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ColinMacLaurinColinsCosmos.com
UniversityofQueensland
CCGRAA16,Vancouver8th July2016
StaticvsFalling:Timeslicings of
Schwarzschildblackholes
Outline
• RadialdistanceinSchwarzschildspacetime• Adaptedcoordinates• Spatialprojector• Tetrads• Radarmetric
• Flamm’s paraboloid• Volume• r coordinatevector• “Length-contraction”• Bonus:
• Coordinatechoices• Tidalforces(elementaryderivation)• r as“reducedcircumference”• Observeratinfinity
Purposes:• Conceptualfoundations• Pedagogy• Avoidmisconceptions
Slicingbyfamilyofobservers• Parametriseradially-fallingobserversbye,dubbed“energyperunitmass”:
• Measurementsstrictlylocal
LorentzboostfromSchwarzschildtoGullstrand-Painlevecoordinates.
Taylor&Wheeler,ExploringBlackHoles(2000),§B4
Adaptedcoordinates:Lorentzboost
• Generalisationto• Orthonormaldualbasis:
• LorentzboosttonewtimecoordinateT:
• TheoriginalGullstrand-Painleve coordinatesarefor“rain”,e=1
• (Bonus:generalisedLemaitrecoordinates)
GeneralisedGullstrand-Painlevé coordinates
Coordinateshistory
rain (e=1)Gullstrand(1922)Painleve (1921)Lemaitre(1932)Robertson,etc.
drips (0<e<1)Gautreau &Hoffmann(1978)
hail &drips(e>0)Martel&Poisson(2001)
Finch(2015)
Radialdistancefromadaptedcoordinates
Gautreau &Hoffman(1978),0<e<1
Set:
Spatialprojector
• Givensomemetricg andobserver4-velocityu,thespatialprojectortensoris:
• Spatialdistancemeasuredbythisobserveris:
• Indirectionr,radialdistanceis:
Spatialprojector:Schwarzschildcoordinates
Spatialprojector:generalisedGullstrand-Painleve coordinates
• SametensorP,samecoordinater,butdifferentresult!(Duetodifferenttimeslicings,infactther-coordinatevectorsaredifferent…)• The2formulaeconcurforstaticobservers,forwhom:• Staticobserversmeasuretheusualdistance:
Observerframe:orthonormaltetrad
• Orthonormalè 1timelike,3spacelike vectors
source
Tetraddecomposition
ingeneralisedGullstrand-Painleve coordinates
inSchwarzschildcoordinates
Schwarzschildcoordinates
Radarmetric
(Landau&Lifshitz 1971,§84)
source
Observerswiththeradargunmustbecomoving.Butthespatialprojectorreducestothissameformwhenu iscomoving!Henceitisthesameastheradarmetric.
TextbooksonSchwarzschildr:
• Ofr:“Itisnot thedistancefromany‘center.’”(Hartle §9.1)• Of
“radialrulerdistance”(Rindler 2006,p230)“physicaldistance”,“actualradialdistance”(Moore2012,p106-7)• Newtonian:acoordinateiseitherdistanceoritisnot,andthisisthesameforallobservers• GR:acoordinateisnotthedistance,norisitnot thedistance.Rather,itdependsonwhoismeasuring
Space
• 3-dimensional“space”partofspacetime• InSchw.coords forr>2M,takedt=0andequatorialslicedφ=0:
• Representasa2-dimensionalsurfacez=z(r)inEuclideanspace,withthesamecurvature:
• Flamm’s paraboloid(1916):
Space
• IngeneralisedG-Pcoordinates,setdT = dφ =0:
• Then
• Aconefor|e|<1.|e|=1givesaflatplane.|e|>1cannotberepresentedbythismethod.
• Moore:spacecannotberepresentedinsidethehorizon
Volume
• Spatial3-volume• Butwhat“space”isdependsontheobserver• Volumeinsider=2M iscloselyrelatedtothevolumeofaEuclideanballofradiusr=2M.Finchgivesthisfore>0
• Morerigorously:
Timeandspaceinsidetheeventhorizon
• Dotimeandspaceswaproles?• “Spaceandtimethemselvesdonotinterchangeroles.Coordinatesdo…”(Taylor&Wheeler§3.7)• “Themostobviouspathologyatr =2M isthereversalthereoftherolesoftandr astimelike andspacelike”(MTW§31.3)• “First,thecoordinater forr <r_g playstheroleofatimecoordinate,andtactsasaspatialcoordinate.”(Frolov &Novikov §14.2)
Timeandspaceinsidetheeventhorizon
• Acoordinateistimelike/null/spacelike dependingonitscoordinatevector• Ther-coordinatevectorpointsinthedirectionofincreasingr:
• Thisvectorhasnorm-squared:• ForSchwarzschildcoordinates,thisisspacelike (negativeinourconvention)forr<2M• ButforgeneralisedGullstrand-Painlevé coordinates,itisspacelikeeverywhere
Timeandspaceinsidetheeventhorizon
Allthesecoordinatesystemsusethesame
Timecoordinate
• Unlikeforr the“time”coordinatesaredistinct.Buttheyarenotalwaystimelike:
Importantimplications:theyaredifferent vectors!
