affine connection etc good notes

Some topics inGeometry andGravitation EefvanBeveren CentrodeFsicaTeorica Departamen todeFsicadaFaculdade deCi^encias eTecnologia Universidade deCoimbra(Portugal) http://cft.s.uc.pt/eef 7deJulho de2007 Contents IIntroduction 1 1Theaftermath ofaphysicsexam 1 2Movingreference system 4 3Transformations 7 4Galilei transformations 8 4.1Thekinetic energy ..................................9 5Pioncar etransformations 9 5.1Velocities neverexceed thevelocityoflight.....................11 5.2Small velocities ....................................11 5.3Energy andmomen tum................................12 5.4Totalinvariantmass .................................13 6Relativistic kinematics 14 6.1+!K+K..................................16 6.2Elastic Scattering inthecenter-of-mass system ...................17 6.3Elastic Scattering inthelabsystem .........................19 IIGeneralities 20 7Covariantandcontravariantcomponents 20 8Themetrical tensor 22 9Localbasis 23 10Themetric ofthelocalcoordinates 24 11Dieren tiation withrespecttothelocalbasis 24 12Christoel symbols(ane connection) 25 13Therelation betweentheane connection andthemetric 26 14Thederivativesofavector eld 27 15Covariantderivative 27 i IIIThree dimensions 28 16Curvesinthree dimensions 28 17Thenatural localbasisofacurve 31 18Thederivativesofthenatural basis 33 18.1ProofoftheFrenet relations .............................33 18.2Darbouxvector ....................................34 19Atwo-dimensional surface inthree dimensions 35 20Thederivativesofthelocalbasis 37 21Curvesonthesurface 39 22Thecurvature ofthecurvesonthesurface 41 23Thecurvature ofthesurface 42 24Thelocalprinciple axes 44 25Egregium theorem ofGauss 46 26Thecurvature tensor 48 27Geodesics 50 IVExamples ofRiemann surfaces 51 28Thecylinder 51 29TheEllipsoid 53 30Geodesics onthesphere 57 31Thetorus 60 VThedescription ofgravitation bycurvature 63 32Rectilinear motion 63 33Parallel transp ort 65 34Parallel transp ortalong curvesatthesphere 66 35LocalEuclidean geometry 68 36Tidal forces 71 ii 37Theprinciple ofequivalence 73 38Minkowskian space-time 74 39Gravitational forces 76 40TheSchwarzschildmetric 77 41Planetary orbits 79 iii Some topics inGeometry andGravitation These notes arebornasaresult ofaseries ofdiscussions withmycolleague HumbertoPascoal fromtheCentreofTheoretical PhysicsattheUniversityofCoimbra,Portugal, ontheprinciples ofthedescription ofacurvedtwo-dimensional surface embedded inthreedimensions. Theideas contained inthese notes arenotnew.Onthecontrary,thesimple andelegan tmetho dswhich leadGauss tohisEgregium Theorem, leadustostudy inmoredetail asubjectwhichalmost two centuries agohasbeenstudied byCarlFriedric hGauss (1777-1855), Janos Bolyai(1802-1860) andNikolayIvanovichLobac hevski (1793-1856). Thenotation whichIhaveusedinthisnotes, istheonewhichismost popular amongst physicists, inorder nottodiscourage rstandsecond yearphysicsstuden ts. Moreo ver,Idonotintendtoberigorous, neither complete. Belowyoundtheliterature whichIconsulted. Decem ber,1988. EefvanBeveren References [1]A.Einstein, ZurElektrodynamik bewegterKorper,Annalen derPhysik,(1905). [2]A.Duschek,Dierentialge ometrie ,Handbuc hderPhysik,Band III,edited byH.Thirring, VerlagvonJulius Springer, Berlin, 1928, page153. [3]Horst Tietz, Elementar eDierentialge ometrie ,Handbuc hderPhysik,Band II,edited byS. Flugge, Springer-V erlag, Berlin, 1955, page146. [4]H.S.M. Coxeter, F.R.S., Introduction toGeometry ,John Wiley andSons, Inc,NewYork, 1961. [5]StevenWeinberg,Gravitation andCosmolo gy,JohnWiley andSons, Inc,NewYork,1972. PartI Introduction Einstein's discoverythattheLorentztransformations followfromasimple principle, namely the constancy ofthelightvelocityinallinertial reference frames, wasanimportantbreakthrough forthedevelopmen tofafulltheory ofGravity.But,manyoldconcepts hadtobereplaced aswell.Inthisnotes wewillpaysome attentiontoimportantcontributions fromdieren tial geometry ,inparticular, tothenotion ofafreely movingparticle inacurvedmanifold. Firstwepassthrough thedenition ofcoordinates andthedescription ofmovingobjectsby acoordinate system. Then wepaysome attentiontoGalilei, LorentzandPoincar etransfor- mations. InpartIIwecome tointroducethemain dish. 1Theaftermath ofaphysicsexam Praia Perp^etuahasaverylongandperfectly straigh troadforbicycles alongside thebeach. Onabeautiful sunnydayjustafewweeksbeforethebeginning ofthebeachseason, ayoung man,named Alex, isseated attheveranda ofhisbeachapartmen tjustnexttotheroad. From there hewatchesthemovementsonthebeachandenjoystheneverending sound ofthewaves breaking intofoam atthesandofthebeach.Itisalovelyday,butonlyaveryfewpeople arewalking inthesandortaking abathintheocean, probably sinceitisonlynineo'clockin themorning. Heobserv esthatthismorning theice-cream vendor installed hiscarnexttothe roadatonly43meters tohisright.Athisleftat39meters distance there isasmall restauran t where youcaneatfreshsheverydayoftheyear. Since Alexconsiders himself thecenteroftheUniverse,wherev erheisseated, healways represen tstheorigin ofhiscoordinate system. Moreo ver,sinceheisnevertheless areasonably welleducated person, hemeasures distances inmeters. Asfarasthebikeroadisconcerned we willconsider onlyonedimension andindicate thecoordinate system ofAlexbyx(A).Forthe ice-cream carwendthen x(A) ice-cream car=+43; (1) whereas, fortheshrestauran tweobtain x(A) shrestauran t=39: (2) Theplusandminussigns havenothing todowithaclassication ofAlextowardseating ice cream orsh,heactually lovesbothicecream andfreshsh,butindicate thatwehavechosen 1 thepositivesenseofthecoordinates inthedirection oftheice-cream car.Theoppositedirection isthenautomatically inthenegativ esense. Onpurpose,wehavenotindicated unitsinformulas (1)and(2),sincewehavestated theunitsystem ofAlexbefore. Ayoung couple onabicycle passes by.They areBruno andClara whobothgreet Alex. Bruno doesthepedaling, whereas seated onthebackthebike,hisarms around thewaistof Clara whoisseated onthesaddle anddoesthesteering. They areheading towardstheice- cream car.However,justsome meters afterpassing Alex, Bruno stops thebikeandsitswith Clara inthesandnexttotheroad, Clara atBruno's lap.They aretalking aboutthephysics exam ofyesterda yand,inparticular, discussing aproblem aboutrelativ emotion, while, inthe mean time, hugging andkissing eachother. After somehalfanhourtheydecide tobuyanice cream, butnotwithout measuring thedistance fromwhere theyareseated tillthecar.Bruno measures distances inlargesteps, whichheknowsarejustequal toonemeter. So,hesetsout towardsthecarandarrivesafter26steps. There, hepatien tlywaitsforClara. Clara measures theworld,whichconsists outofeverything whichisclosetoher,inthewell- shapedpalms ofherslimhands. Consequen tly,sheprefers tomeasure distances withthesizeof herhands. Bruno enjoysobserving her,busyonherknees, slowlyapproac hinghim,byputting onehand nexttotheother inthesandwhile thesunisshining onherback.Threeh undred twentyveshecountswhen sheendsatBruno's feet.Welldone, Bruno agrees, Iknowthat thepalms ofthose tender hands ofyoursmeasure precisely eightcentimeters. Hegiveshera bigkissandorders theiricecreams. Onegirlish, withstrawberries, forhisClara, andonemore serious, withchocolate, forhim.They walkbacktotheirbicycle while eating theiricecreams. Inthecoordinate system ofBruno, whichweindicate byx(B),wendforthecoordinates of theice-cream carandtherestauran trespectively x(B) ice-cream car=+26; (3) and x(B) shrestauran t=56: (4) Furthermore, inthecoordinate system ofClara, whichweindicate byx(C),wendforthe coordinates ofthesame respectively x(C) ice-cream car=+325; (5) and x(C) shrestauran t=700: (6) Notice thattheorigins ofthecoordinate systems ofBruno andClara areattheposition where theystoppedtheirbicycle. While stillwalking, Bruno andClara seefarbehind therestauran thowblond beautyDiane withherelegan tlegsispedaling herbicycle towardsthem. Clara isnotveryfondofmeetings withBruno's former girlfriend. Inparticular not,when sheisdriving herbicycle dressed in ascandalous minibikini. Butthen, Bruno putshisarmaround Clara's shoulders andgives alongkisswithalittlebitofchocolate icecream inhercurly darkhairs fullofsand. Both startcalculating howmuchtimewillelapse beforeDiane willpassbytheplace where theywere seated. Bruno starts hisstopwatchwhen Diane ispassing bytheshermen monumentat56meters beyondtheshrestauran t.Hendsthatittakesherexactly 20seconds fromthere tothe restauran t.So,heconcludes thatitwilltakeyetanother 20seconds beforeshewillreachthem. 2 Enough timetositdownwithClara, facing theocean, theirbacksturned towardstheroad. Clara countstimewiththebeatsofherheart which,actually ,isbeating asbitfaster than normal rightnow.Measuring exactly thesame distance asBruno hadselected, shecomes at 28beatsintotal. They sitdowninthesand, enjoyingtheiricecreams andpretending notto notice whoispassing byonherbike. Inthemean time,Alexalsohadcomeawareofwhowasapproac hinghimatherbicycle. Ever sinceDiane hadbrokenupwithBruno, heassumed thathischances withherwerewritten inthe stars. Therefor, healsomeasured herspeedathisstopwatchtocome tothesameconclusion as Bruno, andprepared himself foragoodimpression onhis,asyetsecret, love.Inhiscoordinate system hefound thatherposition coordinate attimet(A)isgivenby x(A) Diane t(A) =95+2:8t(A): (7) But,although hisformulacorrectly corresp ondstoDiane's motion, unfortunately forAlex, Diane hardly responded tohis\howareyouDiane" when shepassed by. Disapp ointed,heisstillwatchingher,when heassists howClara cannot resist togiveDiane a provocativelookoverBruno's shoulder. Itisresponded byaslowandsensual helloofDiane. On hearing herattractiv evoice,Bruno turns hishead, utters asoundless helloand,likehypnotised, keepswatchingthemostpleasan tscene ofthesun-tanned bodyofthebeautyfulgirl,pedaling onherbicycle, herblond hair oating intheairlikethetailofacomet when itpasses closeto thesun.However,Clara, disturb edbyBruno's confusion, reacts byputting what isleftofher icecream onhisforehead. Thatgesture rapidly brings himbacktoreality.While joking about herattackofjealousy ,heholds Clara rmly inhisarmsandkisses herdelicately .Both start laughing andsoontheinciden tseems tobelong tothepast. While refering tothemotion ofherbicycle, Bruno arms thatDiane's motion iswellde- scribedbythefollowingformulainhiscoordinate system. x(B) Diane t(B) =112+2:8t(B): (8) Clara, stillnotcompletely secure ofherspellofcharm onBruno, isnotexactly eager toshare herresult onthisparticular eventwithBruno. But,ontheother hand, shealsodoesnotwant himtonotice herinsecurit y.So,shetellsBruno thatshefound x(C) Diane t(C) =1400+25t(C): (9) Bruno, stillabsentminded, saysthatheisnotcompletely surewhether thisagrees withhis result (8).Itistobeadmired thatClara manages nottorespondthatheapparen tlyisnot sureaboutseveralissues, but,instead, proposestovisitAlex. They decide togotoAlexand checkwithhisndings. Alex, gladtohavesome compan ytodistract himfromhisdisapp ointment,happily shares thendings onthemotion ofhissecret love,withClara andBruno. Soonthethreestarttalking aboutDiane, eachwithdieren tfeelings withregard tothefascinating girl.Bruno withsome sadness aboutthewayhelosther,Alexregretting hisrecentfailure toattract herattention andClara withsome latentboiling fury. Based inequations (1)to(6),theyhadcome totheconclusion x(B)=x(A)17;x(C)=12:5x(B)and x(C)=12:5x(A)212:5:(10) Notice thatallrelations arelinear inthecoordinates, whereb ytherstjuststems fromthe translation oftheorigin ofBruno andClara withrespecttotheorigin ofAlex, thesecond from 3 thedieren tunitsofBruno andClara, whereas thethirdrelation comes fromacombination of thedierence inunits andthetranslation oforigins. With respecttoinstan tsoftimetheyfound t(B)=t(A);t(C)=1:4t(A)and t(C)=1:4t(B): (11) Clara's heart beatwasabitfaster thanonepersecond. Hence, herunitoftimeisabitsmaller thantheunitoftimeforBruno andAlex. Thelatter unitisthesecond. 2Movingreference system Alex, Clara andBruno arestilltalking aboutDiane, when theysuddenly observ eherreturning fromherbicycle trip.But,fromthelanguage ofherwell-shap edbody,Alexdeduces thatshe isnotashappyaswhen shepassed byhisveranda earlier. Hedecides tocallher. Thistimeheissuccesful, Diane stops herbicycle andaccepts hisinvitation tojointhem. They greeteachother withhugsandkisses. Butjustafterthiswarmreception ceremon y,Clara suddenly remem bersthatsheandBruno hadsomething urgenttoberesolved.Bruno stillhas thenervetoaskwhat itwasagain thatisnowsuddenly sourgent.But,fortunately ,fromthe expression atClara's faceheunderstands thatitisbetternottoinsist onhavinghisignorance claried. Alexisdeligh tedwithClara's attitude andimpatien tlyawaitsthedeparture ofthe couple. Asbyamiracle hehashisblond beautyDiane seated withhimathissofaonhis veranda. Muchmore thanhehadbargained forearlier thismorning. Without payingmuchattentiontoit,Alexhadnoticed thatDiane wasnotdriving herown bicycle when shepassed byhisveranda forthersttime. Actually ,itisherfather's bicycle she isusing. Thereason forthat,aswewillsee,isasubjectwhichshebynomeans isprepared to confess toAlex. Diane livesinasmall house neartheshermen monument.Inthemorning, when Diane awakes,sheusually opensthewindo wofherbedroomcompletely andstaysthere forawhile inhernightdress, leaning onherelbows,watchingthesea,thebeach,theseagulls andthe people passing byattheroad. Thismorning, when shesawBruno andClara passing by,Diane started toslipintoherdream world,thinking onhowmuchshehadenjoyedsuchtripswith Bruno onhisbicycle. Inparticular, when he,liketodaywithClara, acted asthemotor ofthe bicycle, whereas she,withhisarmsaround her,could justgotoanydirection whichcame to hermind. Veryoften shehadsteered bothofthem toaquiet sunnyvalleyinthedunes, where theyhadoncediscoveredaperfect siteforlovers,hidden bypleasan tvegetation ofredand white ourishing oleanders andyellowbroomwiththeirintense perfume. Shestarted feeling likereturning toherbed.But,thenshesawaglimpse ofEricwhocomes regularly tothebeach foraweekendorforafewdays,andwholivesinanapartmen tjusttwoblocksdowntheroad. Eric,whoisprobably somewhere inhisearly twenties,hasastrong muscular bodyanda pleasan tfacewithblond curlyhair.Heusestopractice jogging when heisatthebeach.When Diana sawhimmovingtowardstheshermen monument,shecalculated fromhisspeedthatif shecould manage, within veminutes,tobeonherbicycle infrontoftheshermen monument, shecould justcatchupwithhimattheplace where thebikeroadturns awayfromthebeach, towardstheinland, andwhere heprobably wouldrestalittlebeforereturning tohisapartmen t. Thedistance tothatplace is2.1kilometers measured fromtheshermen monument. Diane already started imagining Ericonthebackofherbikewithhismuscular armsaround herbody.But,thensheremem beredthatherbike,unfortunately ,cannot carry persons inthe back.Hence, shedecided totakeherfather's bikeinstead. 