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Chapter 4 Falling into a black hole（第1页）

Chapter4Fallingintoablackhole...

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Howcloseistooclose?

Beforewesiderienifyingshadthemisfortuofallintoablackhole，itisimportaheeffeobserver'sparticularperspeeofreferehismeansthatdifferentobserversseeverydifferentthilyerspectiveisofallingintoablackholedependsonhowfarawayyouarefromthatobjedeedwhetheryouarethatobjesideraparticleoflight，aphoton，thatisoutsidetheeventhorizonofablackhole:siisoutsidethehorizon，itprincipleessidetheeventhorizonitwouldbeadifferentstory-thephotonotescapethegravitatioheblackhole.Buteveheeventhorizon，aphotonthatistravellingawayfromtheblackholewillelyuhephotonsuffersalossiotheworkithastodoagainstgravity.Thisisanexampleofagravitatioialwell;justasenergywouldbeohaulyourselfupwardsoutofadeepwell，sothephotooexpeopullitselfawayfrionnearamassiveobject.Theeffecthasevenbeenmeasuredforphotonsmovih'sgravity.Theenergyofaphotonisiedtoitswavelength:ahigh-eonhasashortwavelengthwhereasalyphotonhasalohephotonlyasitretreatsawayfromtheblackhole，soitswavelehisgesthecolourofthelight，movingfromtheblue（shortwavelength）towardsthered（loh）erum（thiseffectiscalledredshift）.Thissortofredshift，knownasgravitatio，ariseswherespacetimeitselfstretchesout，orisplebytheeffeassivebodysuchasablaotethatJohnMichell，despitehavingsignifitinalthoughtsaboutdarkstars，wasihinkingthatthevelocityoflightdecreasesasitclimbsoutofthepotentialknowthatitisthewavelength（hency）oflightthatisaffectedbythepresenassivestar.

enstotimenearablackhole?

InChapters1and2Idescribedhowspacetimeisdistortedbythepresenass（i.e.somethingwhichproducesitsowngravitatiohismeansthatnotjustspace，butalsotimeisaffectedclosetoablackhole.

ImagitokeepasafedistanaSchwarzschildblackholebutyouwanttolearhowtimebehaveshusyedfortwenty-sixfixedobserverstobestatioheblackhole'seventhorizoelysafelyoutsideit.TheseobserversareoZ，andarearrangedihAclosesttotheeventhorizonandwithZbeioyou，safelyfaraway.EachobserverfromAtoZhasagoodclockwithwhieasuretheirlocaltime，attheirparticularlo.AspartofthedealtopersuadeAtoZtopartithisexperiment，youhadofferedthemeaduthefiftofanadditional，unusualclockthathadbeethetimeonitwouldreadthesameasthetimeonyourclockatyoursafedistaitZ，closesttoyou，wouldfindthatthetwohispossessihtlydifferenttimesbecausehisownclock，whichmeasureslocaltime（propertime'inthejargon），wslightlymoreslowlythaclockwhichmatchesthetimeyoumeasureatyourrathersaferaaedresultsofpartitsZtoAwoulddisplayaremarkableeffect:closertoablackhole，aeasuringtimerunsmoreslowly'paredwiththedistanttimeasreportedoits'speciallyadjustedgiftclocks.Thiseffect，describedbyEinstein'stheey，isknowiowouldbegreateraheobserversofthealphabetwhoareheblackhole.Thegreatertheproximitytoablaoreslowlyalocalcloykind:atomiical）willruoaclockusedbyadistantobserver.

Supposeyouweremulti-taskiswithadifferewenty-sixobserversatthesamedistanadifferentblackhole.Theyarearrathesamewayastheirhefirstblackhole.However，inthissedcase，theblackholehastwicethemassoftheblayourfirstexperiment.Theunusualclocksyouhadpreparedasgiftsforthissedsetofobserverswouldoberadicallyalteredasfi，buttherateatwhiusualclockhastobeadjustedisexactlydoublethatoftheratehedingthefirstsetofgiftclocksattheexactsamedistahetreofthefirstblackholewhichhashalfthemassofthesed.Thesetimedilatiosarelargeriftheblaassislarger，andalsobeetheclettotheeventhorizon.

&histimedilationisnotaeadditionallight-traveltimeforaclockclosertothebladheherfromyou，thesafely-distahereisimeoffsetforaherawayfromtheblackhole.Thecloseraclockistoablackhole，thesloweristherateatwhichtimeismeasuredtoflowonthatatterutablemeaomeasurethatflowoftime.Timeitselfisstretched（or，ied）.

Whatisthecorollaryoftimedilationnearablackhole?Thiscauseseffectsthathappenintheframeofanobserververylocaltotheblaeasuredtobeverydifferentfromthoseintheframeofanobserverwhoisverydistant，artinfact.

