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The Unstoppable Human Species, The Emergence of Homo Sapiens in Prehistory John J. Shea
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ToSarah,Dane,andDaisy
Chapter 6 definesthegeneralLorentztransformationusingmatricesand,with thehelpofAppendix B,extendsthetwo-dimensionalLorentztransformations tothefullfour-dimensionalspacetime.Theconceptof4-vectorsandtheir dotproductareintroducedasarethenotionsof4-velocity,4-acceleration, 4-momentum,4-force,andthesecondlawofmotion.Variousexamplesillustratethepowerof4-vectorsandtheirdotproducts.
Chapter 7 is,asfarasIknow,uniquetothisbook.Itisthearenainwhich thepowerofLorentztransformationandthelimitation,eventheincorrectness,oftheuseoftheconceptsoftimedilationandlengthcontractionis illustrated(seeProblem 7.1 forapopularmistake).Conditionsunderwhich lengthcontractioncanbe photographed areinvestigated,andthereasonwhy ordinaryphotographyofrelativisticallymovingobjectsdoesnotrevealthis lengthcontractioniselaborated.Oneoftheoutstandingresultsofthischapteristhe rigorousproof thatthephotographofarelativisticallymovingsphere inapinholecameraisanellipse elongated alongthedirectionofmotion,and thattheimageofsuchasphereisacircleif(andonlyif)itmovesdirectlytowardorawayfromthecamera.ThisdoesnotcontradicttheresultsofPenrose andTerrell,whodemonstratethatthe cone convergingona pointobserver is circular.Byactuallyconstructingthesecircularcones,thechaptershowsthat, ifthepinholecamerapointsatthecenterofthesphere,theimageformedon itsphotographicplatecannotbecircular.Infact,thechapterprovesinthree differentwaysthattheimageisanellipsestretchedinthedirectionofmotion.
Chapter 8 seesanimportantapplicationof4-vectorsandfour-dimensionaldot producttoparticleinteractions.Thenotionsofcenterofmassandlabframes areintroducedandusedinvariousexamplestoinvestigateparticlecollisions anddecays.ThechapterendswithasimpleintroductiontotheDirac’sdiscoveryofantimatterandtheconnectionbetweenspecialrelativityandspin.
Chapter 9,anotherchapteruniquetothisbook,appliesfour-momentumconservationofChapter 8 torocketpropulsion.Themotionoffuelcarryingspacecraftisanalyzedformassiveaswellasphotonexhaustcasesandtheirimpracticalitydemonstrated.Thealternativeground-basedlaserpropulsioniscovered inlengthydetailinlightoftherecentinterestinsuchprojects.Certainpeculiaritiesofrelativisticmotioninonedimension,suchasthefactthatforceisan invariantquantity,arepointedoutandtheimportanceofLorentzcovariance, acrucialprerequisiteoftheapplicationofspecialrelativity,isemphasized.It isthenshownthattherequirementofLorentzcovarianceinvalidatesmany existingapproachestotheproblem.
Chapter 10 isanelementaryintroductiontotensoralgebraasappliedtospecialrelativityandusedmainlyinChapter 11 onrelativisticelectrodynamics. ThelatterderivestheLorentztransformationforelectricandmagneticfields andcalculatestheelectricandmagneticfieldsofauniformlymovingcharge startingfromtheCoulombforceofastaticcharge.Electromagneticfieldtensor
leadsnaturallytotheLorentzforcelawandanelegantwayofwritingMaxwell’s equations.
Althoughsomewhatoutsideitsmainthrust,thebookendswithaveryaccessibleintroductiontothestandardcosmologicalmodel.Afewofthetopics developedinthebookareusedtoillustratetheimportanceofrelativityinthe physicsoftheearlyuniverse.Alongtheway,anumberofideas,notrelatedto relativity,arediscussedandtherelevantequationsderivedindetail.So,Chapter 12 isalsoanaccessibleexposuretosuchtopicsasEMradiationincavities, Planckformulaoftheblackbodyradiation,Stefan-Boltzmannlaw,Friedmann equation,andtheevolutionoftheearlyuniverse.
Discussions(sometimesheated!)withsomeofmycolleaguesclarifiedmany subtleideasofrelativityandconsolidatedmybeliefthatLorentztransformationisthebedrockofspecialrelativity,andshouldbeatthecoreofteaching thesubjecttothenovice.IwouldliketothankRobertSchrockforreadingan earlyversionofChapter 7 andgivingconstructivecommentsonit.Ialsothank SörenHolstandJohnMallinckrodtformanyexchangesofemail,whichhelped medeciphertheintricaciesofapplyingLorentztransformationstovariousinterestingexamples,someofwhichappearthroughoutthebook.Needlessto say,Iamsolelyresponsiblefortheaccuracyofthecontentofthebook.