4 When shestarted outattheshermen monument,sheknew fromEric's speedthathewas some 600meters far.Fromhervelocityshededuced thatinherreference frame, withher father's bicycle attheorigin, thedistance toEricwasdecreasing at0.8meters persecond, hence, thatEricwasapproac hingheratthatspeed.Itwouldthustake12.5minutesbeforeshe wouldmeetEricrightattheplace whichshehadcalculated. Inhercoordinate system Diane describ esEric's position by x(D) Eric t(D) =6000:8t(D): (12) AlexalsohadseenEricpassing bythismorning. Hehadmethimafewtimes inthe discoth eque. So,theygreeted eachother. Alex, curious toknowatwhichplace Ericwould turnaround toreturn home, measured hisspeedat2.0meters persecond. Inthecoordinates ofAlex, Eric's position isgivenby x(A) Eric t(A) =505+2:0t(A); (13) aslongashecontinuestomoveinthepositivesense withthesamevelocity.Thelatter formula isexplained inmoredetail below. Alexrestarted hisstopwatchwhen hesawDiane passing bytheshermen monument.Con- sequen tly,hethereb ydenes forhimself anewbeginning oftimet(A)=0,atthesame time that,coinciden tly,Diane started herstopwatch.Hence, t(A)=t(D): (14) ButAlex' choice ofanewinstan toftimet(A)=0,occurred 4minutesand12.5seconds after Erichadpassed byhisveranda. InthatintervaloftimeEricmoved505meters. Consequen tly, att(A)=0Ericisinposition x(A) Eric t(A)=0 =505 inthecoordinates ofAlex. Thisexplains theconstan tinformula(13)and,moreo ver,agrees withtheobserv ation ofDiane, x(D) Eric t(D)=0 =600; sinceDiane andAlexare95meters apart att(A)=t(D)=0. Alex, whoisaverycurious person, asksDiane whysheisdriving herfather's bicycle. But Diane, whohasofcourse nointentiontoshare herstory withAlex, tellshimthatherbikeis brokenandhas attires. Aleximmediatly oers himself torepair herbike.However,justwhen Diane starts explaining thatherfather isalready taking careoftheissue, sheseesEricpassing by,heading forhisapartmen t.Ericgreets Alexand,furthermore, givesabigsmile toDiane. Alexnotices herblushing andfeelshisheart brokenforthesecond timethismorning. But,heignores thisfeelings. Instead, hetellsDiane abouthisformula(13)onEric's dis- placemen tinfunction oftime, whichhehaddetermined earlier thismorning when Ericwas jogging towardstheother endoftheroad. Diane decides toalsoshare herformula(12)with Alex. Forawhile theyarebothpuzzled aboutthedierences. ButthenAlexnds thesolution. Hestarts explaining ittoDiane, bytelling herabout hisformula(7)onhermotion, when shewascycling towardstheroadexit. Diane reacts surprised andrstlooksawhile atAlexbefore sheaskshimwhichespionage service makes himcalculating everybody'smovementsonthebicycle road. Now,Alexstarts blushing and confesses thatinhercase,thecalculus hadbeendoneforhimtoknowhowmuchtimehehad 5 toprepare himself formaking agoodimpression onher.Healsotellsherthathehadbeen verydisapp ointedwhen shehardly tooknotice ofhim.Diane's mouth stayshalfopenwhile she continuesstaring atAlex, inspecting thenoble regular faceofthetallslimboy,outstanding in physics, mathematics andchemistry .Then, suddenly shemovesslowlyasclosetohimasshe canmanage andaskshimtoplease holdherverytight. Atthispoint,itseems thatitmighttakeawhile beforeAlexwillexplain hissolution to Diane, ifever.So,webetterdoitourselv es. While Diane wasdriving herbicycle, herreference frame ofx(D)coordinates wasmoving withher.Sheconsidered herself theorigin ofthatreference frame andwasmoreo vermoving inthepositivesense. Informula(7)Alexhaddetermined herspeedat2.8(m/s) withrespect tohiscoordinate system whichisthereference frame attachedtotheEarth. Alexhasdetermined thespeedofEricat2.0(m/s) inthepositivesense informula(13). Diane found informula(12)thatEricmovedinthenegativ esense, towardsher,withaspeedof 0.8(m/s). Hence, when wedenote speedbyvthenarelation forthevarious velocities involved, isgivenby v(D) Eric=v(A) Ericv(A) Diane ; (15) orinwords: Although running intheoppositedirection, Ericisapproac hingDiana sincetheir distance isdecreasing withtime. Consequen tly,thevelocityofEricwithrespecttoDiane is inthenegativ esense inDiane's coordinate system andequals thedierence ofthevelocityof EricwithrespecttoAlex, whichisinthepositivesense, andthevelocityofDiane withrespect toAlex, whichisalsointhepositivesense. Now,howcantherelation (15)befullyrecoveredfromtherelations (13)and(12)? Fromformula(7)weobtain inthereference frame ofAlextheposition oftheorigin ofthe reference frame ofDiane (whichisherfather's bicycle) infunction oftheinstan toftimet(A) measured atthewatchofAlex. Hence, wecandetermine theposition ofEric,x(A),atacertain instan toftimet(A)inthecoordinate system ofAlex, starting fromhispostion inthecoordinate system ofDiane andtheposition ofherorigin withrespecttotheorigin ofAlex: x(A) Eric t(A) =x(A) Diane t(A) +x(D) Eric t(A) : (16) Here, wesubstitute equation (7),tond x(A) Eric t(A) =95+2:8t(A)+x(D) Eric t(A) : (17) Next, when wesubstitute formula(12),alsousing therelation (14),thenwerecoverrelation (13). Now,fromformula(16)wemaydeduce dx(A) Eric t(A) dt(A)=dx(A) Diane t(A) dt(A)+dx(D) Eric t(A) dt(A): (18) Inthesecond termontherighthand sideofformula(18),wemaysubstitute relation (14). We obtain thenindeed relation (15)intheform v(A) Eric=v(A) Diane+v(D) Eric: (19) Thelinear transformations (14)and(17)preserv etherelation (19),orequivalentlyrelation (15). Transformations whichpreserv ethose relations arecalled Galilei transformations. 6 3Transformations Fromtheprevious section wehaveundersto odthatitisveryimportanttodistinguish between acoordinate system andthecoordinates ofamovingobject.Furthermore, weshould dene well what wereally,reallywant(Spice girls,1994) when weconstruct coordinate transformations. 1.Coordinate system (reference frame) Here, wedene acoordinate system asacontinuoussetofpointswhichrepresen tthe positions ofpointlikeobjects. Weassume thatsuchpointsllupthewhole space. To eachpointwemayassociateasetofrealnumbers,called vectors.Inonedimension each pointischaracterized byonerealnumber,intwodimensions bytworealnumbers,etc. Weassume furthermore thatneighbouring pointsarecharacterized byneighbouring sets ofrealnumbers.Usually weerectasetofcoordinate axeswhichintersect allinonepoint, theorigin ofthecoordinate system. Theorigin isrepresen tedbytheset(0,0,...,0).All other pointsarerepresen tedbytheirprojections onthose axes. Inthatcasethewhole space isdened oncetheunitvectors oneachaxisaredened. 2.Thecoordinates ofamovingobject Amovingobjectisdescrib edbyasetofcoordinates whichvariesinfunction ofatime parameter: (x1(t),x2(t),...,xn(t))inann-dimensional space. Theresulting space isa one-dimensional subspace ofthefullcoordinate space. Inonedimension onedoesnot notice wellthedierence betweenthespace andthesubspace whichdescrib esamoving object.However,inhigher dimensions itisobvious, sincethespace describ edbyamoving pointparticle isjustaline. 3.What wereally,really want Wewanttostudy coordinate transformations whichdescrib ereference frames whicharein relativ emotion withconstan tvelocity.Inparticular, wewanttostudy whatispreserv ed under suchtransformations. 7 Inthefollowingwewillconsider twoone-dimensional reference systems AandBandindicate thecoordinates andtimeparameters byrespectivelyx(A)andt(A)inreference frame Aandby respectivelyx(B)andt(B)inreference frame B. 4Galilei transformations Besides possible scaletransformations bychoosing dieren tunitsystems andwhichwewillnot further discuss here,theGalilei transformations consist outof 1.translations Wehavespacetranslations ,givenby x(B)=x(A)+constan t; (20) whichstemfromchoosing adieren torigin ofspace, andtimetranslations ,givenby t(B)=t(A)+constan t; (21) whichstemfromchoosing adieren tbeginning oftimecounting. 2.rotations Inonedimension there arenorelevantrotations. 3.inversion Wehavespaceinversion ,givenby x(B)=x(A); (22) whichstemfromchanging positiveandnegativ esense, andtimeinversion ,givenby t(B)=t(A); (23) whichstemfromcountingbackwardintime. 4.boosts Under aboostweunderstand acoordinate transformation forrelativ elymovingreference frames. SupposethatBismovinginthepositivesense withrespecttoAwithaconstan t velocitygivenbyVandassume alsothatattimet(A)=0theorigins coincide and, moreo ver,t(B)=t(A).Then thisboostisgivenbythecoordinate transformation x(B)=x(A)Vt(A)=x(A)Vt(B): (24) Alltheabovetransformations preserv edistanc eandrelativ emotion. Takeforexample the boosttransformation (24)andtwopointlikeobjectsaandbinmotion. Lettheirmotion be describ edby x(A) a t(A) and x(A) b t(A) ; (25) 8 inreference frame A.There arenorestrictions onthedependence oftheirargumen ttofthe functions x(A) a(t)andx(A) b(t).Onemayconsider anymotion, like x(A) a t(A) =43+24cos t(A) and x(A) b t(A) =512+318t(A)5; (26) orwhatev erother complicated motion. Insystem B,whichishereconsidered tobeinmotion withrespecttosystem Awithconstan t velocityV,using thetransformation (24)andt(B)=t(A),wendforthedescription ofthe motion ofobjectsaandb x(B) a t(B) =x(A) a t(B) Vt(B)and x(B) b t(B) =x(A) b t(B) Vt(B):(27) Fortheirrelativ edistance inreference frame Bweobtain x(B) b t(B) x(B) a t(B)2=x(A) b t(B) x(A) a t(B)2=x(A) b t(A) x(A) a t(A)2: (28) Hence, therelativ edistance ofaandbisequal inbothreference frames AandBforalltimes. Fortheirrelativ evelocityweobtain dx(B) b t(B) dt(B)dx(B) a t(B) dt(B)=dx(A) b t(A) dt(A)dx(A) a t(A) dt(A): (29) Hence, wendthatrelativ emotion isnotaected bytheGalilei transformations. Newtonian physicsisthesame inreference frame Aasinreference frame B. 4.1Thekinetic energy Thekinetic energy ofamovingpointparticle isrelated toitsmass, m,anditsvelocity,v,(or, alternativ ely,itslinear momen tum), by E(kinetic )=1 2mv2: (30) Now,sincemassisinvariantunder Galilei transformations, andsince, moreo ver,velocities add up(seeformula19),itisclear thatthekinetic energy ofaparticle isdieren tindieren t inertial frames, under Galilei transformations. 5Pioncar etransformations Instead ofleavingjxbxaj2invariant,asunder Galilei transformations, under Poincar etrans- formations theeventdistanc e,forc=1givenby (xbxa)2(tbta)2; (31) isleftinvariant. Therstquestion is,ofcourse, todene wellwhat thisquantityrepresen ts.Inorder to answerthatquestion, westartbydening thenotion ofanevent.Aneventisthehappening ofsomething atacertain place andatacertain time. Forexample, when Bruno stoppedthe bicycle at9h02at17meters distance fromAlex, orwhen Ericpassed bytheveranda ofAlexat 9 9h36. These aretwoevents.Wecandetermine thequantity(31)forthose twoevents.Under Poincar etransformations thisgivesthesame result forAlex, sitting inhissofaathisveranda, asitgivesforDiane driving herfather's bicycle. Forspace andtimetranslations andinversions itisobviousthatthequantity(31)isinvariant, aswellasforspace rotations, because space rotations donottouchthetimeanddoleave invariantdistances inspace. So,weonlyhavetoconsider Poincar etransformations forreference frames whicharemovingwithrelativ econstan tvelocity=V=c,i.e.boosts.Thelatter transformations arealsocalled Lorentztransformations. Hence, thetaskistondatransformation formovingframes whichleavesthequantity(31) invariant.ThishasbeendonebyAlbertEinstein (1905). Heobtained x(B)=  x(A)t(A) and t(B)=  t(A)x(A) ; (32) where =1 q 12: (33) Itiseasytodemonstrate thatthecoordinate transformation (32)satises thecondition that thequantity(31)isinvariant.Belowweexpress interms ofthecoordinates ofreference frame A,theeventdistance (31)forthetwoeventsaandbasdetermined inreference frame B.We makethereb yuseofexpressions (32)and(33). n x(B) ax(B) bo2n t(B) at(B) bo2= (34) =n  x(A) at(A) a  x(A) bt(A) bo2n  t(A) ax(A) a  t(A) bx(A) bo2 = 2n x(A) ax(A) b  t(A) at(A) bo2 2n t(A) at(A) b  x(A) ax(A) bo2 = 2 12 x(A) ax(A) b2+ 21 t(A) at(A) b2 = x(A) ax(A) b2 t(A) at(A) b2: Once more using relation (32), wededuce belowarelation fortheaddition ofvelocities. When reference frame Bmoveswithconstan tvelocity=V=cwithrespecttoreference frame A,andanobjectmoveswithvelocitydx(A)=dt(A)inreference frame A,thenwendforits velocityasmeasured withrespecttothecoordinates usedinreference frame Bthefollowing. dx(B) dt(B)=dx(B) dt(A) dt(B) dt(A)= dx(A) dt(A)! 1dx(A) dt(A)!=dx(A) dt(A) 1dx(A) dt(A): (35) Inparticular foranobjectwhichtravelswiththespeedoflight(c=1)inframe A,wehavein frame B c(A)=dx(A) light dt(A)=1()c(B)=dx(B) light dt(B)=1 1=1: (36) 10 Fromthisresult wemayconclude thatobjectswhichmovewiththespeedoflightwithrespect tothecoordinates ofonereference frame, alsomovewiththespeedoflightasobserv edby using thecoordinates ofareference frame whichisinmotion withaconstan tvelocitywith respecttotherstreference system. Orinother words, lightmoveswiththesame velocity withrespecttoobserv ersindieren tinertial systems. 5.1Velocities neverexceed thevelocityoflight Ininertial frame Awestudy anobjectwithvelocityv(inunitsc)withrespecttothecoordinates ofreference frame A.Weassume thatvissmaller thanthevelocityoflight,i.e. jvj<1()1+v>0and 1v>0: (37) Letusfurthermore consider areference frame Bwhichmoveswithconstan tvelocity,also smaller thanthevelocityoflight,withrespecttoreference frame A.Now,since bothvand aresmaller thanthevelocityoflight,wehave,bytheuseofrelations (37),thefollowing inequalities. 1v>0 (1+v)(1)<0()(1v)<v 0<(1v)(1+)()v<1v: Putting things together, wend 1<v 1v=v(B)<+1: (38) Thisproofsthatv(B),whichisthevelocityoftheobjectunder study,asmeasured inthe coordinate system B,isalwayssmaller thanthevelocityoflightc=1,forthecasethat velocities inAdonotexceed thevelocityoflight. Velocities larger thanthevelocityoflight,while not(yet?)observ ed,donotseemtomake partofourphysical world,hence, donothavetobeconsidered (yet?).Thisdoesnotmean that itisforbidden tostudy theproperties oftachyons,but,justmeans thatwedonotyetneedto study them. 5.2Small velocities Here, wewillconcen trate onall-dayvelocities, likejogging Ericandbicycling Diane. With respecttoAlexthevelocityofjogging Ericisgivenby(seeformula13) dx(A) Eric dt(A)=2:0m/s 3:0108m/s=2 3108: (39) Furthermore, thereference frame ofDiane moveswithaconstan tvelocity(seeformula7)given by dx(A) Diane dt(A)=2:8m/s 3:0108m/s=2:8 3108: (40) 11 Hence, bytheuseofformula(35),wendforthevelocityofEricinthereference frame of Diane thefollowing. 