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Chapter2Navigatingthroughspacetime
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&icsistheexquisitelyperfeguageneededfhowthetheoryofrelativityappliestothephysiiverseandallofspadthatdesirahatoearblaathematicaldes，whilepowerfula，evensoethingnandflahosewithouttheappropriateteiiivewords，howevereloquent，lacktherigourandpowerofamathematicalequationandbeimpredlimiting.Picturesh（itissaid）worthathousandwords，otonlyausefulisebutaveryhelpfulwaytovisualizewhatisgoingon.Forthisreason，itiswellworthspendingalittleefforttouandaparticulartypeofpicture，calledaspacetimediagraThiswillhelpiaureofspacetimearoundblackholes.
&imediagrams
&ooninFigure3sholespacetimediagraFollowingtraditioime-like'axisistheoisvertithepageandthespace-like'axisisdrawnperpendiculartothis.Ofcourse，wereallyneedfouraxestodescribespacetimebecausetherearethreespace-likeaxes（usuallydenotedx，y，andz）aime-likeaxis.Hoillsufficeforourpurpose（andofutuallyperpendicularaxesareimpossibletodraw!）.Wherethesetwoaxesiiscalledthein，andthismayberegardedasthepointofhereandnow'fortheobserverwhohasstructedtheirspacetimediagraAaheerashutter，occursatapartieaparticularlospace.Sustaisrepresentedbyadotoimediagram，appropriatetothetimeandspatialloquestiowodotsinFigure3，atiallyseparated（theydonotoccuratthesamepointonthespaceaxis）buttheyaresimultaheidenticalateoimeaxis）.Youagisdtothesimultaerpressesoftwophotographerswhoarestaafromothesamespectacle.Ifpoievents，whatdolinesiimediagramrepresent?Alinesimplyshoathofahroughspacetime.Asweliveourlives，wejhspadthepathweleavebehindus（somewhatasaseningtrailofslimebehindit）isalineiime，ahisiscalledaworldline.Ifyoustayathomeallday，yourworldliicalpaththroughspacetime（withspaate=22Aue'，forexample）.Youmoveforwardiarefixediheotherhandyoumadealongjourney，yourworldlisoverbecauseyourdistaime，beoveinspaceaswellastime.
3.Asimplespacetimediagra
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.
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