&'snowsiderensifinyourfirstexperiment，observerAbecamealittlecarelessanddroppedhisfirstcloewithwhieasurepropertimeathislo）sothatitfelltowardstheblackhole.Despitethisdisaster，hewouldbehelesssafelygrippingoclockwithwhichyouhadeopartitheexperiment.BothyouandAwouldseehisfirstovetowardsthehole.Theclockwouldfinditselfmovingintotheblaoreandmorerapidly.YouandAwouldgraduallyhetimeyoureadoingesevenmoredistwiththetimeonA'sotherelytheclockthatwasadjustedtoruhelocalorderthatitwouldreadthesametimeasthetoyourtime）.AfterawhilebothyouandAwouldbegintoimestopsfortheplummetingcloittedattheeventhorizontowardsadistantobserverappearstostaythereihappenstoanythingthatfallsintoablackholeafterithaspassedwithiicalradiusoftheeventhorizonisuoaernalobserver.Sotheeventhorizonmayberegardedasaholeiime.Nolightwillemergefromwithihorizon，aster1.Thatiswhyitisblack.However，inthereferehedroppedetingthroughtheeventhorizon，lifeisveryfarfromung.Fromtheclock'sperspective，itwouldtraveltothesingularityien-thousandthofaseingthattheblackholehadamassofteofourSun.IftheclockhadthemisfortuofallintoasupermassiveblackholewithamassonebillioofourSun（suchasestudyquasarsisjourimeihevastlylargereventhorizonandthesingularitywouldbeamoreleisurelyfewhours.

Tidalforearablackhole

Supposei，persojumpiowardstheblackhole，inhopesofbeiheclockhedropped.Whaten?Suchaleapwouldprovetobeabigmistake，asthesurvivaloutewouldbezero.Thedifferehegravitationalforhisfeetandtheforhisheadwouldbee.Thisisafeatureofanyinverse-squareforcefield，suchasgravityfromamassivebody.TheEarthisratheralongwayfromthemoohesmalldifferengravitatiohemoonexperienoppositesidesoftheEarth，knownastidalforces，areattherootofwhythetideseandgoabouttwiceperday.Iheseffromdifferengravityiplacesarecalledtidalforces.Thereareadditionalfactorsthateailsandfallingoftidessuchasthegravitatioherelativeahemooailedshapesoftalmasses.ButevenifthesurfaceofEarthwereentirelycoveredbyowithoutlaillbetideslitudeofthesealevelvaryingbyabout20cmtwiceperday，simplybecauseofthedifferentialgravitationalforceexperiencedbypoiatdifferentdistaheSun.

&'snowsiderthesmallerdistaweehetreoftheEarth.AsIsittypingthischapter，myheadissomewhatoverametrehigherthawhithefloorofmystudy.MyfeetarethusclosertothetreoftheEarththanmyheadis.Becausethegravitationalforcefollowsaninverse-squarelawbehavihallthemassoftheEarthwerelocatedattheverytreoftheEarth，ahaveasmallerdistahistretheyfeelastrongerforce，orpull，tothetreoftheEarththanmyheaddoes.Butactually，thediffereherslender:fhtdifferehedifferengravitatiohreepartsintenmillion.ThisissuchaslightdifferencebecauseIamabout6，400kmfromthetreoftheEarth.Muchclosertoapointmasssuchasablackhole，thedifferengravitationalforceexperiepointsjustametreapartiowardstheblackholewouldbevastlymoreextreme.SoextremethatclosetothesingularityA'sfeetwouldbestretchedawayfromhiskherestofhisbodybeyondwhathistendonsandmusclescouldholdtogether，andhewouldbeelooslonottojump.

Dynamicspacetime

&ionofablaakesanimportantdifferenceregardinghowatterorbitaroundit，aohowmuergybeextrait.FromtheworkofRoyKerrandhissolutioeiiohatthesmallestorbitthatapartihavearoundablackholewithoutfallingindependsonjusthowfasttheholeisspierablackholeisspihemattergetbeforetheholeswallowsit，asillustratedinFigure13.Ifyoudropsomethingstraightdoinningblackhole，itwillstarttheholeeventhoughthereisnothiyspacetimeoutsidethehole.Outsidetheergosphere，itispossibletooverethisframedraggingusiheregioatingblackhole&#osphere，justoutsideitseventhorizon，nothingdstill.Thespiuallydragsthespadheentsaroundwithit.Afurtheraspectingisthateveniflightitselfisgoingagaiiontheblackh，itwillbethereversedirearoundthehole.