2 31082:8 3108 12 32:8 310160:8m/s 3108m/s 1+5:6 91016 : (41) Fromformula(41)weobtain theresult thatonlyinthe16-th decimal wemaynotice adierence fromtheaddition rule(15). Inpractice itisimpossible toverifythisresult, sincealready the errors inthemeasuremen tsofthevelocities ofEricandDiane areorders ofmagnitude larger. Consequen tly,forall-dayvelocities wewillnotnotice anydierence betweenGalilei and Poincar etransformations. Evenwiththevelocities ofsupersonic airplanes, whicharemore thantwoorders ofmagnitude larger thanrunning ordriving abicycle, theeects areonly insomewhere the11-th decimal. However,atcosmic scales orinparticle accellerators where objectsreachvelocities close tothelightvelocity,thedierences betweenthetwotypesof transformations areverywellobserv able. Millions ofexperimen tseverysingle dayconrm that Einstein's basic assumption (1905) ontheconstancy ofthelightvelocitywasveryclever. 5.3Energy andmomen tum Letusdene forapointparticle withvelocityv "=1 p 1v2and ~u=~v p 1v2; (42) whichinonedimension reduces to "=1 p 1v2and u=v p 1v2: (43) First, bytheuseofrelations (33)and(35),wedetermine "(B)=1 q 1v(B)2= 1v(A) q 1v(A)2and u(B)=v(B) q 1v(B)2= v(A) q 1v(A)2; (44) whichcanbewritten intheform (B)=  (A)u(A) and u(B)=  u(A)(A) : (45) Oncomparison offormulas(45)withformulas(32),weobserv ethatthepair(;u)trans- forms thesame wayasthepair(t;x).Consequen tly,thequantity 2u2; (46) isaninvariantunder Lorentztransformations. Moreo ver,from 1 p 121 11 221+1 22; (47) 12 wendthattolowestorder m"m+1 2mv2and mumv: (48) Therstexpression informula(48)corresp ondstoaconstan t,m,whichrepresen tsthemass ofaparticle, added tothekinetic energy oftheparticle. InNewtonian mechanics thezeroof energy isanyhownotwelldened. Hence, m"represen tsforlowvelocities thekinetic energy oftheparticle, sincetheconstan ttermisofnoimportance. Furthermore, thesecond termin formula(48)represen tsthelinear momen tumoftheparticle forlowvelocities. Those concepts canbegeneralized. Here, wedene forthetotalenergy,E,andthelinear momentum ,pofaparticle whichhasamassmandwhichmoveswithvelocityvinacertain reference frame E=m p 1v2and p=mv p 1v2: (49) Under Lorentztransformations Eandptransform thesame wayas"andu.Fromformula (49)weobserv ethatwhen vapproac hesthelightvelocityc=1,thenEtends toinnit y.This corresp ondsverywelltoexperimen talobserv ation inparticle accellerators. When aparticle isatrest,itsenergy equals m.Thisisexactly themostfamous formulaof physics: E=mc2.Therestmassmofaparticle isinvariantunder Lorentztransformations, i.e. E2p2=m2: (50) 5.4Totalinvariantmass Forasystem oftwoparticles aandbthetotalenergy isgivenby E=Ea+Eb=q m2a+p2a+q m2 b+p2 b; (51) andthetotallinear momen tum,obviously ,by p=pa+pb: (52) Itisnotdicult todemonstrate thatthetotalinvariant mass,ps,squared, dened by s=fEa+Ebg2fpa+pbg2; (53) isinvariantunder Lorentztransformations. 13 6Relativistic kinematics Most oftheconcepts whichwestudied intheforegoing, canstraigh tforwardly beextended to three dimensions. Inthefollowing, westudy scattering processes inthree dimensions. For scattering processes onecanavoidtocarry outexplicitly theLorentztransformations bythe useoftheso-called Mandelstam variables. Thelatter areLorentzinvariant,hence thesame in anyinertial frame. Thetotalenergy ,Etotal,forasystem oftwonon-in teracting on-mass-shell particles ofmasses m1andm2,whicharefreely movingwithlinear momen tarespectively~p1and~p2,isgivenby thesumoftheindividual energies, E(~p1)andE(~p2)respectively,according to Etotal=E(~p1)+E(~p2)=q ~p12+m12+q ~p22+m22: (54) Inthecenter-of-mass frame, where ~p1=~p2=~p,onehasthefollowingrelations: s=(ECM,total)2=2~p2+m12+m22+2q ~p2+m12q ~p2+m22;  s2~p2m12m222=4 ~p2+m12 ~p2+m22 ; 4s~p2=s22s m12+m22 +m14+m242m12m22; and~p2=1 4snh s(m1+m2)2ih s(m1m2)2io : (55) TheMandelstam variables, s,tandu,fortheprocess 1+2!3+4 (56) aredened by s=(p1+p2)2;t=(p1p3)2;u=(p1p4)2; (57) or,alternativ ely,byusing totalfour-momen tumconserv ation whichisgivenby p1+p2=p3+p4; (58) onealsohas s=(p3+p4)2;t=(p2p4)2;u=(p2p3)2; (59) Notice thatweuseherethemetric (+;;;),whichforsgives s=(p1+p2)2=(E(~p1)+E(~p2))2(~p1+~p2)2: (60) Inthethecenter-of-mass frame, where ~p1=~p2,oneobtains, moreo ver s=(E(~p1)+E(~p2))2=(ECM)2; (61) whichequals thetotalinvariantmass, asalready anticipated informula(55). Furthermore, from their denition oneobserv esthattheMandelstam variables (57)are Lorentzinvariantandhence invariantswithrespecttoanyLorentztransformation. 14 Bytotalmomen tumconserv ation (58),onededuces s+t+u=3p12+p22+p32+p42+2p1
(p2p3p4) =3p12+p22+p32+p422p12 =p12+p22+p32+p42 =m12+m22+m32+m42: (62) Consequen tly,foron-mass-shell processes s,tanduarenotindependen t. InFig.1wevisualize things forthecenter-of-mass frame. CM1 2 ~p1 ~p2 m1 m2CM 43 ~p4~p3 m4m3 #CM Before collision After collision Figure 1:Collision inthecenter-of-mass system. Before collision particle 1andparticle 2move towardstheircenterofmass withequal andoppositethree-momen ta,~p1and~p2respectively.After collision particle 3andparticle 4moveawayfromtheircenterofmasswithequal andoppositethree- momen ta,~p3and~p4respectively. Theangle betweenthedirection ofmotion oftheoutgoing particle 3andthedirection of motion oftheincoming particle 1isdened astheangle#CMofthescattering processofformula (56)inthecenter-of-mass system. Ithasthefollowingrelation withtheMandelstam variable t. t=(p1p3)2(63) =(p1)2+(p3)22p1
p3 =(p1)2+(p3)22E(~p1)E(~p3)+2~p1
~p3 =(m1)2+(m3)22q (~p1)2+(m1)2q (~p3)2+(m3)2+2j~p1jj~p3jcos(#CM): 15 6.1 +!K+K Consider a+meson whichannihilates withameson, resulting intwooutgoing Kaon mesons, aK+meson andaKmeson. Themeson isatrestinthelaboratory ,whereas the+meson hasatotalenergy of9.0GeV. TheKaon meson comes outwithanangle of60 withrespecttothedirection oftheincoming pion,inthecenter-of-mass system. Giventhisinformation, wemaydetermine theother kinematical quantities. Forthemasses oftheparticles wetake m=0:14GeV ;mK=0:50GeV ;mK=0:89GeV : Letusrstdetermine thetotalinvariantmassofthesystem. ps=q 2m2+2E+m=1:60GeV : With thatresult, alsousing formula(55),wemaydetermine ~p2inthecenter-of-mass frame. (~p)2=1 4h s4m2 i =0:62(GeV) : Next, wecandetermine (~pK)2inthecenter-of-mass frame, asfollows (~pK)2=1 4snh s(mK+mK)2ih s(mKmK)2io =0:148(GeV) : Thelinear momen tumofKisinthecenter-of-mass frame oppositeto~pK,ofcourse. Conse- quently,wecancheckourcalculations bydetermining thetotalinvariantmasspsaftercollision. Thisgives ps=q ~pK2+m2 K+q ~pK2+m2 K=0:63(GeV) +0:97(GeV) =1:60(GeV) ; whichisindeed what weobtained forthesituation beforecollision. Weobtain herethusthat thetotalenergy beforeandaftercollision isthesame, hence, totalenergy isconserv ed. Then wemaydetermine tandu t=(m)2+(mK)22EEK+2j~pjj~pKjcos(#CM)=0:437GeV2: and u=(m)2+(mK)22EEK+2j~pjj~pKjcos(#CM)=1:04GeV2: Wewilldenote thetotal momen tumofaparticle inthelaboratory byqanditslinear momen tumby~q.Themeson isatrest,hence ~q=0.Forthe+wehave E(~q+)=q (~q+)2+m2 +=9(GeV) ()~q+9(GeV) : Moreo ver,since~q=0,wehaveforttherelation t=(qqK)2=m2 +m2 K2mq ~qK2+m2 K; hence ~qK2= m2 +m2 Kt 2m!2 m2 K=4:37GeV2: 16 Similarly u=(qqK)2=m2 +m2 K2mq ~qK2+m2 K; hence ~qK2= m2 +m2 Ku 2m!2 m2 K=4:66GeV2: Also E(~qK)=q ~qK2+m2 K=4:68(GeV) ; and E(~qK)=q ~qK2+m2 K=4:46(GeV) : Finally ,wedetermine theangle intheframe ofthelaboratory ,ofthedirection oftheoutgoing Kmeson withrespecttothedirection oftheincoming +meson. cos(K;lab)=tm2 m2 K+2E(~q+)E(~qK) 2j~q+jj~qKj=0:9974 ; corresp onding toanangle of4.1degrees. Although hidden, bytheuseofMandelstam variables, theabovecalculus isbased inLorentz transformations andtheconstancy ofthelightvelocity.Itis,furthermore based inthedeni- tions(49)fortheenergy andlinear momen tumofamovingparticle. Theresults areveried byexperimen t,theultimate judge onourguesses. Everysingle day,invarious particle ac- cellerators, millions ofsuchscattering processes, involving two,ormanymore particles, are performed. Uptilltoday,nothing hasbeenfound whichcon icts withEinstein's assumptions. 6.2Elastic Scattering inthecenter-of-mass system Inelastic scattering theoutgoing particles areidenticaltotheincoming particles, whichatthis levelamoun tstom3=m1andm4=m2.When wedene thethree-momen tabeforeandafter collision byrespectively~pand~p0,thenwehave ~p1=~p2=~pand ~p3=~p4=~p0; (64) hence, byusing formula(55), ~p2=1 4snh s(m1+m2)2ih s(m1m2)2io and ~p02=1 4snh s(m3+m4)2ih s(m3m4)2io : (65) Since, moreo ver,m3=m1andm4=m2forelastic scattering, wehaveforthecenter-of-mass three-momen tainthatcase ~p2=~p02: (66) Substitution oftheresult (66)inexpressions (61)and(63)givestheresults s=q (~p)2+(m1)2+q (~p)2+(m2)22 =(~p)2+(m1)2+(~p)2+(m2)2+2q (~p)2+(m1)2q (~p)2+(m2)2 =2(~p)2+(m1)2+(m2)2+2q (~p)2+(m1)2q (~p)2+(m2)2; (67) 17 and t=(m1)2+(m3)22q ~p2+(m1)2q ~p02+(m3)2+2j~pj~p0cos(#CM) =2(m1)22 ~p2+(m1)2 +2~p2cos(#CM) =2~p2f1+cos(#CM)g: (68) Furthermore u=(m1)2+(m4)22q ~p2+(m1)2q ~p02+(m4)2+2~p1
~p4 =(m1)2+(m2)22q (~p)2+(m1)2q (~p)2+(m2)22~p2cos(#CM):(69) Asisobvious fromthedenitions of#CMinFig.1and~pinformula(64),onehas ~p20and 1cos(#CM)+1; (70) hence fortheMandelstam variables s(formula67)andt(formula68),wend s(m1+m2)2and t0: (71) 18 6.3Elastic Scattering inthelabsystem Inthelaboratory system particle 2isassumed tobeatrest. Thisisvisualized inFig.2. Wedene thelaboratory four-momen tabyq1,q2,q3andq4,inorder todistinguish fromthe center-of-mass four-momen ta.Westudy hereagain thecaseofelastic scattering, whichimplies m3=m1andm4=m2. 1 2~q1 m1 m23 4~q3 ~q4m3 m4#lab Before collision After collision Figure 2:Collision inthelaboratory system. Before collision particle 1moveswiththree-momen tum ~q1towardsparticle 2atrest(~q2=0)inthecenterofcoordinates. After collision particle 3andparticle 4moveawayfromthecenterofcoordinates withthree-momen ta~q3and~q4respectively. Here, weobtain fortheMandelstam variables (57),which,asmentioned before, areinvariant under Lorentztransformations, hence thesameforthelaboratory system andthecenter-of-mass system, s=(q1+q2)2=(m1)2+(m2)2+2E(~q1)E(~q2)2~q1
~q2 =(m1)2+(m2)2+2E(~q1)m2 t=(q1q3)2=2(m1)22E(~q1)E(~q3)+2~q1
~q3 =2(m1)22E(~q1)E(~q3)+2j~q1jj~q3jcos(#lab) u=(q2q3)2=(m1)2+(m2)22m2E(~q3): (72) Alsousing formula(62),wendfort t=2m1+2m2su=2m2(E(~q3)E(~q1)): (73) Onedenes thekinetic energy oftheincoming particle by T1=E(~q1)m1: (74) Atthreshold, where ~q1=0,weobtain T1=0. 19 PartII Generalities 7Covariantandcontravariantcomponents Inavector space ofNdimensions wedene anarbitrary setofbasisvectors, ei;i=1;:::;N; (75) andtheirinnerpro ducts, givenby ei
ej=gij;i;j=1;:::;N: (76) Anarbitrary vectorvinthisN-dimensional vector space maybecharacterized byitscom- ponents,vi,onthebasis(75),asfollows v=viei; (77) where, asusually ,repeated indices imply summation. Using expression (76),weobtain fortheinnerpro ductoftwovectors theresult v
w=viwjei
ej=gijviwj: (78) Thevectorvofformula(77)mightequally wellbecharacterized byitsinnerpro ducts with thebasisvectors (75). Forthispurposewedene vi=v
ei: (79) There exitsofcourse arelation betweenthetwoquantities viandvi,dened informulas (77)and(79)respectively.Alsousing denition (76),weobtain forthatrelation thefollowing result vi=v
ei=vjej
ei=gijvj: (80) Since wearefreetochooseanybasisintheN-dimensional vectorspace under consideration, letusselect abasisfag,whichisrelated tothebasis(75)byanonsingular transformation A givenby aj=Ajieiandei= A1 ijaj: (81) Thevectorvdened informula(77)hasnewcomponents,sayv0j,atthenewbasisfag.A relation betweenthetwosetsofcomponentsis,bytheuseofthetransformations (81),readily found, trough therelation v0jaj=v=viei=vi A1 ijaj; toyield 20 v0j=vi A1 ij: (82) Wendthusthatthecomponentsofavectortransform withtheinverseofthetransformation ofthebasiselemen ts,forwhichreason those componentsaresaidtobecontra-variant . Thecomponentsofvwhicharedened informula(79)transform under thebasistransfor- mation (81)asfollows v0j=v
aj=v
Ajiei=Ajivi: (83) Aparan tly,thequantities(79)transform inthesamewayasthebasisvectors, forwhichreason theyaresaidtobecovariant . Theinversetransformation isforthecontravariantcomponentsgivenby vi=v0jAji; (84) as,bytheuseofthetransformation property(82)forcontravariantcomponentsandtheusual rulesforthecomponentsofproducts oftransition matrices, caneasily beseenfrom v0jAji=vk A1 kjAji=vk A1A ki=vki k=vi: Anarbitrary pointPintheN-dimensional vectorspace under consideration canbecharac- terized bythecomponentsfxgofitsposition vectorx(P),withrespecttothebasisfegwhich isdened informula(75),butequally wellbythecomponentsfx0gwithrespecttothebasis fagdened informula(81),according to x(P)=xiei=x0jaj: (85) Relations betweenbothsetsofcoordinates fxgandfx0gare,according toformulas(82)and (84),givenby x0j=xi A1 ijandxi=x0jAji: (86) Suchrelations mightalsobeseenasthedenitions ofthebasistransformations (81).Moreo ver, therelations (86)arelinear relations betweenthetwosetsofcoordinates andhence itfollows @x0j @xi= A1 ijand@xi @x0j=Aji; (87) suchthatwealsoobtain x0j=xi@x0j @xiandxi=x0j@xi @x0j: (88) 21 8Themetrical tensor Theobjectgwhichisdened informula(76),willbegiventhenamemetric altensor .Westudy inthissection itstransformation rulesunder acoordinate transformation oftheform(88). Fromrelations (76)and(81)oneobtains forthemetric g0atthefagbasis, thetransformation rule g0 k`=ak
a`=AkiA`jei
ej=AkiA`jgij; whichuponsubstitution ofrelation (87)takestheform g0 k`=@xi @x0k@xj @x0`gij: (89) Relation (89)isoneofthebasic relations indierential geometry . Thecomponentsoftheinverseofthemetrical tensor aredenoted withupperindices, ac- cording to  g1gi j=gikgkj=i j: (90) Those componentstransform under thecoordinate transformation (88)asfollows g0k`=@x0k @xi@x0` @xjgij: (91) Notice, thatbythedenition (76)boththemetric tensor anditsinversearesymmetric intheir indices. 22 9Localbasis Inthissection weconsider anorthonormal basisfeginanN-dimensional Euclidean space and associated withitasetofcoordinates fxg.Theinner productofthebasis vectors (orthe metric ofthespace) isthengivenby ei