Timeandspaceinsidetheeventhorizon
• Whatcausesdecreaseinr?• “…youcannotevenstopyourselffrommovinginthedirectionofdecreasingr,sincethisissimplythetimelike direction.…t becomesspacelike andrbecomestimelike.Thusyoucannomorestopmovingtowardthesingularitythanyoucanstopgettingolder.”(Carroll§5.7)• “Ther =0singularityintheSchwarzschildgeometryisnotaplaceinspace;itisamomentintime.”(Hartle §12.1)
Timeandspaceinsidetheeventhorizon
• Hypersurfacer=const hasnormalvector:• Thisvectorhasnorm-squared:• Thenormalvectoristimelike,sothehypersurfaceisspacelike• Thisholdsforallcoordinatesystemsusingr• Socoordinatevectorsdependontheothercoordinates,butconstantcoordinatesurfacesdonot.
Radialdistance:length-contraction
Conclusion
• Emphasisedrelativityofspace,time,andsimultaneity• Avoidsover-interpretationofSchwarzschildcoordinates.Eisenstaedtdescribedthis“neo-Newtonian”interpretationreigninguntilthe1960s“renaissance”.Howevervestigesremain.Asdomisconceptions.
References
• C.MacLaurin,T.Davis,G.Lewis(2016),inpreparation• Taylor&Wheeler,ExploringBlackHoles (2000)• Gautreau &Hoffman,“TheSchwarzschildradialcoordinateasameasureofproperdistance” (1978)• Martel&Poisson,“RegularcoordinatesystemsforSchwarzschildandothersphericalspacetimes”(2001)• Finch,“CoordinatefamiliesfortheSchwarzschildgeometrybasedonradialtimelike geodesics” (2015)
Mainformula: (&gratuitouseyecandy)
ColinMacLaurinColinsCosmos.com
UniversityofQueensland
Adaptedcoordinates
AllexceptSchwarzschildarenon-singularatr=2M
Bestsuited to Comoving? Comments
Eddington-Finkelstein photons yes
Kruskal-Szekeres photons no(but“lightcone”variantis)
maximalanalyticextension
GeneralisedGullstrand-Painleve
massiveparticles:e-faller
no
GeneralisedLemaitre massiveparticles:e-faller
yes
Schwarzschild massiveparticles:static(r>2M)e=0(r<2M)
yes
Bonus:elementaryderivationoftidalforces
• Allradiallymovingobserversexperiencethesametidalforces:“amazingresult(aconsequenceofspecialalgebraicpropertiesoftheSchwarzschildgeometry…)”(MTW§31.2,32.6)• Usuallylotsoftensoralgebra.Butthelength-contractionresultgivesan“elementary”proof.ThisgeneralisesTaylor&Wheeler(§B.7)e=1
Bonus:r as“reducedcircumference”
• Drawacircle,circumference2pir,socanrecoverr(Droste,1910s)• Onlyforstaticobservers!(Orradialmotion,ifcouldmeasurefastenough)• ComparetherotatingdiskinMinkowski spacetime,akaEhrenfest’sParadox.Thishasanon-Euclideangeometry,withcircumferencenot2pir
Bonus:t timeofobserveratinfinity
• tastimeofobserveratinfinity.• Con:simultaneityisrelative.Especiallyatsuchgreatseparations• E.g.raindroptime(Gullstrand-Painleve coordinates).Thenraindrop timeisthetimeatinfinity!(Howmuchtimetofallpasttheeventhorizon?Distantobserverwouldanswer:exactlythesametimeasthepropertimeoftheraindropitself!)• Pro:extendspatialr-coordinateinwards,andindeedthe4-velocitieslineup• Mixed:Riemannnormalcoordinates.Proasabove.Butconifslicingbytheinducedtimecoordinate,thegeodesicextensionofthetimelike 4-velocity…butweknowwhatthisis;itisraindroptime!• Con:Fermicoordinates.Considerobserverfreefalling• Bettercomparison:staticobservers(atallr>2M).Better,butnotexact.Thespacetime slicings arethesame.Whilet isnottheirpropertime,itisproportionaltoit(sinceconstantr).
• Lemaître (1932):• WeshowthatthesingularityoftheSchwarzschildexteriorisanapparentsingularityduetothefactthatonehasimposedastaticsolutionandthatitcanbeeliminatedbyachangeofcoordinates.
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