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Chapter 2 Navigating through spacetime（第1页）

Chapter2Navigatingthroughspacetime

&icsistheexquisitelyperfeguageneededfhowthetheoryofrelativityappliestothephysiiverseandallofspadthatdesirahatoearblaathematicaldes，whilepowerfula，evensoethingnandflahosewithouttheappropriateteiiivewords，howevereloquent，lacktherigourandpowerofamathematicalequationandbeimpredlimiting。Picturesh（itissaid）worthathousandwords，otonlyausefulisebutaveryhelpfulwaytovisualizewhatisgoingon。Forthisreason，itiswellworthspendingalittleefforttouandaparticulartypeofpicture，calledaspacetimediagram。Thiswillhelpiaureofspacetimearoundblackholes。

&imediagrams

&ooninFigure3sholespacetimediagram。Followingtraditioime-like'axisistheoisvertithepageandthespace-like'axisisdrawnperpendiculartothis。Ofcourse，wereallyneedfouraxestodescribespacetimebecausetherearethreespace-likeaxes（usuallydenotedx，y，andz）aime-likeaxis。Hoillsufficeforourpurpose（andofutuallyperpendicularaxesareimpossibletodraw!）。Wherethesetwoaxesiiscalledthein，andthismayberegardedasthepointofhereandnow'fortheobserverwhohasstructedtheirspacetimediagram。Aaheerashutter，occursatapartieaparticularlospace。Sustaisrepresentedbyadotoimediagram，appropriatetothetimeandspatialloquestiowodotsinFigure3，atiallyseparated（theydonotoccuratthesamepointonthespaceaxis）buttheyaresimultaheidenticalateoimeaxis）。Youagisdtothesimultaerpressesoftwophotographerswhoarestaafromothesamespectacle。Ifpoievents，whatdolinesiimediagramrepresent?Alinesimplyshoathofahroughspacetime。Asweliveourlives，wejhspadthepathweleavebehindus（somewhatasaseningtrailofslimebehindit）isalineiime，ahisiscalledaworldline。Ifyoustayathomeallday，yourworldliicalpaththroughspacetime（withspaate=22Aue'，forexample）。Youmoveforwardiarefixediheotherhandyoumadealongjourney，yourworldlisoverbecauseyourdistaime，beoveinspaceaswellastime。

3。Asimplespacetimediagram。

Forexample，lookattheworldlineshownihelinewhichispartvertifurtherupbeesslanting。Thisdstotheworldliherentity，whichisstationaryforthetimeihevertitoftheline。Abeacamerabelongihephotographers，leftonachair（sothatitsworldliicalbecauseitspositionisn'tg），beforeitwasstolenandheiallogesuously）。Wherethisliisspatialloisgwithtime。Theslopeofthisliellsyouabouttherateofgeofdistaime，whiohespeed。Inthiscasethisisthespeedatwhichthethiefiswhiskingawaythestolehefasterthethiefismakingoffwiththeotherwordsthemroundheisgihelessvertidthemoreslantingthispartofthelihereisofcoursearobustupperlimittothespeedatwhichthethiefoffwithhisillegallygottengainsandthis，asdisChapter1，isthespeedoflight。Thetrajeoflightwouldberepresentedbyamaximallyslantingline（oediimediagramsasbeiothetimeaxisbyusieduhinggofasterthanthatspeed，noworldliagreateraimeaxisthanthis。

Worldlinesoimediagramhavingthismaximallyslantingangle，dingtothismaximalspeed，thespeedoflight，giverisetoanimportacalledalighte。Theideaofthisisverysimple:youlyhaveaheUhefuturebysomepriordthatcausalsequepropagatefasterthanthespeedoflight。Thereforeyoursphereofinfluence&#htnowisediedrangeofspaamelythatpartwhichiswithina45-degreeahepositivetimeaxisasshowninFigure4。Moreover，youlyhavebeeninfluencedbyacausalofeventsthatatedfasterthanthespeedoflight。Thereforeohina45-degreeahebackwardstimeaxisfluenodraacetimediagramwithtwospace-likeaxesaime-likeaxis，therianglesinFigure4beedthesearewhatwemeanbylightes，asshownihelightFigure5deliesregionsofspawhiobserver（deemedtobelocatedatthein，theirhereandnow'）principlereach（orhavereathepast）withouthavingtoiheicspeedlimitandtravellihespeedoflight。Theregiohepositive（future）timeaxisisknowurelightewhilethetredoimeaxis（i。e。pasttimes）isknoastlighte。