SadriHassani
Urbana,IL,USA
December2016
ListofSymbols,Phrases,andAcronyms
ˆ a unitvectorinthedirectionof a antiparticle aparticlewhosemassandspinareexactlythesameasitscorrespondingparticle, butthesignofallits“charges”areopposite.Ifaparticleisrepresentedbytheletter p ,then itiscustomarytodenoteitsantiparticleby p .Ifaparticleisrepresentedbytheletter q (or q + ),thenitiscustomarytodenoteitsantiparticleby q + (or q ). as arcsecond;anarcsecondisanangle 1/3600 ofadegree. baryon ahadronwhosespinisanoddmultipleof /2.Baryonsarecomposedofthreequarks. Examplesofbaryonsareprotonsandneutrons.
β fractionalvelocityofoneobserverrelativetoanother, β = v/c boson aparticlewhosespinisanintegermultipleof .Allgaugeparticlesarebosonsasareall mesons,aswellastheHiggsparticle. causallyconnected referringtotwoevents.Ifanobserveroralightsignalcanbepresentattwo events,thoseeventsaresaidtobecausallyconnected. causallydisconnected referringtotwoevents.Ifanobserveroralightsignalcannotbepresent attwoevents,thoseeventsaresaidtobecausallydisconnected.
CBR CosmicBackgroundRadiation.
CM centerofmass.
CS coordinatesystem.
ˆ ex , ˆ ey , ˆ ez unitvectorsalongthethreeCartesianaxes.
EM electromagneticorelectromagnetism,oneofthefourfundamentalforcesofnature. equilibriumtemperature temperatureoftheuniverseatwhichmatterandradiationdensities areequal.
eV electronvolt,unitofenergyequalto 1 6 × 10 19 J. fermion aparticlewhosespinisanoddmultipleof /2.FermionsobeyPauli’sexclusionprinciple:notwoidenticalfermionscanoccupyasinglequantumstate.Electrons,protons,and neutronsarefermions,soareallleptonsandquarks,aswellasallbaryons.
γ theLorentzfactor, γ = 1/ 1 β 2 = 1/ 1 (v/c)2 . gaugebosons Accordingtothemoderntheoryofforces,fundamentalparticlesinteractviathe exchangeofgaugebosons.Excludinggravity,whosemicroscopicbehaviorisnotwellunderstood,thereare12gaugebosonswhoseexchangeexplainsalltheinteractions: Z 0 , W ± and γ (photon)areresponsibleforelectroweakinteraction,while8gluonsareresponsibleforstrong interaction.
gluons theparticlesresponsibleforstronginteractions:twoormorequarksparticipateinstrong interactionbyexchanginggluons.Therearefourgluons,whichwiththeirantiparticlescomprisetheeightgluonswhoseexchangebindsquarkstogether.
GTR generaltheoryofrelativity;therelativistictheoryofgravity.
QualitativeRelativity
Theseedsofrelativitytheorywereplantedonaspringdayinalecturehall onthecampusoftheUniversityofCopenhagenin1820.AsProfessorHans ChristianÖrstedwasdemonstratingthepoweroflargeelectriccurrents,he notedthedeflectionofanearbycompassneedleassoonasalargecurrent wasestablishedinthecircuit.Thushestumbledononeofnature’sbest-kept secrets,namelythatelectriccurrentscanproducemagneticfields.
1.1ITBEGANWITHMAXWELL
Youmightthinkthatbecausecurrentsareproducedbymovingchargesand becausemotionisarelativeconcept,thereisalreadyaconnectionbetween electromagnetismandrelativity.Afterall,ifyoumovewiththechargesproducingtheelectriccurrent,theywillappearmotionlesstoyouandyoushould notdetectanymagneticfield.1 Whilethisiscertainlytrue,itcontainslittle morethanthefactthatthemagneticfielddisappearsforanobservermoving alongsidetheelectriccharges.Tomakecontactwith the theoryofrelativity, youneedthefullmachineryofelectromagnetismasdescribedbyMaxwell’s equations.