ej=ij; (92) andanypointPinspace byitscomponents,according to x(P)=xiei: (93) Asanexample consider ordinary three-dimensional space.Forthethreebasisvectorswe selectforourexample e1=^x,e2=^yande3=^z,whereasforthecoordinates wetakex, yandz. Now,inthisspace weselect adieren tsetofcoordinates fx0g,suchthateachpointinthe vectorspace canuniquely bedescrib edbythose coordinates. Thismeans ingeneral thatthere exists relations betweenthetwosetsofcoordinates fxgandfx0gwhicharesucien tlywell behaved,suchthatwemaytakeasmanyderivativesasweneedinthefollowing. Takeasanexample thesetofspheric alcoordinates r,#and'which arerelatedtothe ordinary 3Dcoordinates, x,yandz,by x=x(r;#;')=rsin(#)cos('),y=y(r;#;')=rsin(#)sin(')and z=z(r;#;')=rcos(#). Associated withthesetofcoordinates fx0gwechooseateachpointofourspace anewlocal basisfu(x0(x))g,whichbasisservesformeasuremen tsinthedirect vicinit yofthepointunder consideration, buthasnomeaning ataglobal level.Thetransformations whichrelate the global basisfegandthelocalbasisfu(x0(x))g,aregivenby uj(x0)=@xi @x0jeiandei=@x0j @xiuj(x0): (94) Forthe3Dspheric alcoordinates, weobtain foralocalbasisthe^r,^#and^'unitvectors: ^r(r;#;')=f^xcos(')+^ysin(')gsin(#)+^zcos(#), ^#(r;#;')=r[f^xcos(')+^ysin(')gcos(#)^zsin(#)]and ^'(r;#;')=r[^xsin(')+^ycos(')]sin(#). 23 10Themetric ofthelocalcoordinates Themetric gofthenewcoordinates is,byanalogy offormula(76),determined bytheinner products ofthelocalbasisvectors. Using formulas(92)and(94),onends gk`(x0)=uk(x0)
u`(x0)=@xi @x0k@xj @x0`ij; (95) whereas, asinformula(91),fortheinversemetric follows gk`(x0)=@x0k @xi@x0` @xjij: (96) Forthe3Dspheric alcoordinates, wendforthemetric g(r;#;')=0 BBBB@grrgr#gr' g#rg##g#' g'rg'#g''1 CCCCA=0 BBBB@^r
^r^r
^#^r
^' ^#
^r^#
^#^#
^' ^'
^r^'
^#^'
^'1 CCCCA=0 BBBB@10 0 0r20 00r2sin2(#)1 CCCCA (97) 11Dieren tiation withrespecttothelocalbasis Fordieren tiation withrespecttothelocalcoordinates weintroduceanewnotation, inorder tosimplify theformulastocome. Wewrite: F(x0);i=@F(x0) @x0i: (98) Inthisnewnotation wemaycast,forexample, expression (95)forthelocalmetric, intheform gk`(x0)=ijxi;kxj;`: (99) Notice however,thatforthetransformation (96)thenotation remains asitwas. Fordouble derivativeswehavetwoalternativ es,oneofwhichismore compact andwillbe usedinthisnotes, i.e. F(x0);ij=F(x0);i;j=@2F(x0) @x0i@x0j: (100) Since dieren tation doesnotdependontheorder, onehasmoreo verthat F(x0);ij=F(x0);ji: 24 12Christoel symbols(ane connection) Inthissection westudy howthelocalbasisvectors change ifwemoveinspace fromoneplace totheother. Itmightbeclearthatthevariations ofthelocalbasisvectors insuchaprocess arecompletely determined onceweknowthederivativesofthese vectors ineachpoint.So, werestrict ourselv estoinnitesimal small variations inspace. Moreo ver,wemayassume that thederivativeofoneofthelocalbasisvectors canbeexpressed asalinear combination ofthe complete localbasis, since locallythelatter formacomplete basis. Thecoecien tsofsuch expression arecalled ane connections ,symbol.Therelation is,bytheuseofthenotation dened informula(98),givenby: uj(x0);i=k ij(x0)uk(x0): (101) Inorder toexpress the'sinterms ofderivatives,letus,alsousing formulas(94)and(100), consider thefollowingexpression uj(x0);i= x`;je` ;i=x`;ije`=x`;ij@x0k @x`uk(x0); fromwhichwe,bycomparing withformula(101), deduce theidentity k ij(x0)=x`;ij@x0k @x`: (102) Notice thattheane connection issymmetric inthelowerindices, sincedieren tation is,i.e. k ij(x0)=k ji(x0): (103) Thenon-zer ocomponents oftheane connections forthe3Dspheric alcoordinates, are thefollowing r ##=x`;##@r @x`=rsin(#)cos(')x rrsin(#)sin(')y rrcos(#)z r=r; r ''=x`;''@r @x`=rsin(#)cos(')x rrsin(#)sin(')y r=rsin2(#); # ''=x`;''@# @x`=rsin(#)xcos(')+ysin(') r2px2+y2z=cos(#)sin(#); similarly # r#=1 r=# #r;' r'=1 r=' 'r;' #'=cos(#) sin(#)=' '#(104) 25 13Therelation betweentheane connection andthe metric Forcompleteness, werepeatherethederivation oftherelation betweenthemetrical tensor (95) andtheane connection (102). First, wedetermine thederivativeofthecomponentsofthemetrical tensor withrespectto thelocalcoordinates gij(x0);k= `mx`;ixm;j ;k=`mx`;ikxm;j+`mx`;ixm;jk: Next, wenotice thatifwetakealinear combination oftheaboveexpression forthederivatives, obtained bypermuting theindices (i;j;k),thenwecansingle outoneterm, according to kij=1 2 gki;j+gkj;igij;k =`mx`;ijxm;k: (105) Theabovequantities aretheChristoel symbols,whichcanberelated totheane connec- tions(102), alsousing formula(96),by gk``ij=@x0k @xp@x0` @xqpqmnxm;ijxn;` =@x0k @xppqnqmnxm;ij=k ij: (106) Using themetric altensor which isgiven informula (97),weobtain forthenon-zer o components oftheane connections forthe3Dspheric alcoordinates, thefollowing r ##=grrr##=r;# r#=# #r=g###r#=1 r; r ''=grrr''=rsin2(#);' r'=' 'r=g'''r'=1 r; # ''=g###''=sin(#)cos(#);' #'=' '#=g'''#'=cos(#) sin(#): (107) 26 14Thederivativesofavector eld SupposethatintheN-dimensional Euclidean space (92)isdened anarbitrary vector eld v(x).With respecttotheglobal Cartesean basisfegletitscomponentsinsome pointP, whichhascoordinates x,begivenby v(x)=vi(x)ei; (108) andwithrespecttothelocalbasisu(x0)atP,whichhascoordinates x0inthecorresp onding coordinate system, by v(x(x0))=v0i(x0)ui(x0): (109) Inthevicinit yofthepointPonemightwishtodetermine thederivativesofthevector eld withrespecttothelocalcoordinates. However,sincethelocalbasisvectors udier fromplace toplace, alsotheirderivativeswillbeinvolved,i.e. v;i= v0kuk ;i=v0k ;iuk+v0juj;i; which,bytheuseof(101), leads to v;i=( v0k ;i+v0jk ij) uk: (110) Another quantityofinterest isthecovariantcomponentofthederivativeofthevectoreld. Using thedenition (83)ofthecovariantcomponentofavectorandformula(101), oneobtains v;j
ui= v
ui ;jv
ui;j =v0i;jk ijv
uk=v0i;jk ijv0k: (111) 15Covariantderivative Itiscommon practice tointroduceanewnotation forthecomponentsofthederivativeofa vectoreldwithrespecttothelocalbasis. Using theresult (110), wewriteforthose components v;i=v0k ;iukwith v0k ;i=v0k ;i+k ijv0j; (112) andwhicharesometimes called thecovariant derivative ofthecontravariant components ofa vector eld. Forthecovariantcomponentsofavector eld,using theresult (111), wewrite similarly its covariantderivativesby v0i;j=v;j
ui=v0i;jk ijv0k: (113) 27 PartIII Three dimensions Inthefollowing, westudy curvesandsurfaces inthree dimensions. 16Curvesinthree dimensions InthecaseN=3,thebasis vectors fegandthecoordinates fxgrepresen tthewell-kno wn Cartesean coordinates inthree dimensions. Inthatspace weparametrize anarbitrary curveby arealparameter t,suchthatwhen onemovesalong thecurve,tchanges inacontinuousway fromonerealvaluetotheother. Now,informula(93)wecharacterized onepointxinspace byitscoordinates fxgatthebasisfeg.Similarly ,wecharacterize hereawhole curvebyletting thecoordinates dependcontinuously ontheparameter t,i.e. x(t)=xi(t)ei: (114) Specic curvescanbecharacterized bytheadequate choice forthethree functions oft,x1(t), x2(t)andx3(t).Wewillhereonlyallowforcurvesforwhichthose three functions areinnitely manytimes dieren tiable asfunctions oft. Thetangen tvector inanarbitrary pointofthecurveisgivenbytherstderivativeintof thecurve _x(t)=dx(t) dt=dxi(t) dtei=_xi(t)ei: (115) Itslength iseviden tlyrelated totheusual expression ofthesquare ofthelength ofavector (remem berthatwehaveanorthonormal basisfeghere): j_x(t)j2=_x(t)
_x(t)=_xi(t)_xj(t)ij: (116) Thisquantitycanalsobeusedtomeasure adistance s2s1ofalinesegmen talong the curve.Clearly ,onemusttherefor integrate overtheinterval(t1;t2)intheparameter twhich corresp ondstothatsegmen t.Wendthen s2s1=Zt2 t1dtj_x(t)j=Zt2 t1dtr _xi(t)_xj(t)ij: (117) Fromtheaboveexpression onemaydeduce thataninnitesimal distance dsisdetermined by ds2=dxidxjij: (118) Now,sincetheparameter s,whichiscalled theproperlength along thecurve,isamore con- venientparametrization ofthecurve,thananarbitrary choice, wewilluseinthefollowings astheparameter along thecurve.Derivativeswithrespecttoswillbedenoted byaprime, instead ofadot(nottobeconfused withthenotation foranewcoordinate setasdened in section (9),butthatmightbeclearfromthecontext). So,expressed intheproperlength parameter s,thecurveisnowcharacterized by 28 x(s)=xi(s)ei; (119) anditstangen tvector by (s)=x0(s)=dxi(s) dsei: (120) Notice, that, byourchoice (118) ofparametrization, thetangen tvector(s)hasunitlength inanypointofthecurve.Thiscanalsobeseenasfollows: j(s)j2=dxi(s) dsei
dxj(s) dsej=dxi(s) dsdxj(s) dsij=dxidxjij ds2=ds2 ds2=1: (121) Forarstexample, letusconsider acircleofradius Rinthe(x;y)-plane, centeredat theorigin. Weparametrize apoint,P,ofthecirclebytheangle, t,itsposition vector, xP,makes withthex-axis. Below wewriteourparametrization, x(t),oftheposition vectorsofpointsatthecircumfer enceofthecircleandtheirrelatedtangent vectors, _x(t), respectively: x(t)=0 B@Rcos(t) Rsin(t) 01 CAand _x(t)=0 B@Rsin(t) Rcos(t) 01 CA: Thelength ofthetangent vectorequalsR.Forthelength ofalinesegment ofthecurve, wend s2s1=Zt2 t1dtR=R(t2t1); fromwhich wededucethatfortheproperlength parameter wemayselects=Rt.Inthis parametrization wehavethen x(s)=0 B@Rcos(s=R) Rsin(s=R) 01 CAand(s)=x0(s)=0 B@sin(s=R) cos(s=R) 01 CA: (122) Noticethatx0(s)hasunitlength. 29 Forasecondexample, westudyacircularhelix,orscrew,ofradius aandconstant speed bcenteredaroundthez-axis. Fortheparametrization oftheposition vectors, x(t),of pointsatthiscurve andtheirrelatedtangent vectors, _x(t),wechooserespectively: x(t)=0 B@acos(t) asin(t) bt1 CAand _x(t)=0 B@asin(t) acos(t) b1 CA: Thelength ofthetangent vectorequalsp a2+b2.Forthelength ofalinesegment ofthe curve, wend s2s1=Zt2 t1dtp a2+b2=p a2+b2(t2t1); fromwhich wededucethatfortheproperlength parameter wemayselects=p a2+b2t. Inthisparametrization wehavethen x(s)=0 BBBBBBB@acos sp a2+b2 asin sp a2+b2 bsp a2+b21 CCCCCCCAand(s)=0 BBBBBBB@ap a2+b2sin sp a2+b2 ap a2+b2cos sp a2+b2 bp a2+b21 CCCCCCCA: (123) Noticethatx0(s)hasunitlength. 30 17Thenatural localbasisofacurve Inthissection wewillconstruct asetofthree vectors ateachpointP(s)along thecurve,which servesasalocalbasisfortheEuclidean three-dimensional space inthevicinit yofP(s). Therstvectorofthissetistheunittangen tvector(s),dened informula(120). Forthe second vector onemayselect thederivativeofthetangen tvector, normalized tounity h(s)=d(s) dsd(s) ds: (124) Thetangen tvector(s)anditsderivative0(s)areorthogonal inanypointofthecurve,as canbeseen,alsousing formula(121), from 0=d1 ds=df(s)
(s)g ds=20(s)
(s): (125) Thevariation ofthetangen tvector along thecurveinthevicinit yofapointP(s)indicates theamoun tofcurvature ofthecurveatthatpoint.Inorder toseethismore explicitly ,letus determine thedierence betweenthetangen tvector atthepointP(s)andthetangen tvector atthepointP(s+
s).Torstorder thisdierence isgivenby (s+
s)(s)0(s)
s: (126) Now,since thetangen tvectors haveunitlength, themodulus oftheabovedierence yields, alsotorstorder, theangle (s)betweenthetwovectors atlocation P(s),i.e. j(s+
s)(s)j(s): (127) Moreo ver,thecurvature radius ofthecurveatlocation P(s)multiplied bytheangle(s)equals thedistance along thecurvebetweenthetwopointsP(s)andP(s+
s).Whereas, furthermore thelength ofthislinesegmen tequals
s,sincetheparametrization (119) corresp ondstothe length oflinesegmen tsonthecurve.Hence, onends j
sj(s) (s); (128) where (s)stands fortheinverseoftheradius ofcurvature atlocation P(s).Thechoice to parametrize curvature bytheinverseoftheradius, rather thanbytheradius itself, isquite logical, sincethenonehasvanishing parameter intheabsence ofcurvature. Now,byjoining thethree pieces (126), (127) and(128), weobtain therelation j
sj(s) (s)j(s+
s)(s)j (s) j0(s)jj
sj (s); fromwhichweconsequen tlymayconclude thatthederivativeofthetangen tvector (120) and thenormalized derivative(124), arerelated viatheparameter oflocalcurvature, by 31 0(s)=(s)h(s): (129) Thethirdvectorb(s)ofthenatural localbasisatP(s)isgivenbytheouter productof(s) andh(s),i.e. b(s)=(s)h(s): (130) Since both(s)andh(s)areunitvectors andmoreo verorthogonal, itisclear fromthe denition (130) thatalsob(s)mustbeunity.Hence, thenatural localbasisvectors (s),h(s) andb(s)formanorthonormal set. Forthecircleofexample (122) weobtain forthederivative ofthetangent vector 0(s)=0 BBBBB@1 Rcos(s=R) 1 Rsin(s=R) 01 CCCCCAand(s)=j0(s)j=1 R; (131) which leadsforh(s)andb(s)to h(s)=0 B@cos(s=R) sin(s=R) 01 CAandb(s)=0 B@0 0 11 CA=^z: (132) Forthecircularhelixofexample (123) weobtain forthederivative ofthetangent vector 0(s)=a a2+b20 BBBBB@cos sp a2+b2 sin sp a2+b2 01 CCCCCAand(s)=j0(s)j=a a2+b2:(133) Noticethatinthelimitb=0,wereturntothecaseofthecircleforR=a.Inthelimit ofb!1onehasastraightlineparalleltothez-axiswithvanishing curvatur e. Forh(s)andb(s)onehasforthecircularhelix h(s)=0 BBBBB@cos sp a2+b2 sin sp a2+b2 01 CCCCCAandb(s)=1p a2+b20 BBBBB@bsin sp a2+b2 bcos sp a2+b2 a1 CCCCCA: (134) 32 18Thederivativesofthenatural basis Knowledge ofthederivativesofthenatural localbasisvectors (s),h(s)andb(s)informs us aboutthedevelopmen tofthisbasisfordisplacemen tsalong theline.Below,weshowthatthese derivativesaregivenbytheFrenetrelations: 0(s)=(s)h(s) ;(s)=j0(s)j ; h0(s)=(s)(s)+(s)b(s);(s)=j(s)h0(s)j; b0(s)=(s)h(s) :(135) IntheproofoftheFrenetrelations weusethefactthatthederivativeofaunitvectorisalways perpendicular totheunitvector itself (seeformula125). Moreo ver,weknowthatbytheir denitions (120), (124) and(130), (s),h(s)andb(s)areunitvectors (seealsoformula121). Hence, thederivatives0(s),h0(s)andb0(s)areperpendicular torespectively(s),h(s)and b(s). 18.1 ProofoftheFrenet relations Therelation between0(s)andh(s)hasbeenstudied insection (17),where alsothedenition ofthecurvature parameter (s)hasbeengiven. Fortheproofofthethird lineinformula(135), weremem berthatboth(s)andh0(s)are perpendicular toh(s).Hence, theirouter productisparallel toh(s).Using denition (130) andformula(129), wendthen b0(s)=0(s)h(s)+(s)h0(s)=(s)h0(s); whichisavector parallel andoppositetoh(s).Moreo ver,sinceh(s)isunity,oneobtains jb0(s)j=j(s)h0(s)j; fortheconstan tofproportionalit y(s).WhichproofsthethirdoftheFrenet relations. Forthesecond relation, wejuststudy theinnerpro ducts ofh0(s)with(s)andb(s),since thenatural basisisorthonormal ateachpointP(s)ofthecurve.Using formulas(125) andthe rstrelation ofFrenet, weobtain 0=df0(s)
(s)g ds=00(s)
(s)+0(s)
0(s)=00(s)
(s)+2(s); whichleads to h0(s)
(s)= d ds0(s) (s)!