4。Asimplelight。

ThustheassassinationofJulius44BCispartofyourpast，becausethereisaceivablekbetweeandyou。（Ifyouhadtolearnaboutitatschool，thatdemoheexistenceofak!）BecauselightfromtheAndromedaGalaxyreachatelesEarth，ittooispartofyourpast。Hhttakes6milliettous，soitisthe

5。Aspacetimediagramshowieofaparticularobserver。

AndromedaGalaxyof6milliothatispartofyourpastandsitshte。TheAndromedaGalaxyoftoday，oreventheAndromedaGalaxyof44BC，isoutsideyhttshappeningohernoworevenba44Botinfluenceyhtnowbeykwouldhavehadtotravelfasterthanthespeedoflight。

&hreespacetimediagramsthatwehaveseeninthischaptersofarhavetheiraxeslabelledastimea，professionalswouldn'tnormallyincludeaxislabelsoreventheaxesiimediagrams。Thisisn'tsimplythatitissoroutiimegoesupandspacegoesacrossthatprofessionalastrophysicistsgetsloppy（thoughthat'snotanunknowitisbecausetheexactpositionsiimeotbeagreeduponbyallobservers。Intheworldofspecialrelativity，thenotionofsimultaybreaksdowwoeveobesimultaneousforoneobserverdoesn'tatallmeanthattheyaresimultaneousforotherobservers。

Thusthetwophotographerspressiersoftheircameras‘simultaneously'willaravellingiveryfastrelativetothecamerassees。Thatobserverwilldedueraclicksubstaheother。ThetwopointsinFigure3whichIdrewatthesameverticalheight（sinceIclaimedtheeventsoccurredatthesametime）earatdiffereicalpositioimediagramoftherapidlytravelliein'srelativityinsistsherdiagramisjustasvalidasmihepointsoimediagramdependonanobserver'spoiheirframeofrefere'sthereasthem?

Touhis，itishelpfultofotheworldlineofamovingpartidsodraacetimediagraminarticlemhspacetime，takingitslightewithit（thistriwithinthee）。iheparticle'spath（i。e。itsworldline）alwaysstayswithieasitottravelfasterthanthespeedoflight。

&ein'sSpecialTheoryofRelativity，whichisasubsetofhisGeaiedsetofphysicalsituatioceptualframeworkbeyondSpecialRelativityishetextofspacetimewhichisexpanding，thepre-emiexampleofwhichistheexpandihistext，themaionofcausalityissuovefasterthanthespeedoflightwithrespecttoyourlocalbitofspace。

Howdoobjeo?

Althoughphotonshavenomass，itturnsoutthattheyarestillinfluencedbygravity。Itisbestnottothinkofthisasduetoaforce，butratherthatthisesaboutbecauseofthecurvatureofspacetime。Aphotonisusuallythoughttotravelinastraightline，egetthenotionofalightray'。Hhacurvedspacetimeitwillfolloathknownasageodesic。