Ifyouhavesomefamiliaritywithvectoranalysisandhavenotseenhow Maxwellderivedhisfamousequations,you haveto readAppendix A.Ittruly affirmsthe“powerofthemind,”notinthewaythatmystichealthguruswant youtobelieve,butthepowerinstigatedbymathematicsandvindicatedby countlesspracticalusesofitsimplications.Italsolaysthefoundationofrelativityinastatementrepeatedhere: SecondPostulateof Relativity
Note1.1.1. Electromagneticwavestravelat c = 299792458 m/s,thespeedof lightinvacuum,regardlessofthemotionoftheirsource.
1 Inarealwire,themotionofthenegativeelectronscreatesacurrent.Ifyoumovewiththeelectrons, thenthepositivebackgroundchargesmoveintheoppositedirection,andthemagneticfieldwillnot disappear.Iamnotconsideringrealwires,butthecaseinwhichonlychargesofonesignarepresent. 1
NowsupposethatSonya,ridingonasupertrainandholdingherlaserpointer outsidethetrain,sendsabeamoflighttowardthefrontofthetrain.Sam,who isstandingoutside,hundredsofkilometersawayfromthetrain(therefore,not evenseeingthesourceofthebeam),detectsthelaserlightandmeasuresits speedtobe299792458m/s,regardlessofhowfastthetrainismoving!That’s theessenceofthesecondpostulateofrelativityin Note1.1.1.Ontheother hand,LAVinstructsourintuitiontoaddthespeedofthetraintothespeed ofthelightrelativetothelasergun.ThisistheconflictbetweenLAVandthe secondpostulate.
TheLAVwassofirmlyestablishedinthenineteenthcenturythatthenotionof lighttravelingwiththesamespeedrelativetotwodifferentRFs—andtherefore, thesecondpostulateofrelativity—wasconsideredalogicalfallacy!So,what wasthealternative?OnepossibilitywastoconsidertheEarthasthespecialRF inwhichMaxwell’sequationshold.ButthiswasimmediatelydismissedbecauseCopernicus,threecenturiesearlier,hadalreadycautionedaboutthedangersofbestowingupontheEarthaspecialpositionintheuniverse.Moreover, theorbitalmotionoftheEarth,withitssemi-annual180-degreedirectional flip,wouldhavetohavedistortingeffectsonthelightofstarsandplanets, whichwouldbeeasilydetectable.Lackofsuchdistortionsimmediatelyrules outtheEarthasaprivilegedframe.IftheEarthisnottheprivilegedRF,then, forthesamereason,nootherplanet,northeSunnoranyotherstarorgalaxy canbeprivileged.
Theonlyremainingalternativewasthe“medium”surroundingthecelestial bodiesandfillingthespacebetweenthem.Justaswaterwaveistheoscillation
Theriseandfallofether theory. ofwaterandsoundistheoscillationofair,theundulationofthismedium, called ether,manifesteditselfaslight.Theideaofetherwasinventedandused byHuygensasearlyas1690,butitbecamethesubjectofintensescrutinyin thenineteenthcenturyafterMaxwell’spredictionoftheelectromagnetic(EM) waves.Ethertheorydemandedsomanystrangeandcomplicatednotionsthat bythelastquarterofthenineteenthcentury,physicistsstartedtodoubtitsexistence.ThefinalblowtoethercamethroughtheworksofAlbertMichelson (1852–1931)andEdwardMorley(1838–1923),oneofwhosecrucialexperimentsshowedthattherewasnodetectablemotionofEarthrelativetoether, therebyunderminingthetheoryandtheconceptofether.
1.1.1EinsteinandSignalVelocity
Whenhewasonly16yearsoldin1895,Einsteinwonderedhowtheworld wouldappeartohimifhecouldcatchupwithlight;inparticular,whatwould lightitselflooklike.Youcanappreciatetheintriguingnatureofthisquestion byananalogy.Imaginethatyouarequietlymovingovercircularwaterwaves atexactlythesamespeedastheirexpansionrate.Howwillthewavesappear Howwouldwaterwaves lookifyoucaughtupwith them? toyou?Concentrateonlyonthelocalwaves2 andassumethatyouridejust
2 Clearlyyoucannotmovewithallthewavesastheygoindifferentdirections.
FIGURE1.2 (a)Twoadjacentfirecrackersaresaidtohaveexplodedatthesametimeifthe observerreceivestheirsignalsatthesametime.(b)Twowidelyseparatedfirecrackersaresaid tohaveexplodedatthesametimeiftheobserverreceivestheirsignalsatthesametimewhile beinglocatedontheperpendicularbisectorofthelinesegmentjoiningthefirecrackers.
occuratthesametime?If A and B wereatthesamepointwhentheyexploded, itiseasytoanswerthisquestion:Ifyoureceivetheirlightsignalsatthesame time,theymusthaveoccurredatthesametime[Figure1.2(a)].Becauseof thespecialpropertyoflightstatedinthesecondpostulate,thisconclusion isindependentofthestateofmotionofthefirecrackers: A and B couldbe movingatveryhighspeedsinarbitrarydirectionsatthetimetheyreachthe commonpointinthefigure.
