The Black Hole Scatterer, Absorber and Emitter of Particles

THEBLACKHOLE:

SCATTERER,ABSORBERANDEMITTEROFPARTICLES

N.SANCHEZ

ObservatoiredeParis-DEMIRM,

61Avenuedel’Observatoire,75014PARIS,FRANCE

Abstract

Accurateandpowerfulcomputationalmethodsdeveloppedbytheau-thor,basedontheanalyticresolutionofthewaveequationintheblackholebackground,allowtoobtainthehighlynontrivialtotalabsorptionspectrumoftheBlackHole.Aswellasphaseshiftsandcrosssections(elasticandinelastic)forawiderangeofenergyandangularmomentum,theangulardistributionofabsorbedandscatteredwaves,andtheHawk-ingemissionrates.ThetotalabsorptionspectrumofwavesbytheBlackHoleisknownexactly.Itpresentsasafunctionoffrequencyaremarkableoscillatorybehaviourcharacteristicofadiffractionpattern.Itoscillatesarounditsopticalgeometriclimit(27

Contents

I

INTRODUCTIONANDRESULTS

IIPARTIALWAVEANALYSIS

II.1Absorptioncrosssections.......................

Lowfrequencies............................Highfrequencies............................II.2ElasticScattering...........................

Lowfrequencies............................Highfrequencies............................HighAngularMomenta........................

IIIDIFFERENTIALELASTICCROSSSECTIONIVHAWKINGEMISSIONRATESVREMARKSONAPPROXIMATIONSVICONCLUSIONS

2

3799101011111213141517

THEBLACKHOLE:

SCATTERER,ABSORBERANDEMITTEROFPARTICLES

N.SANCHEZ

ObservatoiredeParis-DEMIRM

61,Avenuedel’Observatoire,75014Paris-FRANCE

IINTRODUCTIONANDRESULTS

Ishallreporthereaboutsomeofmyresultsonthephysicsofblackholesandthedynamicsoffieldsinthevicinityofsuchobjects,describingatthesametime,theBlackHoleunderitstripleaspectofScatterer,AbsorberandEmitterofparticles.

IshallfirstreportabouttheAbsorption,itappearsintheconceptofblackholeitself,thegravitationalfieldbeingsointensethatevenlightcannotescapeofit.Absorptionisoneofthepropertiesthatcharacterizestheblackholedescriptioninclassicalphysics:blackholesabsorbwavesbuttheycannotemitthem.Ifaquantumdescriptionofperturbationfieldsisconsidered,blackholesalsoemitparticles.Forastaticblackhole,thequantumparticleemissionrateH(k),andtheclassicalwaveabsorptioncrosssectionσA(k)arerelatedbytheHawking’sformula(1975,[1])

σA(k)

H(k)=

σA(k)=27πM−2

2

√

27πkM)

√(

3

√

.Theemissionspectrum(Fig.1)doesnotshowanyof

theinterferenceoscillationscharacteristicoftheabsorptioncrosssection,becausethecontributionoftheS-wavedominatestheHawkingradiation.TherapidlydecreaseofthePlanckfactorforkM≥1supressesthecontributionofhigherpartialwaves.

M

Thus,forablackholetheemissionfollowsaplanckianspectrum,givenbyeq.(1),(Fig.1),andtheabsorptionfollowsanoscillatoryspectrum,givenbyeq.(2),(Fig.2).

1086420

0.10.20.30.40.5

262422201816

1

2

3

4

5

6

7

8

M

)(l+

1

.Inwhatconcernstheabsorptionspectrum

itisnotpossibletoassociatearefractionindextotheblackhole.Foropticalmaterials,theabsorptiontakesplaceinthewholevolume,whereasfortheblackhole,ittakesplaceonlyattheorigin.

(8πM)

Itisalsointerestingtocalculatetheangulardistributionofabsorbedwaves.Foritonemuststudytootheblackholeaselasticscatterer.

