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.