Youcanappreciatethesignificanceoflightsignalsusedincommunicating simultaneitybylookingatevents A and B whichoccurin moving reference frames.Supposebulletsareusedinsteadoflightsignals.SamandSonyaare standingontwotrainsparkednexttoeachotheratpoint C adistanceof 300mfromJames.Theyfirebulletssimultaneouslyfromtheirriflesmoving at100m/s toward James.4 Threesecondslater,Jamesseesthebulletspassby himtogether.Heconcludesthatthetwoeventsoffiringthebulletsmusthave occurredsimultaneouslybecauseheknowsthatthetworifleswereatthesame locationandthespeedofthebulletswereequal.
NowsupposethatSamismovingawayfromJameswhileSonyaismoving towardJamesbothat50m/s.Whentheyreach C atthesametime,theyfire bulletstowardJames.Sonya’sbullethasaspeedof150m/swhileSam’sbulletspeedis50m/s,bothrelativetoJames.So,althoughfiredsimultaneously,
4 ...whois(hopefully)notdirectlyinfrontoftherifles!
FIGURE1.3 Differentobserversdisagreeonthesimultaneityoftwoevents.Thegroundobserver (Sam)seesthetwoeventsassimultaneous,whilethetrainobserver(Sonya)doesnot.Themarks labeled A and B areleftonthegroundbytheexplosionofthefirecrackers A and B ,respectively. ThesemarksareasseenbySam.
A and B atthesametime. Figure1.3 showsthesituationasseenbySam.The topfigureisthemomenttheexplosionsoccur,as calculated bySamafterhe receivesthesignals(allowingforthesignals’traveltime).Themiddlefigure showsthewavefrontsmovingawayfromtheirsources.NotethatSonyahas receivedthesignalfrom B ,andthatthetwowavefrontsareequidistantfrom Samandfromthepositions—accordingtoSam—ofthefirecrackers atthetime ofexplosions (i.e.,positionsof A and B inthetopfigure,orequivalently,positionsof A and B ).Finally,inthebottomfigurethetwowavefrontshave reachedSamwhileSonyaisstillwaitingforthesignalfrom A . Afterreceivingthetwosignalssimultaneously,Samwalksto A countinghis steps,walksto B countinghissteps,noticesthatthetwodistancesareequal, andconcludesthatevents A and B wereindeedsimultaneous.Whatabout Sonya?Afterreceivingthetwosignals,shemeasuresherdistancefrom A and B andverifiesthatshewashalfwaybetweenthem.Shethereforeconcludesthat B mustdefinitelyhaveoccurredbefore A .
Althoughsimultaneityoftwoeventsisnotuniversal,thefactthatSamsees A and B assimultaneous,is.Notonlydoeshesay“ A and B aresimultaneousfor me,”butalsoSonyaandallotherobserversintheuniversesay“ A and B are simultaneousforSam,”despitethefactthattheythemselvesdonotmeasure theoccurrenceof A and B tobesimultaneous.Wecanactuallydemonstrate thisbyequippingSamwithaspecialfirecracker C thathecantriggerwhenthe twosignalsreachhimatthesametime.Thus,theexplosionof C heraldsthe simultaneityof A and B asseenbySam,andallobserverskeepinganeyeon
FIGURE1.5 Sonyaseesthetwoeventsassimultaneousanddecidesthatthelength AB isthe distancebetweenthetwomarksonthetrain.Samsees A before B anddecidesthatthedistance betweenthemarksisshorterthan AB
notarrivedyet,heconcludesthatthefrontendofthetrainhasnotreachedthe markonthegroundyet.He,therefore,decidesthatthe trainisshorter thanthe distancebetweenthegroundmarks!Itdoesn’tmatterwhoseesthetwoevents atthesametime; thetrainisshorterforSam.
