlybuthavenolong-termmemory,and(c)toprovideanillustrationofthemethodsofthe-oreticalphysicsforwhichtheeleganceandsimplicityofspecialrelativityareideallysuited.Historyisapartofallscience|Iwillmentionsomeoftherelevanteventsinthedevelopmentofspecialrelativity|butthereisnoattempttopresentthematerialinahistoricalway.Acommonconfusionforstudentsofspecialrelativityisbetweenthatwhichisrealandthatwhichisappar-ent.Forinstance,lengthcontractionisoftenmistakenlythoughttobesomeopticalillusion.Butmovingthingsdonot\appearshortened,theyactuallyareshortened.Howtheyappeardependsontheparticularsoftheobser-vation,includingdistancetotheobserver,viewingangles,times,etc.Theobserverndsthattheyareshortenedonlyaftercorrectingforthesenon-fundamentaldetailsoftheobservationalprocedure.Iattempttoemphasizethisdistinction:Allapparenteects,includingtheDopplerShift,stellaraberration,andsuperluminalmotion,arerelegatedtoChapter7.Ithinktheseareveryimpor-tantaspectsofspecialrelativity,butfromapedagogicalstandpointitispreferabletoseparatethemfromtheba-sics,whicharenotdependentonthepropertiesoftheobserver.Ilovethedescriptionofspecialrelativityintermsofframe-independent,geometricobjects,suchasscalarsand4-vectors.TheseareintroducedinChapter6andusedthereafter.Butevenbeforethis,thegeometricproper-tiesofspacetimeareemphasized.Mostproblemscanbesolvedwithaminimumofalgebra;thisisoneofthemanybeautifulaspectsofthesubject.Thesenotes,rstwrittenwhileteachingsectionsofrst-yearphysicsatCaltech,trulyrepresentaworkinprogress.Istronglyencouragea