(s)=( 0(s)0(s) 2(s)+00(s) (s))
(s)=(s);(136) whichproofstherstpartofthesecond Frenet relation. Forthesecond part, weusethefactthath(s)andb(s)areperpendicular andthethird Frenet relation, resulting to h0(s)
b(s)=dfh(s)
b(s)g dsh(s)
b0(s)=(s); (137) whichproofsthesecond partofthesecond Frenet relation and,hence, completes theproofof theFrenet relations. 33 18.2 Darbouxvector Forcompleteness, westudy herethevariation ofthenatural localbasis vectors (s),h(s) andb(s)fordisplacemen tsalong thecurve.Wewillndthatthose vectors rotate around the so-called Darbouxvector, dened by d(s)=(s)(s)+(s)b(s): (138) Under asmall displacemen t
salong thecurveonemaywrite, torstorder, thetransfor- mation ofthenatural localbasisvectors, by 0 BBBB@(s+
s) h(s+
s) b(s+
s)1 CCCCA0 BBBB@(s) h(s) b(s)1 CCCCA+0 BBBB@0(s) h0(s) b0(s)1 CCCCA
s; which,using formula(135), canbecasted intheform 0 BBBB@(s+
s) h(s+
s) b(s+
s)1 CCCCA0 BBBB@(s) h(s) b(s)1 CCCCA+0 BBBB@
(s)
(s)
(s)
(s)
1 CCCCA0 BBBB@(s) h(s) b(s)1 CCCCA
s: Thematrix intheaboveexpression represen tsaninnitesimal rotation around theso-called Darbouxvector, whichisgiveninformula(138), whereas theangle oftheinnitesimal rotation followsfrom rotation anglejd(s)j
s=q 2(s)+2(s)
s: (139) When passing fromoneposition toanearbyposition along thecurve,thenatural basis vectors, while remaining arighthand orthonormal set,changes itsorientation following the abovedescrib edrotation. 34 19Atwo-dimensional surface inthree dimensions Inthefollowing, westudy atwo-dimensional arbitrarily curvedsurface embedded inathree- dimensional Euclidean space. Thethree-dimensional space weendowwithasetoforthonormal basis vectors fegandcoordinates fxg.Weassume thatthetwo-dimensional surface ischar- acterized byasetoftwocoordinates, fug.Thismeans thatweassume thateachpointPin thesurface isassociated withaunique setoftwocoordinates u1andu2.Weassume moreo ver thatthose coordinates varyinacontinuouswaywhen onemovesfromoneplace toanearby position. Thecomponentsofthethree-dimensional vectors x(P)whichconnect theorigin of thethree-dimensional embedding space topointsPonthesurface, arethiswayfunctions of thetwocoordinates whichcharacterize thesurface, i.e. xP u1;u2 =xi u1(P);u2(P) ei=xi(u)ei: (140) Weassume thenthatthethree functions xi(u)areinnitely manytimes dieren tiable func- tionsofthevariables u1andu2. Now,ateachpointofthesurface wecandene atangen tplane byselecting twovectors which uniquely characterize thisplane. Suchlocaltangen tbasisvectors canbeconstructed fromthe three-dimensional vectors associated tothepointsonthesurface, sincetheirderivativeswith respecttothesurface coordinates fugindicate locallythetangen tdirections, i.e. a(u)=x(u);=@xi(u) @uei: (141) where =1or2,andwhere irunsover1,2and3. Notice thatweindicate herewithacomma dieren tation withrespecttothearbitrary surface coordinates fug,sinceweconsider those thelocalcoordinates ofthetwo-dimensional surface (compare section 11). With thedierence thatherewedenote thecoordinates byfugandthelocalbasisvectors byfag,wecanuseformula(95)inorder todetermine thelocalmetric inthevicinit yofany pointP.Weobtain ingeneral g(u)=a(u)
a(u)=x(u);
x(u);=xi;xj;ij: (142) Being useful lateron,wecomplete herethesetoflocalbasis vectors byathird unitvector, normal totheplane, whichthusallowsforthedescription ofpointsoutside thetwo-dimensional surface inthevicinit yofapointPonthesurface. Wedene therefor n(u)=a1(u)a2(u) ja1(u)a2(u)j: (143) Bytheuseofformula(141) wemayrewrite theouter productoftheaboveexpression asfollows: a1(u)a2(u)=xi ;1xj ;2eiej=xi ;1xj ;2ijkek: (144) Where, ijkrepresen tstheLevi-Civita symbol,whichhasthefollowingproperty ijk`mk=3X k=1ijk`mk=i`jmimj`: (145) 35 Using thisequalit yandtheexpression (142), wendforthesquare ofthemodulus oftheouter product(144) theresult ja1(u)a2(u)j2=xi ;1xj ;2ijkek
x` ;1xm ;2`mnen =jx;1j2jx;2j2(x;1
x;2)2 =g11(u)g22(u)g12(u)g21(u) =det(metrical tensor )=g(u): (146) Using thisresult, weobtain forthelocalnormal vector (143) theform n(u)=a1(u)a2(u)q g(u): (147) Notice, asmaybeobvious, that n(u)
a1(u)=n(u)
a2(u)=0: (148) Alsoforlaterusewedetermine heretheinnerpro ductofthelocalnormal vector andan arbirary three-v ector, i.e. q g(u)v
n(u)=vixj ;1xk ;2jk`e`
ei=v
(x;1x;2): (149) 36 20Thederivativesofthelocalbasis Inthissection westudy thederivativesofthelocalbasisonthetwo-dimensional surface with respecttothelocalcoordinates fug.Itisworthwhile tonotice fromthestartthat,because of thewaythetangen tvecorsfagaredened informula(141), onehasthat a;(u)=x(u);; (150) issymmetric intheindices and. Fromformula(142) wededuce that g;(u)=a;(u)
a(u)+a(u)
a;(u) =x(u);
x(u);+x(u);
x(u);; forwhich,moreo verusing expression (105), weobtain fortheChristoel symbolsofthetwo- dimensional surface theexpression (u)=1 2 g;+g;g; =x;(u)
x;(u)=a;(u)
a(u): (151) Inorder todetermine theane connections fromformula(102), weneedmoreo vetheinverse ofthelocalmetric inthetwo-dimensional surface, thecomponentsofwhichwedenote here withupperindices asbefore(seeformula90).Wewrite then  (u)=g(u)(u): (152) Now,letuswrite forthederivativeofalocaltangen tvector (141), thefollowingdecomp o- sition interms ofthethree localbasisvectors a;=A a+Bn: Thecoecien tsA arereadily determined bytheuseofformulas(79),(80),(90),(151), (152) and(148). Oneobtains A = A =ggA =ga
aA  =ga
a;=g= : Remem bering thatthelocalunitnormal vectornisnormal tobothlocaltangen tvectors, we dene herethetorsion tensor Lby L(u)=a;(u)
n(u)=B: (153) 37 Notice, thatbecause ofproperty(150) thetorsion tensor issymmetric initsindices. Further- more, bytheuseofformulas(141) and(149), wemaycasttheexpression (153) inthefollowing form L(u)=x;(u)
n(u)=x;(u)
x;1(u)x;2(u)q g(u): (154) Weobtain thenforthedecomp ostion ofthederivativeofthelocaltangen tvectors, the expression a;(u)= (u)a(u)+L(u)n(u): (155) Forthederivativesofthelocalnormal werstremem berthatnisaunitvector, which according toformula(125) implies thatnisperpendicular toitsderivativeandhence onlyhas componentsinthetangen tplane. Let n;=N a; then, using oncemore formulas(79)and(80),andmoreo verdenition (153), onededuces N =gN=gn;
a=gn (n
a);n
a;o =gn
a;=gL: Hence, n;(u)=gLa(u): (156) 38 21Curvesonthesurface Aswehaveseenbefore, insection (16),wegivethenamecurve toaone-dimensional subspace, whichiscontinuousandthuscanbeparametrized byrealnumbersandwhichmoreo veris dieren tiable, asmanytimes aswedesire. What wehaveinmind hereissomething likea smoothlinedrawnwithaninnitesimally sharp pencilonapieceofpaper.Furthermore, what weactually onlyneeded fromthose curvesisasmall segmen tintheneighbourho odofawell dened pointofspace, asexplored insection (17). Acurveembedded inthetwo-dimensional surface dened insection (19),canobviously be parametrized byaparameter, t,suchthatthesurface coordinates, u,whichindicate points ofthecurve,become functions, u(t),oft.Weconsider herecurvesinthesurface whichare innitely manytimes dieren tiable. Acurveatthetwo-dimensional surface isthuscharaterized by x(t)=x(u(t))=xi(u(t))ei: (157) Thetangen ttothecurveatthepointP(t)is,bytheuseofequation (141), givenby _x(t)=du dtxi;(u(t))ei=du dta(u(t)): (158) Expression (158) showsnicely thefactthatthevectors fa(u)gspanlocallythecomplete tangen tspace, asindeed anarbitrary tangen tvector, tangen ttoanarbitrary curveinthe surface, canbewritten asalinear combination ofthevectors (141). Insection (16)weintroduced thenotion oftheproperlength, whichwedenoted bys(see formula118)andwhichmeasures distances oflinesegmen tsalong thecurvebymeans ofthe integral overthelengths ofinnitesimal lineelemen tsinthedirections ofthelocaltangen t vectors ofthecurve.Here, werepeatthisprocedure. Thelength ofthelocaltangen tvector (158) ishere,bytheuseofformula(142), givenby j_x(t)j2=du dtdu dta(u(t))
a(u(t))=du dtdu dtg(u(t)): (159) So,followingtheprocedure ofsection (16),weobtain fortheproperlength oftheabovedened curvetheexpression ds2=dudug(u(t)): (160) Clearly ,wewereuptointroducetheproperlength sforparametrizing thecurve. Asbefore(seesection 16),wedene herethelocalunittangen tvectoratthepointP(s) by (s)=x0(u(s))=du dsa(u(s)): (161) Notice that,since(s)hasunitlength andis,moreo ver,avector inthetangen tplane ofthe surface atpointP(s),allcurveswhichhaveatP(s)thesame tangen tialdirection, share the same tangen tvector(s)when parametrized bytheirproperlengths. Furthermore 1=j(s)j2=jx0(u(s))j2=du dsdu dsg(u(s)): (162) 39 Insection (17)wesawthatthevariations ofthelocaltangen tvectorforsmall displacemen ts, measure thecurvature ofthecurve.Fromequation (161), alsousing formula(155) forthe derivativesofthelocalbasisvectors, weobtain forthederivativesofthelocaltangen tvector thefollowing 0(s)=d2u ds2a(u(s))+du dsa0(u(s)) (163) =d2u ds2a(u(s))+du dsdu dsa;(u) =d2u ds2a(u(s))+du dsdu ds  (u)a(u)+L(u)n(u) =8 < :d2u ds2+ (u)du dsdu ds9 = ;a(u)+L(u)du dsdu dsn(u): whichisingeneral neither avector ofthetangen tplane, tangen ttothesurface atP(s),nora vector normal tothatplane. Aquantityofinterest istheinnerpro ductofthelocalnormal, n(u(s)),normal tothetangen t plane, and0(s).Wend,using formula(163) andthefactthatn(u(s))isnormal tothelocal tangen tvectors fag,thefollowing 0(s)
n(u(s))=du dsdu dsL(u(s)): (164) 40 22Thecurvature ofthecurvesonthesurface Intheprevious section westudied thelocaltangen tvectors, ,tangen ttocurvesonthetwo- dimensional surface whichpassatlocation Pandtheirderivatives,0.Fromexpression (163) it isclearthatthederivativesare,ingeneral, notnormal tothelocaltangen tplane. Furthermore, byitsdenition (143), thelocalnormal vector, n,normal tothelocaltangen tplane, isunity. Moreo ver,fromsection (17)welearn, asexpressed informula(129), thatthelength ofthe derivativeofthelocaltangen tvectorisequal totheinverse,,oftheradius ofcurvature ofthe curveatlocation P.Consequen tly,theinnerpro ductof0andnmustbeequal totimes the cosine oftheangle, say#,whichthetwovectors make.Alsousing formula(164), weobtain for aspecic curve,parametrized byitsproperlength parameter s,thefollowing k(curve)=cos(#)=L(u(s))du dsdu ds: (165) Fromitsdenition (154), wemustconclude thatthetorsion matrix doesnotdependon aspecic choice ofcurveatPonthetwo-dimensional surface, butonlyontheproperties of thesurface itselfatlocation P.Now,asdiscussed before(seethetextfollowingformula161), innitely manycurvesonthesurface share atPthesametangen tvector.Consequen tly,when werestrict ourselv estoallcurveswhith thesametangen tialdirection inP,which,according to formula(161), means aparticular choice forthetwovalues ofdu=ds ,thenthese curvesshare thesame valueforthequantitykasdened informula(165). Thecurvature, ,ofsuchclass ofcurvesinPdiers fromonecurvetotheother. But,thenalsotheangle #isdieren tfor eachcurve,insuchawaythatkisconstan t. Hence, wemayconclude thatkisafunction ofthetangen tialdirection only,notofa particular choice ofcurveoutoftheabovedescrib edclassofcurvesatthepointP.Wewrite therefor: k=k(): (166) Outoftheclassofcurveswhichbelong toacertain tangen tialdirection, ,weselect one represen tative:thecurvewhichinthevicinit yofPisdened bythecross section between thetwo-dimensional surface andtheplane dened bythelocalnormal vectornandthelocal tangen tvector.Forthatcurve,when parametrized byitsproperlength parameter s,0(s) mustatP(s)beparallel (orantiparallel) tothenormal vector, sinceitwillnothaveacomponent outoftheabovedened plane, when thecurveisentirely inside thatplane, and0(s)alsohas tobeperpendicular to(s).Thisrepresen tativecurvecorresp ondstoageodesic atthepoint P,aswewillseelateron. So,forageodesicatP,wehavefromformula(165) thefollowingresult #=0or;andk((s))=: (167) 41 23Thecurvature ofthesurface Fromtheprevious section welearnthattheradius ofcurvature, R((s)),ofthegeodesicinthe direction (s)atlocation P,isgivenby 1 R((s))=k((s))=L(u(s))du dsdu ds: (168) Wehaveinserted thesignbecause atthisstage itisnotclearwhether theangle #offormula (165) equals 0orforourchoice ofthelocalnormal n(s). Byinspection ofallpossible tangen tdirections, wendthatthecorresp onding values ofk havetwoextrema, whichwedenote byk1andk2.Westudy thisinthefollowing. Inorder to simplify thenecessary algebra, wedene r=du1 ds du2 ds!1 ; (169) L=L11(u);M=L12(u)=L21(u);N=L22(u); (170) and E=g11(u);F=g12(u)=g21(u);G=g22(u): (171) When wesubstitute theabovedenitions (169- 171)informula(168), using moreo verrelation (162), thenweobtain k()=L(u(s))du dsdu ds g(u(s))du dsdu ds=Lr2+2Mr+N Er2+2Fr+G: (172) Theparameter r,dened informula(169), parametrizes thevarious directions oftangen tvector. So,apparan tly,ourproblem isnowreduced tonding theextrema ofexpression (172). Atthe values ofr,whichwedenote by,forwhichthose extrema occur,onehas dk drr==0; whichleads tothefollowingquadratic equations for (a) (L+M) E2+2F+G (E+F) L2+2M+N =0; (b) (FLEM)2+(GLEN)+(GMFN)=0; (c) FL+M E+FM!2 EL+M E+FL! GL+M E+FN! =0: When, next, wedenote anextrem umvalueforkbyk,thenwehave,bysubstituting asolution oftheaboveequation informula(172), theexpression 42 k=L2+2M+N E2+2F+G=L+M E+F; anditsinverserelation =FkM EkL; (173) which,when inserted inthequadratic equation (c)for,leads toaquadratic equation forthe extrema ofk,i.e. (FkM)2(EkL)(GkN)=0: (174) Theequation (174) hasingeneral twosolutions, k1andk2,which,alsousing formulas(170) and(171), arecharacterized by K=k1k2=LNM2 EGF2=det(L) det(g)=det g1L ; (175) andby 2H=k1+k2=GL2FM+EN EGF2=gL=Tr g1L : (176) Itiscommon practice todene thecurvature ofthesurface atapointPbytheproduct (175) ofthetwoextrem umvalues fork.Theparameter Hiscalled theaverage curvature at P. Forthetangen tialdirection parameters r1andr2,whichcorresp ondrespectivelytothe extrema ofcurvature k1andk2,wemayproofthefollowingrelations: Er1r2+F(r1+r2)+G=0andLr1r2+M(r1+r2)+N=0: (177) Theproofofthose relations isamatter ofstraigh tforwardalgebra: First, onesubstitutes relation (173) forr1andr2andthenoneusestheexpressions (175) and(176) toobtain relations (177). 43 24Thelocalprinciple axes Thelocalprinciple axesofthesurface atlocation Paredened bythelocaltangen tialdirections, 1and2,ofthose geodesics atP,whichcorresp ondtotheextrem umdirections, r1andr2, forwhichkhasrespectivelytheextrem umvaluesk1andk2. Inthissection westudy those axesaswellasthedependence ofkonatthebasis1and 2forthelocaltangen tplane. According toitsdenition informula(161) wemaywrite anarbitrary tangen tvector of thetangen tplane atPasalinear combination ofthelocaltangen tvectors (141). Alsousing formulas(142), (169) and(171), weobtain =N(ra1+a2) (178) with 1=jj2=N2n Er2+Fr+Go : First, letusdemonstrate thattheprinciple axesareperpendicular, i.e. 1
2=0: (179) Proof: Fromformula(178) wededuce thattheaboveinnerpro ductmaybewritten intheform 1
2=N1N2fEr1r2+F(r1+r2)+Gg; forwhich,bytheuseoftherstofrelations (177), wendvanishing result, whichcompletes theproof. Hence, 1and2formanorthonormal basisforthelocaltangen tplane atP.Their directions arecalled thelocalprinciple axes.Anarbitrary tangen tialdirection, atPcanthusbewritten as =1cos(')+2sin('); (180) where theangle 'denes thedirection ofwithrespecttotherstprinciple axis. Thecurvature ofthegeodesic inthedirection ofischaracterized byk(')asdened in (168). Itsrelation withk1andk2isgivenby k(')=k1cos2(')+k2sin2('): (181) Proof: Westartfromexpression (178), tondfor(180) theform =N1(r1a1+a2)cos(')+N2(r2a1+a2)sin(') =[N1r1cos(')+N2r2sin(')]a1+[N1cos(')+N2sin(')]a2: Bycomparing thisexpression toequation (178), weconclude thefollowingrelations 44 Nr=N1r1cos(')+N2r2sin(')andN=N1cos(')+N2sin('): (182) Thecurvature parameter kforthetangen tialdirection (180) canbeobtained bysubstituting thenormNofformula(178) intoequation (172), togive k=N2n Lr2+2Mr+No =L(Nr)2+2MN(Nr)+NN2; (183) which,bysubstitution ofmoreo verrelations (182), leads to k=n L(N1r1)2+2MN1(N1r1)+NN12o cos2(')+ +n L(N2r2)2+2MN2(N2r2)+NN22o sin2(')+ +2N1N2fLr1r2+M(r1+r2)+Ngcos(')sin('): When wesubstitute hereequation (183) fork1inthersttermandfork2inthesecond term, andmoreo verthesecond relation offormula(177) inthethird term, thenweobtain result (181), whichcompletes theproof. 45 25Egregium theorem ofGauss Relation (175) forthecurvature ofthesurface canbeentirely expressed interms ofthemetrical tensor (142) anditsderivatives,aswewillstudy inthissection. Informula(155) wendthedecomp osition oftherstderivativesofthelocaltangen tial vectorsfa(u)g,interms ofthetwocomponentsinthelocaltangen tialplane andthecomponent perpendicular tothatplane. Fortheinnerpro ductoftwosuchobjects, alsousing formula(142), wend a;(u)