Isthereanythingspecialaboutthetrain?Isitbecauseitismovingandthe platformisnot?Itcan’tbe,becausemotionisrelative.ToSonyaitistheplatformthatismoving,andifrelativityiscorrect,lengthsontheplatformshould appearshortertoSonya.Isthisindeedthecase?Placethefirecrackersonthe platforminsteadofthetrain(leavingmarksonthetrainwhentheyexplode) andletSonyaandSammeasurethedistancebetweenthem(ortheirmarks). Sincetheprecedingdiscussionshaveshownthatthedistancemeasurement (whetheritisshorterorlonger)isindependentofwhoseestheexplosions simultaneously,assumethatSonya,ridingonthetrain,seesthefirecrackers explodesimultaneously.Sam,ontheotherhand,receivesthesignalfrom A beforethatfrom B Figure1.5 explainsthissituation.
Asusual,assumethatanidenticalexperimenthasbeendonebefore,anditis nowbeingrepeated.So,therearealreadyblackmarksonthetrainfromthe previousexplosionsindicatingthedistancebetweenthefirecrackers asmeasuredbySonya.Samisstandinginthemiddleofthetwofirecrackersonthe platform,anticipatingtheoccurrenceof A .Heeventuallyreceivesthesignal from A andconcludesthatafractionofasecondearliertheblackmarkatthe rearendofthetrainmusthavecoincidedwith A .Immediatelyhelookstothe front,andsincethesignalfrom B hasnotarrivedyet,heconcludesthatthe markinfrontofthetrainhasnotreached B yet.He,therefore,decidesthatthe distancebetweenthetwofirecrackersontheplatformislargerthanthemarks
onthetrain.Therefore, Sonyameasuresthedistancebetween A and B tobeless thanwhatSammeasures.
Notethesymmetrybetweenthetwoobservers.Neitherisanymorespecial thantheother.BothareclaimingthatlengthsmeasuredintheirRFarelonger thanthemovinglengths.8 Inotherwords,theybothconcludethat moving Movinglengthsshrink. lengthsshrink.Andtheshrinkageisnotduetosomekindofamechanicalcompressionofthetrainortheplatform(orcars,trucks,planes,ormetersticks). Thetwoendsofastickmerelyindicatetwopointsinspace.Forinstance,these twopointscouldbethelocationsoftwostars.AsseenfromEarth,thesetwo stars,saytheSunandAlphaCentauri,whicharealmostfixedrelativetothe Earth’sRF,9 areseentobeabout4lightyearsawayfromoneanother.Thecrew ofaspaceship,travelingatveryhighspeedtowardAlphaCentauri,seethis distancecontractedbecause,tothem,thelengthbetweentheSunandAlpha Centauriisinmotion.Thus, motionaffectsthespaceitself
Allthecontractionstreatedsofaroccurforlengthsthatarealongthepathof motion.Wouldthesameeffectoccurifthelengthmovedalongapaththat wasperpendiculartoitself?SincethemotionofSonya’strainhashelpedus somuch,I’lluseitonemoretime.Assumethatperpendicularlengthsalso shrinkwheninmotion.10 WhenSonya’strainisstationary,itswheelsreston thetracks,sothatthedistancebetweenthewheelsisequaltothedistance betweenthetracks.Whenthetrainmoves,Sonyaseesthetracksmovingrelativetoher.Sosheconcludesthatthetracksgetclosertogether,andtherefore, thetrainwillfall outside ofthetracks.Sam,ontheotherhand,seesthetrain movingandconcludesthatthedistancebetweenthewheelsmustshrinkand, therefore,thatthetrainmustfall inside ofthetracks.Butderailingofatrainis anactualphysicalprocess(anevent)andthusmustbeindependentofSonya andSam.Theonlysoundconclusionistosaythatneithershrinks.Summarizing,weconcludethat Onlylengthsparallelto motionshrink.
Note1.3.2. Allmovinglengthsparalleltothedirectionofmotionshrink.Lengths perpendiculartothedirectionofmotionarenotaffected.
1.4PROBLEMS
1.1. Arodoflength L emitslightfromallofitspointssimultaneously(inits restframe)whenaremoteswitchisturnedon.Itscenterisonthe x -axisand ismovingontheaxisinaplaneparalleltoaverylargephotographicplateand
8 Rememberthatitis theplatform thatismovingrelativetoSonya!
9 AlthoughtheSunandAlphaCentauriaremovingrelativetoEarth,theirspeedissosmallcomparedto lightspeedthattheirmotioncanbeignored.
10 Itdoesn’tmatterwhethertheyshrinkorexpand.Theconclusionwillbethesame.