6。Aspacetimediagramofapartigalongitsworldliisalwaysedwithinitsfuturelighte。

&sEarth-basedotations，ageodesiegeodesy，i。e。measurihelandofourpla'ssurfaimportadesgthenatureofspacetimethroughouttheUniverse。Ifspaotcurved（meairelytwitheverydaygeometrythatwemayhavelearsEueofhissuccessors），thenageodesicwouldbethestraightlih'thatalightraywouldtravel。Buttheshortestdistawopoints，whichistheroutethatalightraywants'totake，isknowermnullgeodesic'。Iheshortestdistawopointsisn'twhatwethinkht，butgehtlinesincurvedspaces'。Astraightlinealsobecharacterizedasthepathyoufollowbykeepingmovingiion。Anexampleofhowgeometryisseriouslydifferentonacurvedsurfaglinesoflongitudeowoadjaesoflongitude（aralleltooheequator）willmeetatapointatthepole，asshowninFigure7。However，inflatspaceparallellineswillmeetonlyatinfinity（asperEuclid'slastaxiom）。

Actually，wherespacetimeisplebecauseofthepresenass，thatcurvatureismahepaththatalightrayor（amentaldeviceusedbyphysicists）atestparticle'freelyabletomovewithnoinfluenyexternalforovealowoevesshardedastwopointsin4-Dspacetime，eaotedi，x，y，z）。

7。Linesoflongitudeonasphereareparallelattheequator，aapointatthepoles。

Arulecalledametrictellsushowdrulersmeasuretheseparatiosiimeahebasisfoutproblemsiry。AverysimpleexampleofametricisPythagoras'theorem，whichtellsushowtoputethedistawopointsthatlieihesolutioein'sfieldequationstellushowtocalculatethemetricofspacetimewheributionofmatteriskhisteodesicsfortherealUniverse。Forexample，opiecesofobservationalevideneralRelativitywasthebendinghtbytheSun，measuredduringasolareclipse（agoodtimetoexamipositionsofstarsclosetotheSun'sdiscbecauselightfromthediscisblockedoutbytheMoon，anopportunityseizeduponbySirArthurEddingtonin1919）。TheSun'smasscurvesspacetime。Thustheshortestpath（thegeodesiadistantstartoatelesEarthisraightliisbentroundbytheSun'sgravitationalfield，asshowninFigure8。

Thebendinghtdemospaceiscurved，butEinstein'sGeellsusitisactuallyspacetimethatiscurved。Therefhtexpectthatmassalsohasseeffee。IheEarth'sgravitationalfieldissuffiakeEarth-boundclockstickabitslowerthantheywoulddoindeepspace，althoughtheeffectissmall（roughlyoinabillion）butmeasurable。Thegravitatioshorizonofablauger。Thus，evecaseofanon-spinningblaerulyclosetotheblaparedtohowitrunsatahugedistaheblackhole。Thisisarealeffeddoesnotdependoimeismeasured（forexamplebyanatomicclitalwatch）。Itfollowsdirethecurvatureofspaducedbythemasswhichtipsthelightestowardsthemass。Figure9ihege。

8。AmasssuchastheSuion，orcurvature，iime。

Blackholesprofouheorientatioes。Asaparticleapproachesablackhole，itsfuturelightetiltsmoreaowardstheblackhole，sothattheblaesmoreaofitsiure。Wheiclecrossestheeventhorizon，allofitspossiblefuturetrajediheblackhole。Justwithihorizoetiltingissogreatthatonesidebeesparallelwiththeeventhorizoureliesehihorizoheblackholeisnotpossible。Figure9alsoillustratesthispoint:itisessentiallyarepresentationoflocalspacetimediagrams'，becausetheassemblyoflightesallowsyoutouheloditionsexperieestparticlelocatedatdifferentpositions。Inthisfigure，timeihepageandsothisdiagramalsogivesasenseofhowablasandgrowsduetoinfallingmatter。

9。Diagramofthespacetimesurroundingablackholeshowiurelightesforobjetheeventhorizoheeventhorizon。

JustasforthedarkstarsofMidLaplacedisChapter1whichcouldhavesustaiemsinorbitaroundthemmuchlikeourSolarSystem，soitisthatweonlyknowthatablackholeissgravitationalpull。Thismightleadyoutothinkthattheocharacterizesablackholeisitsmass。Iherornotablackhhasadramaticeffeitsproperties，andIwillexplainhowthisesaboutinChapter3。