a;(u)= (u) (u)a(u)
a(u)+L(u)L(u)n(u)
n(u) = (u) (u)g(u)+L(u)L(u): Fromthisresult wededuce that L(u)L(u)=a;(u)
a;(u) (u) (u)g(u): (184) Relation (184) givesusthepossibilit ytoexpress thedeterminan tofthelocaltorsion tensor as follows det(L)=L11L22L12L21 (185) =a1;1
a2;2a1;2
a2;1  11 22 12 21 g Aseasily canbeveried, forinstance bytaking thedouble derivativesofthelocalmetrical tensor (142) thereb yremem bering thata;=a;(seedenition 141andformula150), thatthersttwoterms attherighthand sideofformula(185) canbeexpressed asalinear combination ofsecond order derivativesofthelocalmetrical tensor, i.e. a1;1
a2;2a1;2
a2;1=1 2 g11;222g12;12+g22;11 ; Substitution ofthisresult informula(185) forthedeterminan tofthelocaltorsion tensor L, leads forthelocalcurvature parameter (175) tothefollowingexpression: K=det(L) det(g)=1 g 1 2 g11;222g12;12+g22;11   11 22 12 21 g (186) Thisisanexpression whichonlyrefers tothelocalsurface coordinates fu1;u2gviathelocal metrical tensor anditsrstandsecond order derivatives.Asnoreference toanembedding space isinvolved,expression (186) usesonlytheinnerproperties ofthesurface forthedenition ofcurvature (EgregiumtheoremofGauss ). Almost vanishing curvature isdicult tomeasure, asmankind struggled formillions of yearstodetect signsofthecurvature oftheEarth's surface andeventhendecided toonlybe 46 really convinced aftersailors made theirtripsaround theGlobe.However,those voyageswere notnecessary astheGreek civilization already hadknowledge ofthecurvature oftheEarth's surface andrough estimates ofitsradius. Themetrical tensor canbecomposedbyprecise measuremen tsonmedium longdistances andfromthatinformation alone, curvature canbe determined. Thisisthepractical result oftheworkofGauss, BolyaiandLobac hevski. 47 26Thecurvature tensor Using therelation (155), wemaydetermine thesecond derivativesofthelocalbasis vectors fa(u)gofthelocaltangen tplane, i.e. a;= ;a+ a;+L;n+Ln;: When wetaketheinnerpro ductofthisexpression andoneofthelocaltangen tialbasis vectors, then, alsousing formulas(142), (151) and(156), andthefactthatnisnormal tothe localtangen tplane, thenweobtain a;
a= ;g+ LL: When, next, wetakethedierence oftheaboveequation withthesame expression for andinterchanged, then, alsousing thesymmetry properties oftheaneconnection andthe Christoel symbol,wend L(u)L(u)L(u)L(u)=R(u); (187) where thecurvature isdened by R(u)=g(u)R (u); (188) andwhere R = ; ;+    : (189) Thecurvature tensor (188) hasthefollowingsymmetry properties: R=R R=R R=0;for= R=0;for=: (190) Withthose symmetry relations, wehaveforthecurvature tensor atatwo-dimensional surface onlyoneindependen tnonzero elemen t,i.e. R1212=R1221=R2121=R2112: (191) Thelocalcurvatur escalarisingeneral dened by R(u)=g(u)g(u)R(u): (192) Foratwo-dimensional surface theonlynon-zero contributions come fromthecomponents collected informula(191), whichresults forthecurvature scalar thenin 48 R(u)=n g11g22g12g21+g22g11g21g12o R1212: So,using alsoformulas(175) and(187), wendforthelocalcurvature scalar intwodimensions R(u)=2det g1 R1212=2det g1 (L21L12L11L22) =2det g1L =2K: (193) Fortheextension toarbitrary dimensions onechoosesthecurvature scalar asdened in(192) fortheparameter whichcharacterizes thedeviation ofthemetrical space fromEuclidean. 49 27Geodesics Aspreviously discussed inthecontextofthecurvature ofcurvesonthetwo-dimensional surface embedded inCartesian three dimensions (seesection 22),oneofthemanycurveswhichshare thesame tangen tialdirection, ,atacertain location Pofthesurface, isthegeodesiccurve. Asbeforewedenote theproperlength atthiscurvebys.Thiscurveisspecial, because its related derivativeofthetangen tialdirection, 0(s),isparallel tothelocalnormal, n(u(s)),of thesurface atpointP(s)(seeformula167) Since 0(s)isnormal tothetangen tplane inthecaseofageodesic, theinnerpro ductof 0(s)withthelocaltangen tbasis vectors fagvanishes. Using formulas(142) and(163) and thefactthatn(u)isalsonormal tothetangen tplane, weobtain fortheinnerpro ductof0(s) anda(u)theresult: 0=0(s)
a(u)=d2u ds2g(u)+du dsdu ds (u)g(u): Notice thatwehavetwoequations here, oneforeachvalueof.For=1wemultiply this expression withthematrix elemen tg1oftheinverseofthelocalmetrical tensor, where can beanyofthetwopossibilities, andfor=2wemultiply theexpression withg2.Thesumof thetworesults 0=d2u ds2g(u)g(u)+du dsdu ds (u)g(u)g(u); iscalled acontraction .When wemoreo veruseformula(90),thenwend 0=d2u ds2+du dsdu ds (u); (194) whichrelation iscalled thegeodesicequation . 50 PartIV Examples ofRiemann surfaces Inthefollowing westudy some properties oftwo-dimensional surfaces, embedded inthree dimensions, orRiemann surfaces. ThebookofCoxeter[4]hasalotmoredetails andexamples. Here, wejustrestrict ourselv estothemostobvious parametrizations forthose surfaces, their tangen tspaces, thegeodesicequations and,when easily possible, thegeodesics. 28Thecylinder Asarstexample, letusstudy thetwo-dimensional surface ofacylinder, embedded inan Euclidean three-dimensional space. Welettheaxisofthecylinder coincide withthez-axis. Foritsradius wetakeaandforitslocalcoordinates weselect theazimuthal angle 'andz. Thesurface ofthecylinder isthengivenby x(';z)=0 BBBB@acos(') asin(') z1 CCCCA: (195) Setting u1='andu2=zandusing formula(141), wendforthelocaltangen tvectors at location P(';z) a1(';z)=0 BBBB@asin(') acos(') 01 CCCCAanda2(';z)=0 BBBB@0 0 11 CCCCA: (196) Thelocalmetrical tensor forthesurface ofthecylinder isthenfound byusing theprocedure whichisgiveninformula(142). Hence g(';z)=0 @g''g'z gz'gzz1 A=0 @a20 011 A: (197) Fromexpression (197) forthelocalmetric itfollows,bytheuseofrelations (105) and(106), thatthelocalane connection vanishes. Consequen tly,alsousing formulas(188) and(189), we mustconclude thatthecylinder surface hasnocurvature. Thismaybeasurprising conclusion, but,afteralittlethinking itbecomes eviden tthattheresult iscorrect: A atpiece ofpaper hasnocurvature. When werollupthissheet ofpapertoacylinder, thenthatprocedure does notaltertheintrinsic properties ofthesurface. Forthegeodesicequations, using formula(194), weobtain accordingly d2' ds2=d2z ds2=0: (198) Hence, both'(s)andz(s)arelinear functions insalong ageodesic curve.Suchsolutions arethecircular helices, represen tedbythevectorx(s)offormula(123). Slightlymore general solutions aregivenby 51 x(s)=0 BBBBBBB@acos sp a2+b2+'0 asin sp a2+b2+'0 bs+cp a2+b21 CCCCCCCA: (199) where arepresen tstheradius ofthecylinder andwhere b,thespeedofthecircular helix, cand '0areconstan tparameters. callowsfortranslations inthevertical (z)direction and'0for rotations around thez-axis. Inthelimitb!1oneobtains x(s)!0 BBBB@acos('0) asin('0) s1 CCCCA; whichparametrizes straigh tvertical linesatthesurface ofthecylinder parallel tothez-axis. Itisillustrativ etodrawaskewstraigh tlineonasheet ofpaperandrollupthesheet, tond thatindeed oneobtains acircular helix. Thisshowsthenthatageodesicisindeed theclosest approximation toastraigh tline,ormoreprecisely ,the"shortest" connection along thesurface betweentwopointsatthesurface. 52 29TheEllipsoid Letusconsider thetwo-dimensional surface ofanellipsoid embedded inanEuclidean three- dimensional space. Thecoordinates ofthree-space aretheusual, givenbyx,yandz.An ellipsoid, centered attheorigin, ismost convenientlyparametrized bythespherical angles # and',according to x=asin(#)cos(');y=asin(#)sin(')andz=bcos(#): (200) These relations represen tasurface whichisrotationally symmetric around thez-axis, whereas itscross-section withthe(x;z)-plane forms anellips withprinciple axesoflength 2ainthe x-direction and2binthez-direction. Forthistwo-dimensional surface weselect, naturally ,the followingcoordinates u1=#andu2=': (201) Using formula(141), wendforthelocaltangen tialvectors atacertain pointP(#;'),the result a1(u)=0 BBBB@acos(#)cos(') acos(#)sin(') bsin(#)1 CCCCAanda2(u)=0 BBBB@asin(#)sin(') acos(#)cos(') 01 CCCCA: (202) Themetrical tensor forthesurface ofellipsoid (200isobtained bytheprocedure offormula (142). Onends g(u)=0 @g11g12 g21g221 A=0 @a2cos2(#)+b2sin2(#) 0 0 a2sin2(#)1 A: (203) Thelocalaneconnection followsfromtherelations (105) and(106). Onendsfromequation (203) forthenon-zero components 1 11=(b2a2)sin(#)cos(#) a2cos2(#)+b2sin2(#); 1 22=a2sin(#)cos(#) a2cos2(#)+b2sin2(#)and 2 12=2 12=cos(#) sin(#): (204) Aswemayobserv efromformula(203), thelocalbasisvectorsfa(u)gofthetangen tplane are orthogonal toeachother. Hence, when weconsider ageodesiccurvex(s)atP(u),parametrized byitsproperlength, s,thenwecanparametrize thecorresp onding tangen tvector, (s),at P(u)bytheangle, ,itmakeswitha1,according to (s)=a1 ja1jcos()+a2 ja2jsin(): (205) 53 Consequen tly,followingdenition (161), wemustconclude thatalong thegeodesiccurveone hasforthederivativesofthelocalcoordinates, alsousing formula(202) thefollowing d# ds=cos() ja1j=cos()q a2cos2(#)+b2sin2(#)and d' ds=sin() ja2j=sin() asin(#): (206) Now,wealsoknowthatthegeodesics followthegeodesic equation (194), whichallowsus tondthesecond order derivativesofthelocalcoordinates along thegeodesiccurve,fromthe rstorder derivatives(206), alsousing equations (204) asfollows d2# ds2=1 11 d# ds!2 1 22 d' ds!2 =cotg(#)sin2() a2cos2(#)+b2sin2(#)(b2a2)sin(#)cos(#)cos2() n a2cos2(#)+b2sin2(#)o2 and d2' ds2=22 12d# dsd' ds=2cos(#)sin()cos() asin2(#)q a2cos2(#)+b2sin2(#): (207) Therelation of(s)withthelocalbasis vectors ofthetangen tplane isgivenbyformulas (205) and(206). Hence forthederivativeof(s),wemaywrite: 0(s)=d ds(d# dsa1+d' dsa2) (208) =d2# ds2a1+d# ds d# dsa1;1+d' dsa1;2! +d2' ds2a2+d' ds d# dsa2;1+d' dsa2;2! : After some algebra, bysubstituting formulas(202), (206) and(207) intotheaboveexpression (208), wendforthederivativeofthefollowing 0(s)=b a(a2b2)sin2(#)sin2()a2 n a2cos2(#)+b2sin2(#)o20 BBBB@bsin(#)cos(') bsin(#)sin(') acos(#)1 CCCCA: (209) Thelength ofthisvector is,according toformulas(126) to(129) ofsection (17),equal to theinverseofthelocalradius ofcurvature inthedirection of.Hence, wend k((s))=0(s)=b a(b2a2)sin2(#)sin2()+a2 n a2cos2(#)+b2sin2(#)o3=2: (210) 54 Theextrem umvalues ofk((s))fordieren tchoices ofthetangen tialdirection parameter ,are k1=ab n a2cos2(#)+b2sin2(#)o3=2; (211) for=0,whichcorresp ondstothetangen tialdirection '=0,and k2=b=aq a2cos2(#)+b2sin2(#); (212) for=,whichcorresp ondstothetangen tialdirection #=0. Thecurvature parameter K(u)=K(#;')ofthelocalcurvature ofthesurface atlocation P(#;')isdened informula(175) bytheproductofk1andk2.Consequen tly,applying the results (211) and(212), onends K(u)=k1k2=b2 n a2cos2(#)+b2sin2(#)o2: (213) Notice thatthecurvature whichfollowsfromformula(213) forthecasea=b,resulting in K=1=a2,corresp ondstothenaveexpectation forthecurvature ofasphere. Theaverage localcurvature parameter H(u)isdened informula(176) byhalfthesumof k1andk2.Hence, fromtheresults (211) and(212), onededuces 2H(u)=k1+k2=b a(b2a2)sin2(#)+2a2 n a2cos2(#)+b2sin2(#)o3=2: (214) Thelocalnormal vectorn(u)isdened informula(143) bytheouter product, normalized tounity,ofa1anda2.So,fromtheresults showninformula(202), oneobtains n(u)=1q a2cos2(#)+b2sin2(#)0 BBBB@bsin(#)cos(') bsin(#)sin(') acos(#)1 CCCCA: (215) Notice, bycomparing withexpression (209), that0(s)isindeed parallel tothenormal ofthe localtangen tplane. Thelocaltorsion tensor L(u)isdened informula(153) bytheinnerpro ductofa;(u) andn(u).Consequen tly,exploiting theresults showninformulas(202) and(215), onegets L(u)=0 @L11L12 L21L221 A=abq a2cos2(#)+b2sin2(#)0 @10 011 A: (216) Consequen tly,wecanverifythattheexpressions (213) and(214) areinageemen twiththe formulas(175) and(176). Using formulas(203) and(216), wend K(u)=L11L22 g11g22=det g1L and 55 2H(u)=L11 g11L22 g22=Tr g1L : Except foraminussigninthelatter result, wendperfect agreemen t.Thissigndierence iscaused bytheangle ambiguit ywhichwediscussed informula(167). Bycomparing formulas (209) and(215), weobserv ethat0(s)andn(u)areantiparallel andhence thereferred angle ishererather than0,whichcauses aminussigninformula(165) forthedenition ofthe curvature. Wemayalsoverifyformula(193) forthisparticular surface. Bytheuseofformulas (188), (189), (203) and(204), wendforthecurvature tensor R1212(#;')=g11 1 21;21 22;1+1 222 211 111 22 =a2b2sin2(#) a2cos2(#)+b2sin2(#)(217) When wesubstitute thisexpression informula(193), thenwendindeed K(u)=R1212(#;') det(g): 56 30Geodesics onthesphere Since asphere withradius Rcentered attheorigin canbeconsidered asaspecialcaseofthe ellipsoid dened informula(200), wemayuseallofthematerial developedinsection (29),by setting a=b=R. Thegeodesic equations canbeobtained fromformula(194). Using alsorelations (204), wendforthecoordinates #(s)and'(s)ofpointsatageodesic onthesphere, whichis parametrized byitsproperlength s,thefollowingdieren tialequations: d2# ds2sin(#)cos(#) d' ds!2 =0andd2' ds2+2cos(#) sin(#)d# dsd' ds=0: (218) Onepossible strategy ofsolving theequations (218), istoexpress oneofthecoordinates as afunction oftheother along thegeodesiccurve,forexample let '(s)='(#(s)): Inorder tosimplify theformulastocome, wedene '0=d' d#and'00=d2' d#2; inwhichnotation weobtain forthederivativesof'withrespecttotheproperlength parameter s,theexpressions: d' ds=d# ds'0andd2' ds2=d2# ds2'0+ d# ds!2 '00; andhence fortheequations (218) d2# ds2sin(#)cos(#) d# ds'0!2 =0and d2# ds2'0+ d# ds!2 '00+2cos(#) sin(#) d# ds!2 '0=0: (219) When wesubstitue moreo vertherstofthegeodesicequations (219) intothesecond, then wendthedieren tialequation d# ds!2( '00+sin(#)cos(#)('0)3+2cos(#) sin(#)'0) =0: (220) Onetypeofsolutions canreadily befound. Namely those forwhich d# ds=0i.e.#(s)=#constan t; andforwhich,followingtheequations (218), onehasmoreo ver sin(#)cos(#) d' ds!2 =0andd2' ds2=0: 57 Thesolutions ofthelatter equation canbeclassied bysingular pointsonthesphere: #=0;; i.e.North andSouth polesonthesphere; #and'constan t; i.e.singular pointsonthesphere; (221) andbytheEquator ,corresp onding tothecircumference ofthecircle inthe(x;y)-plane (see alsoexample 122), givenby #= 2and'(s)=s R: (222) So,besides singular points,whichobviously aresolutions oftheequations (218), weonly found onenon-trivial geodesiccurve,theEquator, actually theonlysolution of(218) forwhich thecoordinate #isconstan talong thecurve. Other solutions of(220) arefound fromthesecond piece ofthisequation, i.e. '00+sin(#)cos(#)('0)3+2cos(#) sin(#)'0=0: (223) Inorder tosolvethisequation, wewillstudy hereanarbitrary intersection ofthesphere withaplane through theorigin. Suchintersections arecalled largecircles. Wewillshowthen thatalllargecircles onthesphere aregeodesiccurves. Anyplane through theorigin canbecharacterized byitsorientation, forwhichonemay select itsnormal vector, sayN=p^x+q^y+r^z.Theposition vectorxP=x^x+y^y+z^z ofapointPintheplane andthenormal totheplane areperpendicular bydenition. Hence, onehas 0=xP