13.Gasorbitclosertoaspinningblackholethantone.

aroundablackhole

ItisiopowouldbethesequesifourSuaamorphoseintoablackhhthatyouorIowaboutitwouldbeeightmihebeautifulSpringsunlightbywhichIamwritingwouldetoanabrupthalt.AlthoughtheluminosityofthesiarwecallourSunistinybyparisonwiththequasarsandmicroquasarsdisChapter8，itissuffitlyclosetotheEarththatitprovideseaboutakiloersquaremetreofpowertoourpla.Remarkably，thishasbeeosustaihepla，allowingplantstogrowaenbyanimalsthataretheheranimals.TheSuheeall.ButiffusioheSunararytoallexpe）tocollapseintoablackhole，thenitwouldgoverydarkandwewouldalleventuallydie.（Thisisabitofagloomyoutlook，butIehereadertoholdfastuntilChapter7，wherewelearnthatourSunisnotthekindofstartoformablackhole-it'stoolightweightforthat.）However，dynamicallyspeaking，asfarasplahandthewholeSolarSystemofplas，dlas，andasteroidsareed，nothingwillgeatall.AllmassivebodiesinorbitaroundtheSunwillueiymuchthesameorbits.ThewaythatgravityworksisthatwhethertheSuentthatithasnow，orwhetheritcollapsestoasingularitywithihorizonof3km，thegravitationalattraoutsidetheSunwouldremaihesphericalcollapseuytoablackholewouldheahebodiesatall，sothepatternsandprogressionsahintheSolarSystemwouldueutterlyuhelashine.

Someswouldbepossiblehowever，muchclosertotheblack-holeSunthanwerepossiblepreviouslywhenthesolarplasmawasintheway.However，theseorbitsotgettooclosetotheeventhorizoailsofspacetimebyamasssingularitymeanthatitisnotpossibletoorbitjustoutsidetheeventhorizoemptingacircularorbittherewouldrequirecorrectiveabyroai.Infact，themathematicsshowsthattheclosestthatweoranyothermassparticlecouldexistonastablecircularorbitionaryblackholewouldbeatadistahatoftheSchwarzschildradiusaway.Youhavebeen>

Actually，unstablecircularorbitsarepossibleuptohalfthisdistanceawayfromaSchwarzs-spinning）blackhole.Thisdistanesasphericalsurfacethatissometimescalledthephotonsphere.Evenforaphotosareunstable，aoolphotoherslitherintowardstheblaevertoreturn，orindeedaace.

ForaKerrblackholethough，ohasspiiofortheorbitsheblackhole.Inparticular，therearetwophotonspheres，intrastwiththeoonspherearoundastationarySchwarzschildblackhole.Theoutermostsphereisforphotonsthatareoppositelytothedireofrotationoftheblackhole（theoneswesayareradeorbits）.Ihisisthephotointhesameseheblackholeasitisrotating（radeorbits）.Foraveryslblackholethatisn'tsoverydifferentfromaSchwarzschildblackhole，thesetwophotonspheresareverynearlyco-spatial.Forblackholesspin，thesesurfaglyfurtherapart.

Moviblackhole，thereisaantsurface（disChapter3），calledthestaticlimit.Thisisthesurfaceatwhigstaticwithrespecttoadistaisjustimpossibletositstillthisclblaatterhowpowerfultheroightbeequippedwith.Atthissurfaretrhtraysaredraggedaloionofrotation.Itisstillpossibletoesthisclblackhole，withsuffitpropulsion，butit'sjustnotpossibleftoremainstationaryandnhere.Movinginwardsstillfurther，thesurfaificeistheeventhorizoiheohatwemetinallyiofSchwarzschildblackhthisoutwardsisn'tpossibleandgitinwardshasaabledestiny，justasforthestaticblackhole.

AnorbitaroundaKerrblackholeisnotgenerallyedtoaplaheonlyorbitsedtoaplahoseihatstheequator（i.e.theplaneofmirrorsymmetryofthespinningblackhole）.Orbitsoutofthisequatorialplahreedimensiosareedtoavolumethatislimitedbyamaximumandminimumradiusandbyamaximumangleawayfromtheequatorialplane.

&ailsofthespinofablackholehaveadramaticeffehowcloseparticlesmayeheblackhole，whichitselfdepeioivetothespin.Foramaximallyspinningblackhole，thephhtraysinthesamesense（prograde）astheblackholespinhasaradiusthatishalfofwhattheSchwarzschildradiuswouldbe.Fhtraysradeorbits，theradiusoftheirphotowicetheSchwarzschildradius.Forparticleswithmassthatareradeorbits，theiablecircularorbitonwhichtheymoveisagainathalfoftheSchwarzschildradius.Forthradeorbits，suchaclosedistancewouldbeuheiriablecircularorbitisat4.5timestheSchwarzschildradius.Thus，arotatingblaablespartiprogradeorbitstoorbitmorecloselywiththepointofheeventhorizon，morecloselythaniftheblackholeweren.InChapter7，wesidertheimportahowatterorbitbefontoabladhowmuergymaybetlyleveraged.