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Chapter 2 Navigating through spacetime（第1页）

Chapter2Navigatingthroughspacetime

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&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.