N=px+qy+rz: (224) When Pismoreo veronthesurface ofthesphere, thenweobtain, using relations (200) with a=b=R,forequation (224) theform psin(#)cos(')+qsin(#)sin(')+rcos(#)=0; or,sincethecases sin(#)=0andcos(#)=0havebeenstudied informulas(221) and(222), forsin(#)6=0theform pcos(')+qsin(')=rcos(#) sin(#): (225) When wedetermine therstandsecond order derivativesofequation (225) withrespectto#, thenwend '0fpsin(')+qcos(')g=r sin2(#)(226) and '00fpsin(')+qcos(')g('0)2fpcos(')+qsin(')g=2rcos(#) sin3(#) 58 Bymultiplying thesecond equation offormula(226) with'0andbysubstitution ofrelation (225) andtherstequation offormula(226) intheresulting expression, weobtain '00r sin2(#)+('0)3rcos(#) sin(#)+2'0rcos(#) sin3(#)=0; (227) whichis,forsin(#)6=0iscompletely equivalenttothegeodesic equation (223). Thisproofs thatlargecircles aregeodesiccurvesofthesurface ofthesphere. 59 31Thetorus Another suchexample, whichisoften referred tointheliterature, isthetorus, thetwo- dimensional surface ofadoughn utshapedobject.Letusconsider thesurface ofawedding-ring withacircular cross section ofradius r.Theradius ofthering,whichismeasured fromthe centeroftheringtothecenterofthecircular crosssection, willbedenoted byR.Fromthis picture ofatorus itmightbeclearthatrisconsidered tobesmaller thanR.Inorder todene coordinates onthesurface ofourtorus, weletthecenterofthewedding-ring coincide withthe origin ofthethree-dimensional Euclidean embedding space. Thecentersofthecircular cross sections ofthewedding-ring, whichtogether formacircle ofradius R,nowcentered attheori- gin,wesupposetoallbeinthe(x;y)-plane. Thiswayweobtain forapossible parametrization ofthesurface ofatorus thefollowingexpression: x(;)=0 BBBB@Rcos() Rsin() 01 CCCCA+0 BBBB@rcos()cos() rcos()sin() rsin()1 CCCCA; where r<R;;2(0;2): (228) Apparan tly,wehavechosen andforthecoordinates atthesurface ofthetorus. Notice, thatthecrosssections ofthissurface withplanes whichcontainthez-axis, arecircles centered atadistance Rfromtheorigin. Suchplanes arecharacterized by=constant andtheircross sections withthetorus, whicharecalled meridians ,bytheusual relations forcircles, i.e. (xcos()+ysin()R)2+z2=r2: (229) Forthelocaltangen tplane, tangen ttothesurface atthepointP(;),wendthen, using alsoformula(141), thebasisvectors givenby ^(;)=@x(;) @=0 BBBB@rsin()cos() rsin()sin() rcos()1 CCCCAand ^(;)=@x(;) @=R r+cos()0 BBBB@rsin() rcos() 01 CCCCA: (230) Inserting theresult (230) intoformula(142) yields forthelocalmetric g(;)=0 B@gg gg1 CA=0 @^
^^
^ ^
^^
^1 A=0 @r20 0(R+rcos())21 A:(231) Thenon-vanishing localane connections are,bytheuseofformulas(151) and(152), collected below 60  = =g=1 2gg;=sin() Rr+cos()(232) and =g=1 2gg;=sin()R r+cos() ; fromwhich,using formula(194), followthegeodesicequations d2 ds2= d ds!2 sin()R r+cos() andd2 ds2=2 d ds! d ds!sin() Rr+cos():(233) Twotypesofsolutions canbediscoveredwithout anydicult y.Those aresolutions for which =constan t)d ds=0)d2 ds2=0)=s r; or (234) =0or)d2 ds2=0)=s R+rors Rr: Thersttypeofgeodesics describ edbyformula(234) arethemeridian circles, represen tedby formula(229). Thesecond typearetheequatorial circles inthe(x;y)-plane andcentered at thez-axiswithextremal radii, respectivelymaxim um,R+r,andminim um,Rr.When both conditions d=dsandd=dsvanish, onehasisolated pointsatthesurface ofthetorus. Inthefollowingweassume thatneither ,nor,isconstan t.Onemaynotice thenthatthe second ofthegeodesicequations (233) canalsobewritten intheform d ds"R r+cos()2 d ds!# =0; whichimplies thattheexpression within thebracketsisconstan t,sayJ,along ageodesiccurve. When thisresult isinserted intherstofthegeodesicequations (233), onendsfor(s)the equation d2 ds2=J2sin() nRr+cos()o3: Wemaysimplify therstofequations (233), byconsidering directly afunction of.Inthat caseonehas d ds=d dsd d;d2 ds2=d2 ds2d d+ d ds!2d2 d2 andd2 ds2=2 d d! d ds!2sin() Rr+cos()=2J2sin() nRr+cos()o5 d d! ; 61 whichformulascanbeputtogether, resulting inasecond order dieren tialequation forasa function of d2 d2+2sin() Rr+cos() d d!2 +sin()R r+cos() =0: (235) whichremains tobesolved. Thecurvature tensor isdened informulas(188) and(189). Bytheuseofformula(232) we ndforthetorus R=rcos()fR+rcos()g; and,hence, forthecurvature scalar, whichisdened informula(192) andinparticular fora two-dimensional surface informula(193), oneobtains R(;)=2cos() rfR+rcos()g: (236) Notice thatattheequatorial circles onendsforthecurvature scalar indeed, asexpressed by formulas(168), (175) and(193), 2divided bytheproductoftheradiioftheequatorial circle andthemeridian circle. Notice moreo verthatattheoutside (=0)theresult ispositive (curvedtowardstheorigin), butattheinside (=)negativ e(curvedawayfromtheorigin). Atthetop(==2)andatthebottom (=3=2),where thecurvature isvanishing, the surface ofthetorus haslocallytheaspectofacylinder. 62 PartV Thedescription ofgravitation by curvature 32Rectilinear motion Therectilinear motion ofafreely movingobjectinordinary threedimensions can,atanyinstan t t,beparametrized by x(t)=at+b; (237) where aandbdependontheinitial conditions ofthemotion: Theconstan tvelocityofthe particle isrepresen tedbythevector a.Whereas thevector brepresen tstheposition ofthe particle atinstan tt=0. Theequations ofmotion fortheparticle's movementcanbecharacterized bythewell-kno wn relation d2xi(t) dt2=0fori=1;2;3: (238) Now,inthelocalcoordinates ofsection (9),theequations ofmotion (238) taketheform 0=d dt0 @dxi dt1 A=d dt8 < :xi ;jdx0j dt9 = ; =xi ;jkdx0k dtdx0j dt+xi ;jd2x0j dt2; or,equivalently @x0` @xixi ;jkdx0k dtdx0j dt+@x0` @xixi ;jd2x0j dt2=0; which,bytheuseofthedenition oftheane connection informula(102), canbewritten as follows: ` jkdx0j dtdx0k dt+d2x0` dt2=0: (239) 63 Thenon-zer ocomponents oftheane connections forthe3Dspheric alcoordinates are giveninformula (107). Substitution intheequations (239) gives 0=r00+r ##(#0)2+r ''('0)2=r00r(#0)2rsin2(#)('0)2 0=#00+2# r#r0#0+# ''('0)2=#00+2 rr0#0sin(#)cos(#)('0)2 0='00+2' r'r0'0+2' #'#0'0='00+2 rr0'0+2cos(#) sin(#)#0'0: (240) Inordertosolvethoseequations forr=r(t),#=#(t)and'='(t),werstobserve thatthethirdequation canbewritten intheform 0=d dt r2'0sin2(#) ; (241) which implies thatr2'0sin2(#)isaconstant alongthecurve. Hence,without much loss ofgeneralitywemayselectcurves inthe(x;y)-plane, forwhich #==2.Theequations ofmotion reducethento 0=r00r('0)2and0='00+2 rr0'0: (242) Ititeasytoverify thatthoseequations aresolvedby r(t)=q (at)2+b2and'(t)=arctg b at! ; which, fortwoperpendicular vectors aandb,justrepresenttheparametrization ofa straightline,as,indeed,parametrizing 3Dspacebyspheric alcoordinates doesnotimply anycurvatur e. Onacurvedsurface weconsider relation (239) asthedenition oflocalrectilinearmotion. Instead ofthetime-parameter t,weusethentheproperlength parameter s,toobtain for \rectilinear" motion atthecurvedsurface (140) theequation ofmotion  (u)du dsdu ds+d2u ds2=0: (243) Thisisactually thesameequation asthegeodesicequation (194), asindeed geodesiccurves aretheclosest approximation tostraigh tlinesonacurvedsurface. Locally,geodesics formexact straigh tlineswithafullEuclidean geometry .Ataglobal scale, geodesics areofcourse notstraigh t,howeverthemostnatural generalization ofthe\straigh t line"concept toacurvedsurface. Wemaysaythata\free" particle whichisconned tomove atacurvedsurface, willfollowageodesic, withequation (243) beingtheequation todescrib e itskinematics. Intheabsence ofcurvature, wereturn toequation (238). 64 33Parallel transp ort Insection (14)thederivativesofvectorelds withrespecttothelocalcoordinates arestudied. Whichisequivalenttothestudy ofvariations ofavector eldforinnitesimal displacemen ts inspace. Itledustothedenition oftheconcept ofthecovariantderivative,formulated in equations (112) and(113). Now,supposethatacertain vector eldv(x0)isconstan t.Then, intheN-dimensional Euclidean space dened insection (9),onehas,forthederivativesofthatvector eldwith respecttothelocalcoordinates, thefollowing dv(x0) dx0i=0: (244) Aconstan tvector eldinspace means thatatanypointinspace thevector v(x0)which represen tstheeldatthepointP(x0),isparallel tothecorresp onding vectoratanyother point inspace. Wecould however,equally wellconsider thevector associated withtheeldatone pointinspace, asthevector oftheeldfromanother pointinspace transp orted parellel to itself. Hence, thevectoreldwhichistheresult oftransp orting parallel toitselfagivenvector atagivenposition toallother positions inspace, isrepresen tedbyequation (244). Moreo ver, inviewofequation (112), onehasforthecomponentsofthevectoreld,when decomp osedat thelocalbasisdictated bythelocalcoordinates, therelation v0k;i(x0)=0: (245) Atacurvedsurface wemightliketoconsider equation (245) asthedenition ofparallel transp ort.Whichmeans thatweassume thenthatthatequation istheclosest analogue tothe global atspace denition. Hence, wearethenassuming thatwearecapable todene aconstan tvector eldatthe surface (140) astheresult ofparallel transp orting onegivenvector atthesurface atacertain pointtoallpointsatthesurface. And,consequen tly,suchvector eldshould satisfy 0=v;(u)=v;(u)+ (u)v(u): (246) Forparallel transp ortalong acurve,parametrized byitsproperlength s,wehavemoreo ver dv(s) ds=v;(u)du(s) ds= (u)v(s)du(s) ds: (247) However,itcanbeshownthatatacurvedsurface onecannot construct avectoreldwhich satises equation (246), sinceparallel transp ortdependsatthecurvedsurface onthepathone chooses(seeexamples insection 34). Hence, wecannot extend thenotion ofaconstant vectoreldtoacurvedsurface,butonly theconcept ofparallel transp ortalong agivenpath, asformulated inexpression (247) 65 34Parallel transp ortalong curvesatthesphere Thesurface ofasphere withradius Randcentered attheorigin isparametrized by#and', asaspecialcaseofrelations (200) forR=a=b.Letacurveatthesurface ofthesphere be parametrized byitsproperlength parameter s,according to#(s)and'(s)andletusmoreo ver consider thevector eldA(#;')atthesurface ofthesphere. Restricted tothecurveAturns afunction ofs. Now,letusassume thatthevectoreldA(s)istheresult ofparallel transp orting thevector A(0),whichisdened atthepointofthecurveparametrized bys=0.Then, ourvector eld satises relations (247), which,alsousing formula(204) forR=a=b=,incomponentsreads d dsA1(s)=sin(#)cos(#)'0A2(s)and (248) d dsA2(s)=cos(#) sin(#)n '0A1(s)+#0A2(s)o ; where, asbefore, '0=d'=ds and#0=d#=ds . Letusconsider thespecialcaseofparallel transp ortalong curvesforwhich#(s)=#is constan t.Inthatcaseitiseasytorelate 'ands.Using equations (118) and(203) for R=a=b,wend ds=Rsin(#)d'; (249) whichequation andthefactthatd#=ds =0,reduce thedieren tialequations (248) to d d'A1(')=sin(#)cos(#)A2(')andd d'A2(')=cos(#) sin(#)A1('): (250) Equations (250) havesolutions oftheform A1(')=Vsin['cos(#)]+Wcos['cos(#)]and A2(')=1 sin(#)fVcos['cos(#)]Wsin['cos(#)]g; withVandWconstan ts. Letusstudy twotypical cases: 1.Thecase#==2 Forparallel displacemen talong theEquator, whichservesasanexample foranygeodesic curveatthesphere, onendsforthevector eld(251): A(')=0 @A1(') A2(')1 A=0 @W V1 A; whichisconstan t.Wemaythusconclude thatforparallel displacemen talong theEquator and,hence, along anygeodesiccurve,thedisplaced vector remains constan t. 66 2.Thecase#==3 Fordisplacemen tontheminor circle for#==3,wendforthevector eld(251) the following A(')=0 BB@Vsin' 2 +Wcos' 2 2p 3n Vcos' 2 Wsin' 2o1 CCA; whichisobviously notconstan talong thecurve.LetA(0)betheoriginal vector, whichhadto bedisplaced parallel toitself. Belowwewrite theoriginal andthenalvectors ofthevector eldafteracomplete roundtrip, i.e.A(0)andA(2).ForthecaseW=1andV=0wend: A(0)=0 @1 01 AandA(2)=0 @1 01 A: Wendthatafteracomplete roundtrip thevectorappearsupside downattheinitial location. Thegeneral conclusion isthatunder parallel transp ortalong non-geo desiccurvesavector elddoesnotremain constan t.And,consequen tly,thatconstan tvectoreldscannot bedened. Forcompleteness letusmentionthatfornottoolargeroundtrips there isarelation between theamoun tofvariation oftheparallel displaced vector, theenclosed areaandthecurvature. Takeforexample thesecond case,where thevector changed over180.Theenclosed areaby theroundtrip isgivenby area=Z=3 0d#Z2 0d'R2sin(#)=R2: When wedevide thisareabytheamoun tofvariation, =,ofthedisplaced vector, thenwe ndthesquare oftheradius ofcurvature ofthesphere. 67 35LocalEuclidean geometry When wemakeamapofasmall village, thenwewillprobably notnotice thattheEarth surface iscurved.Thegeometry ,attheaccuracy withwhichwecanmeasure distances and angles, looksperfectly Euclidean. But,what doweprecisely mean bythislaststatemen t?A possible answertothisquestion could forinstance bethatthesumoftheangles ofanytriangle within thesmall village equals perfectly 1800attheaccuracy ofourmeasuremen ts.Thislatter statemen twasproventobeequivalenttoEuclid's fthpostulate, byAdrien Marie Legendre (1752-1833). However,ifwegotoalarger scaleattheEarth' surface, thenwemightencoun tertriangles whichgiveresults, dieren tfrom1800,forthesums oftheirangles. Asanexample, letus takethetriangle whichmakesattheNorth Poleofthesphere arightangle andhastwomore rightangles attheEquator. Thistriangle encloses oneoctantoftheEarth' surface. Itsangles apparan tlysumupto2700,whichimplies thatatthisscale thegeometry oftheEarth is noticably notEuclidean. Atthemuchsmaller scaleofthelittlevillage itisalsonotEuclidean, butwithin theaccuracy ofmeasuremen twedonotnotice theresulting deviations. InDescartes' analytic geometry ,Euclidean geometry isrelated tothedenition ofthedis- tance betweenanytwopoints,PandQ,inspace, givenby: n dh P x1;x2 ;Q X1;X2io2= X1x12+ X2x22; (251) i.e.thedistance followsthelawofPythagoras. Intheprevious paragraphs westudied atwo-dimensional surface endowedwithcoordinates uandmetrical tensor, g,suchthatinnitesimally adistance isgivenbyformula(160). It wasGauss whoassumed thatinasucien tlysmall neighborhoodaround anypointPonthe surface, wecandene newcoordinates f(P)g,suchthatdistances inthisregion ofspace are givenby: ds2=abda(P)db(P)a;b=1;2: (252) Consequen tly,intheabovedened region ofspace, givenacertain accuracy ofmeasuremen t (orphrased moreformal: tolowestorder), distances obeyPythagoras' law,whichimplies that thegeometry inthatregion isEuclidean. Awaytondthese localEuclidean coordinates f(P)gistocompare witheachother formulas(160) and(252), and,alsousing formula(89),toconclude thatconsequen tlywemust have: gab(P)=ab@a(P) @u@b(P) @u: (253) So,wemustndtwosuchcoordinates f(P)gwhichsatisfy (253) atthepointPofspace, in order tobeallowedtoforget aboutcurvature inasucien tlysmall neighboring region ofspace around P.ItwasGauss whodemonstrated thatthisprocedure isalwayspossible. Notice atthispointthatthelocalEuclidean space hasthegeometry ofthelocaltangen t space. 68 Forexample, atthespherewithradius a(useformula (203) fora=b)themetric al tensor isgivenby: ds2=a2d#2+a2sin2(#)d'2: (254) Now,atacertain pointP(#;')onthesurfaceofthesphere,wedene newcoordinates f(P)g,accordingto: 1(P)=af##(P)gand2(P)=asin(#(P))f''(P)g: (255) Thischoiceofcoordinates implies d1(P)=ad#andd2(P)=asin(#(P))d': (256) AtthepointPwendthusforaninnitesimal distanc e ds2=n 1(P)o2+n 2(P)o2: AndinsmallregionaroundthepointP,wendforaninnitesimal distanc etolower ordertoexpression ds2(#;')=a2d#2+a2sin2(#)d'2 a2d#2+a2sin2(#(P))d'2 n 1(P)o2+n 2(P)o2: (257) Thelatterrelation represents indeedPythagor as'law. Thechoice ofcoordinates (256) can,inamore formal way,bewritten according to a(P)=Va (P)n uu(P)o ; where thetwoobjectsfV(P)garegivenby: V1 1=a;V1 2=V2 1=0andV2 2=asin(#(P)): (258) TheobjectsfV(P)garecalled "Zweibein"(inGerman "Zwei"means two,and"Bein" means leg),andtheirgeneralization tofourdimensions willshowextremely useful forthehandling of particles withspinincurvedspace-time. Interms oftheZweibeinfV(P)g,relations (256) readingeneral da(P)=Va (P)du: (259) Under ageneral coordinate transformation, fug!fu0g,weobtain fortheaboveexpression theform da(P)=Va (P)@u @u0du0; (260) fromwhichequation wemightextract fortheZweibeinthetransformation rule 69 V0a (P)=Va (P)@u @u0: (261) This, bytheuseofformulas(83)and(86),issubsequen tlyfound tobethetransformation rule forcovariantcomponents.So,wemayinterpret theZweibeinasthecovariantcomponentsofa setoftwovectors. 70 36Tidal forces Inthefollowingwewillstudy ourfourdimensional space-time. Wewilldiscoverthattheeects ofagravitational eldcanbedescrib edbyassuming thatthemetric ofspace-time diers from theusual Minkowskian metric. Thisimplies thatweassume thattheeects ofgravitation are equivalenttospace-time beingcurved.However,atthispointwehavetobeverycareful. We onlyndanequivalence inthedescription. Wedonotassume thatourspace-time isembedded insomehigher dimensional space (neither exclude thatthisispossible). Because ofthisreason, wewillnotrefertoanembedding space. Whichmeans thatweonlyhaveatourdisposalthe coordinates ofspace-time andthemetric induced bygravitation, notangen tplane andneither anormal tothetangen tplane. However,theanalogy withthelocalEuclidean space willstill bepresen tinthiscase. AtEarth weobserv etheeects ofgravitation fromthefactthatobjectstendtofalltowards thecenteroftheEarth, whichissometimes rather disturbing. Atthesurface oftheEarth, the eect ofgravitation canbecharacterized byitsconstan tofacceleration g=GM R2; (262) where G,MandRrepresen trespectivelytheGravitational constan tandtheEarth' massand radius. Letx(t)betheposition vector ofaparticle withmassm,whichisfreely falling nearthe surface oftheEarth, withrespecttomebeingcomfortably seated inmyarmchairatthesurface oftheEarth. Theequation ofmotion fortheposition vectorinmycoordinate system isinthis situation givenby md2x(t) dt2=mg: (263) Another situation tostudy themotion ofthisfreely falling particle, mighttobeputmyarmchair inaequally freely falling elevator. LetussupposethatIamcomfortably seated again, and thattheparticle's motion takesplace inside thesame elevator. Thenewcoordinates fx0goftheparticle's position vector withrespecttome,aregivenby x0(t)=x(t)1 2gt2andt0=t: (264) Consequen tly,Ind,bysubstitution ofthenewcoordinates (264) informula(263), forthe equation ofmotion oftheposition vector oftheparticle md2x0(t0) dt02=0: (265) Theparticle appearstobecompletely atrestwithrespecttomeinside thefreely falling elevator. Imightthusconclude, thatnoforces actontheparticle. Apparan tly,itseems thatthecoordinate transformation (264) makestheeect ofthegravitational eldoftheEarth disapp ear. Forasystem oftwoparticles, AandB,withmasses m(A)andm(B)andwithmutual interactions describ edbysomeforce, F,whichonlydependsontherelativ epositions ofAand Bwithrespecttoeachother, onendsasimilar result aswewillseebelow. Inthecoordinate system withrespecttothesurface oftheEarth, Indtheequations of motions 71 mAd2xA(t) dt2=mAg+F(xAxB)and mBd2xB(t) dt2=mBg+F(xBxA): (266) Whereas, inside thefreely falling elevatorIobtain, using thetransformations (264), forthe equations ofmotion mAd2x0A(t0) dt02=F(x0 Ax0 B)and mBd2x0B(t0) dt02 =F(x0 Bx0 A): (267) Aswemaynotice, inthelatter equations theeects ofgravitation haveagain completely disapp eared. However,weshould notconclude thattheeects ofgravitation areabsentinthefreely falling elevator. Atalarger scale, i.e.involving larger distances andlonger periodsoftime, theeects ofgravitation caneasily beobserv edinside thefreely falling elevator,aswewillexplain below forthecasethatthetwoparticles donotmutually interact. Supposethatinitially thetwoparticles haveacertain distance withrespecttoeachother. Then, 1when thetwoparticles havethesame distance withrespecttothecenteroftheEarth, afterawhile wendthattheirrelativ edistance hasshrunk. Thereason isofcourse, thatboth particles arefalling towardsthecenteroftheEarth along dieren tradial directions, which approac heachother when theirdistance totheEarth becomes smaller. 2when, alsoinitially ,thetwoparticles havedieren tdistances withrespecttothecenterof theEarth, butmovealong thesame radial direction, wendthatafterawhile theirrelativ e distance hasgrown.Here, thereason isthattheparticle whichiscloser tothecenterof theEarth feelsaslightlylarger attraction allthetimeandthusisallthetimeslightlymore accelerated. So,ataglobal scaleweobserv eahorizon talattractiv eforcewhichreduces thedistances of particles, andavertical repulsiv eforcewhichincreases thedistances ofparticles. These eects arecalled thetidalforces,andnotransformation canmakethem disapp ear. Thename tidalforces stems fromthetidalmotion oftheoceans duetothepresence ofthe Moon.Ifweconsider theEarth asafreely falling objectintheMoon'sgravitational eld,then theoceanic watersfeelrepulsiv eforces withrespecttotheEarth surface atthesidestowards andoppositetotheMoon(high-tide) andattractiv eforces inbetween(lowtide). Duetothe Earth rotation these positions change intime, whichcauses thetidalmotion oftheseawaters. Inconclusion, wecansaythatlocally(i.e.atsmall spatial distances andinshort periodsof time) wecanndatransformation whichmakestheeects ofgravitation disapp earforagiven accuracy ofmeasuremen t.Globally ,thisisimpossible. 72 37Theprinciple ofequivalence Thetransformation (264) isbased ontheequivalence ofinertial andgravitational mass. The rstperson whoreported thediscoverythatobjectsfallatarateindependen toftheirmasses, wasofcourse Galileo Galilei (1564-1642). Since hisinitial measuremen tsvarious experimen tal impro vementshavebeencarried outtonally come tothesame conclusion. Thetwomost famous results come fromRoland vonEotvosandfromR.H.Dicke.VonEotvosconcluded in 1889thattheinertial andgravitational masses forallmaterials areequal uptohisexperimen tal precision ofninedecimals. Dickeimpro vedthisprecision in1964toelevendecimals. Theaboveexperimen talresults, combined withtheresults ofthesection (36),leadtothe followinggeneralization: Atevery pointinspace-time inanarbitrarygravitational eld,itispossible toselecta locallyinertial coordinate system, suchthatlocallythelawsofnaturetakethesameformasin aninertial system intheabsenc eofgravitation . Notice theresemblance oftheabovestatemen twithwhat issaidinsection (35)aboutthe localEuclidean coordinates atacurvedtwo-dimensional space. Forfurther reading onthissubject,Irecommand theparagraphs 1.2,1.3and3.1ofthebook ofStevenWeinberg[5]. 73 38Minkowskian space-time Fromtheprevious paragraphs wemighthaveundersto od,thatinstead ofthesomewhat boring caseofatwo-dimensional arbitrarely curvedsurface embedded inthree dimensions, wewill study inthefollowing themore exciting caseofafourdimensional curvedsurface without referring toanembedding space. Wedene asetoffourcoordinates fxgwhichcharacterize uniquely events,P(x),inthis space byitscomponents x;=0;1;2;3: Furthermore, weequip thisspace withametric whichrepresen tsthegravitational eldandis characterized bythemetrical tensor g;;=0;1;2;3: Weassociatethecoordinates ofthisspace withthespace-time coordinates (inunitswhere the lightvelocityistakenunity,i.e.c=1),ofevents,i.e. x0=t;x1=x;x2=yandx3=z: Aspecicexample ofsuchspace istheMinkowskian space-time whichrepresen tsspace-time intheabsence ofagravitational eld. Itsmetric tensor isgivenby =0 BBBBBBBB@00010203 10111213 20212223 303132331 CCCCCCCCA=0 BBBBBBBB@1000 0100 0010 00011 CCCCCCCCA: (268) Theposition vectorofacertain particle movingwithaconstan tvelocityinspace, denes atrajectory intheabovespace-time whichincomponentsisspecied by x0=tandx(t)=t: (269) Intherestframe oftheparticle wemightdene coordinates fx0g,whichaccording totherelated Lorentztransformations areassociated withthefxgcoordinates, bytheusual expressions t0= ft
xg andx0=x+ ( 1+ (
x)t) ; (270) where 2=1=(12). Thespatial position oftheparticle itselfis,initsownrestframe, eviden tlygivenby x0(t0)=0: whichresult inagreemen twith(269). 74 Foritspropertimewendconsequen tly,using formula(270), thefollowing t02= 2n t22(
x)t+(
x)2o =t2x2=xx: So,instead ofusing thetimeparameter, t,forthecharacterization ofthetrajectory ,wecould equally wellselect thepropertimeparameter, s,whichwedene by ds2=dxdx: (271) Theaboveexpression reminds usofasimilar denition giveninformula(118) fortheproper lenghtinordinary space. InMinkowskispace therelated quantityiscalled thepropertime. However,wemustbeabitcareful inusing thisparameter, because ofthepeculiar formof themetric (268). Forparticles whichmovewiththevelocityoflightthepropertimevanishes. So,forsuchparticles weareforced touseadieren tparameter forthecharacterization ofits trajectory . Atthispointitmightbeuseful toreadthechapter on"SpecialRelativit y"(chapter 2) ofStevenWeinberg'sbook[5],whichdeals withLorentztransformations, particle dynamics, electromagnetism andtheEnergy-Momen tumtensor. But,becareful enough tonotice that Weinberg'sdenition (2.1.3) ofthemetric inMinkowskian space-time diers aminussignfrom thedenition (268) inthese notes. 75 39Gravitational forces According tothePrinciple ofEquiv alence (seesection 37),there exists alocalfreely falling coordinate system, fg(compare section 35),inwhichtheequation ofmotion ofaparticle whichmovesfreely under thein uence ofpurely gravitational forces, isgivenby(compare formula265) d2(s) ds2=0: (272) andwhere thepropertime,s,canbedened inthesamewayasthepropertimeofMinkowskian space-time (i.e.intheabsence ofgravitational forces, seeformula271). Inadieren tcoordinate system fug,wedene themetric inanalogy withformula(253), by g(u)=@(u) @u@(u) @u: (273) Interms ofthismetric, thepropertimeisgivenby ds2=gdudu: (274) Theequation ofmotion (272) interms ofthecoordinates fug,through thedenition ofthe aneconnection (formulas(105) and(106)), obtains theform 0=d2u ds2+ (u)du dsdu ds; (275) forwhichwerecognize thegeodesicequation (seeformula194). Formassless particles, whichmoveatthespeedoflight,wehave ds2=0: (276) So,formassless particles weareforced touseadieren tparametrization, say,forthe geodesics. Ingeneral, interms ofadieren tparameter, ,wendfortheequation ofmo- tion(275), theexpression 0=d2u d2+ (u)du ddu d; (277) andforthepropertime(274), theexpression: ds2=g(u)du() ddu() d: (278) 76 40TheSchwarzschildmetric Asanexample ofhowtheeect ofagravitational eldcanberepresen tedbycurvature, we study theSchwarzschildmetric. Consider athree-dimensional space withspherical coordinates fug=fr;;'gwhicharerelated totheusual coordinates fx;y;zg,by x=x(r;#;')=rsin(#)cos('); y=y(r;#;')=rsin(#)sin('); andz=z(r;#;')=rcos(#) (279) Concen trated inthecenterofthisspace weassume asource ofgravitational eld(e.g.the Sun)ofmassM.Space-time isinthiscasecharacterized bythecoordinates u0=t;u1=r;u2=#;u3=': (280) Shortly after AlbertEinstein formulated hisgeneral theory ofrelativit y,in1916, Karl Schwarzschildfound asolution toEinstein's equations whichdescrib esprecisely theabove sketchedsituation. TheSchwarzschildmetric isgivenby ds2=A(r)dt2dr2 A(r)r2n d#2+sin2(#)d'2o ; (281) where A(r)=12MGr. Arstinspection oftheaboveformulaforthepropertimeindicates usthatatlargespatial distances theexpression tends towardstheMinkowskian propertimerelation (271) forspherical coordinates. So,atlarge spatial distances weexpectlocallyresults similar tomotion inthe absence ofagravitational eld. Letusstudy herethegeodesics ofthespace-time characterized bythecoordinates (280) andthemetric (281). Forthispurposewerstdetermine thenon-zero elemen tsoftheane connection (seeformulas(105) and(106)), tond t tr=t rt=r rr=A0(r) 2A(r);r tt=1 2A(r)A0(r);r ##=rA(r); r ''=rsin2(#)A(r);# r#=# #r=' r'=' 'r=1 r; # ''=sin(#)cos(#)and' #'=' '#=cotg(#); (282) where A0(r)=dA(r) dr. According toformula(277) thegeodesicequations givehereasetoffourcoupled dieren tial equations, for=0,1,2and3.However,without muchlossofgeneralit ywemay,because of thespherical symmetry of(281), reduce thistoasetofthree coupled dieren tialequations by selecting ==2,whichimplies d=d=0.Itmighthoweverbeinstructiv e,toconsider rst 77 thegeneral caseandthen, afterbeingconvinced thatnoadditional solutions aresurpressed, reduce thesetofgeodesicequations byaspecicchoice for#.Thisisleftasanexercise forthe reader. For=0,wend,using formulas(277) and(282), theequation d2t d2+A0(r) A(r)dt ddr d=0; (283) whichequation forA(r)6=0canberewritten intheform d d( A(r)dt d) =0: (284) Consequen tly,wediscoveredaconstant ofmotion ,forwhich,again without muchlossofgen- erality,wemaychooseunity,i.e. dt d=1 A(r): (285) For=3thegeodesicequation (277), using oncemore formula(282), yields d2' d2+2 rdr dd' d=0; (286) whichequation forr6=0canberewritten intheform d d( r2d' d) =0: (287) Wendasecond constant ofmotion towhichwewillgivethename angular momentum per unitmass,andforwhichwewillintroducethesymbol,j,i.e. d' d=j r2: (288) Finally ,for=1,alsousing formulas(285) and(288), wereachatthegeodesicequation fortheradial distance rasafunction of: 0=d2r d2+1 2A(r)A0(r) dt d!2 A0(r) 2A(r) dr d!2 rA(r) d' d!2 =d2r d2+A0(r) 2A(r)8 < :1 dr d!29 = ;rA(r)j r22 ; (289) whichexpression, forA(r)6=0,canberewritten intheform d d8 < :1 A(r)2 4 dr d!2 13 5+j2 r29 = ;=0: (290) Athirdconstant ofmotion ,forwhichwewillusethesymbol,givenby: 78 =1 A(r)2 41 dr d!23 5j2 r2: (291) Instead ofthegeodesic parameter wecould haveusedtheordinary timeparameter tto parametrize theorbitofageodesic. Thisleads forequation (290), using formula(285), tothe followingexpression d dt8 < :1 A3(r) dr dt!2 1 A(r)+j2 r29 = ;=0: (292) Also, wecould haveusedtheazimuthal angle, ',toparametrize thegeodesic. Asimilar procedure hasbeenstudied insection (30). With thischoice ofparameter wendforequation (290), using theresult (288), therelation d d'8 < :1 A(r)j r22 dr d'!2 1 A(r)+j2 r29 = ;=0; whichalsocanbewritten intheform 2j2 r31 A(r) dr d'!8 < :1 rd2r d'22 r2 dr d'!2" 1+1 4rA0(r) A(r)# +1 2r3 j2A0(r) A(r)A(r)9 = ;=0; Hence, forA(r)6=0,dr=d'6=0andr6=0,moreo verintroducing thenotation r0andr00for respectivelydr=d'andd2r=d'2,andsubstituting theexpression offormula(281) forA(r),we obtain r00 r2 r0 r!2 1+MG 2r +MGr j21+2MG r=0: (293) 41Planetary orbits Inthefollowingwewillmakesome verymildapproximations (only aecting thesolutions to lessthanonepartoutofamillion), inorder tobecapable tostudy without mucheort andin more detail theresults oftheprevious section. Atplanetary distances fromtheSunwehaver2MG.Forexample, forthemassMofthe Sun2MG3103m,whereas forthedistance oftheSuntotheEarth onehasr1;51011 m,whichisadierence ofeightorders ofmagnitude. Consequen tly,wemayexpand A(r) inMG=rforplanetary distances, keeponlythelowestorder terms andyetstillobtain very accurate solutions forplanetary motion. Inthisapproximation weobserv efrom(285) thatthetimeparameter, t,andthegeodesic parameter, ,coincide, sinceA(r)1,whichleads for(288) totherelation r2d' dt=j; (294) whichisthewell-kno wnsecond lawofJohannes Keppler aboutsweeping equal areas inequal time-in tervals. 79 Forthegeodesic equation (293) theaboveapproximation hastheconsequence thatterms MG=r,canbeneglected withrespecttounity.Weobtain thentoanaccuracy ofeightdecimals thefollowing\geodesic" equation forr: r00 r2 r0 r!2 +MGr j21=0; (295) whichhasthewell-kno wnsolutions r(')=j2=MG 1ecos(');e0: (296) Thevarious orbits whicharerepresen tedbyformula(296) canbeclassied bytheexcentricity parameter, e,according to: 1: e=0 circles 2:0<e<1ellipses 3: e=1 parab olas and4: e>1 hyperbolas Inorder tocompare easily relation (296) withgeometric gures, onemaydene R=j2=MG, x=rcos(')andy=rsin(').Relation (296) turns theninto q x2+y2ex=rercos(')=R; whichleads tothefollowingrelation forxandy 0 B@xeR 1e2 R 1e21 CA2 + 1e2y R2 =1: Wendatplanetary distances, where A(r)1,thatthegeodesics ofSchwarzschild's metric (281) represen ttheorbits oftheplanets asobserv edbyKeppler. Newton's equations ofmotion arehererepresen tedbythegeodesicequations (283), (286) and(289) forthe"straigh t"lines atthecurvedsurface, whicharetheshortest connections betweenpointsincurved space-time . Theconserv edquantity,whichhasbeendened informula(291), is,intheapproximation (295), related tothelocalenergy ,E=m,ofthesystem perunitmassoftheplanet. Fromthe Newtonian gravitation formulawendforthetotalenergy perunitmass E m=1 mfE(kinetic )+E(gravitational eld)g=1 2v2MG r; which,bytherelation (294), takestheform 80 E m=j r22n r02+r2o 2MG r: (297) Letuscompare thisexpression withtheapproximation (295). Thelatter could havebeen formulated asfollows d d'8 < :j r22 dr d'!2 12MG r+j2 r29 = ;=0; (298) whichistheapproximation atlargedistances ofthedieren tialequation (290) fortheconserv ed quantity(291). Wendthentherelation =12E m: (299) Forthespecicformofthesolutions (296) wemaydetermine arelation withtheexcentricity parameter, e,andtheabovequantities. Alsousing (298) and(297), wend 2E m=j r22 e21 : (300) Welearn thenthatforfreeparticles, i.e.E>0,weneede2>1,whichgivesthehyperbolic solutions of(296). Forbound planets, i.e.E<0weobtain from(300) thate2<1,which givesthecircles andellipses. Inthelatter casealsoispositive,butthisisgenerally truefor material particles intheSchwarzschildmetric, aswewillshowbelow. Relation (291) canberewritten. Using formulas(285) and(288), weobtain thefollowing formforthisconstan tofmotion =A(r) dt d!2 1 A(r) dr d!2 r2 d' d!2 ; whichexpression, using thedenition (281) ofthepropertime, takes,forthecase==2,the form =ds2 d20: (301) Hence, wecome ingeneral totheconclusion that=0,formassless particles, and>0, formassiv eparticles. TheSchwarzschildmetric (281) notonlyreproduces theorbits ofplanets around theSun, butalsothede ection oflightbytheSunasobserv edduring Suneclipses forstarsbehind the Sun,theperihelion precession oftheplanet Mercury andmore (seeforexample Weinberg's book[5],chapter 8,Classic testsofEinstein 'sTheory).Moreo ver,gravitational collapse and blackholescanbestudied using theSchwarzschildmetric. Thisiswelldescrib edinchapter 11 ofWeinberg'sbook. 81