Thedistributionofscatteredwaves,asafunctionofthescatteringangleθ,hasbeencomputedinawiderangeofthefrequency[3],[4].Itpresentsastrongpeak(∼θ−4)characteristicoflongrangeinteractionsintheforwarddirection,anda“glory”inthebackward,characteristicofthepresenceofstronglyattrac-tiveinteractionsforshortdistances.Forintermediateθ,itshowsacomplicated

6

behaviourwithpeaksanddropsthatdisappearonlyatthegeometrical-opticslimit.

Theangulardistributionofabsorbedwavesisshownin[4].Itisisotropicforlowfrequenciesandgraduallyshowsfeaturesofadiffractionpattern,asthefrequencyincreases.Itpresentsanabsolutemaximumintheforwarddirectionwhichgrowsandnarrowsasthefrequencyincreases.Inthegeometrical-opticslimit,thisresultsinaDiracDeltadistribution.TheanalyticbehaviourexpressesintermsoftheBesselfunctionJ1,asgivenbyeq.(8)below.

Inthecourseofthisresearch,wehavedevelopedaccurateandusefulcompu-tationalmethodsbasedontheanalyticalresolutionofthewaveequation,whichinaddition,haveallowedustodeterminetherangeofvalidityofdifferentapprox-imationsforlowandhighfrequenciesmadebyotherauthors(Starobinsky,Sov.Phys.JETP37,1,1973;Unruh,Phys.Rev.D14,3251,1976),respectively,andbyourselves[5].Itfollowsthattheanalyticalcomputationofelasticscatteringparametersforlowfrequenciesisaratheropenproblem.

Wehavealsoobtainedseveralpropertiesconcerningthescattering,absorp-tionandemissionparametersinapartialwaveanalysis.Theyarerepportedinreferences[2]and[4].Someofthemarealsoreportedinreferences[6]and[7].Theworkpresentedherehasalsoadirectinterestforthefieldandstringquantizationincurvedspace-times,relatedissuesandothercurrentproblems.SeetheConclusionsSectionattheendofthispaper.

IIPARTIALWAVEANALYSIS

Thepartialscatteringmatrixisgivenby

Sl=e2iδlδl=ηl+iβl

Wehavefound([2],[4])thattherealandimaginarypartsoftheBlackHole

functionsofthefrequencyrespectively:phaseshiftsδlareodd

ηl(xs)=−ηl(−xs)

βl(xs)=βl(−xs),xs≡krs=2kM

7

(3)

123

Intermsofthephaseshifts,thepartialelasticandabsorptioncrosssectionsarerespectivelygivenby:

πǫl=

xs

−4βl

(2l+1)(1−e)2

partialabsorptioncrosssection

II.1Absorptioncrosssections

Forallvalueoftheangularmomentuml,theimaginarypartofthephase

shifts,βl(xs),isamonotonicallyincreasingfunctionofxs.

Allβl(xs)arezeroatxs=0andtendtoinfinitylinearlywithxsasxsincreasestoinfinity.Lowfrequencies

[(2l)!][(2l+

2

1)!]2

(4)

WehavefoundforClvaluesinagreementwithStarobinsky’sformulae(Starobin-sky,Sov.Phys.JETP37(1973)1),forxs=0andl=0.However,theStarobinsky’sapproximationisaccurateonlyinasmallneighborhoodofxs=0.Forexample,theratio

β0exact(xs)−C0xs2

Thepresenceofapoleatxs=0forl≥1intheJostfunctionoftheBlackHole([2],[4])meansthatwaveswithverysmallfrequencyandl=0arerepelledoutofthevecinityoftheblackhole.Highfrequencies:

)

xs≫1

(6)

xs3/2

βlasistheasymptoticexpressionderivedwiththeDWBA(DistortedWaveBornApproximation):

βlas(xs)=πxs−1/4ln2−1/16(π/xs)

1/2

π(l+1/2)2−

2

dΩ

=|

isexpressedintermsoftheBesselfunctionJ1as[4]:

dσA(θ)

4xs2

27

∞󰀂l=0

2

(2l+1)gl(xs)Pl(cosθ)|

Lowfrequencies:

k

󰀁

󰀄

∞

dr

r

0

(

Rl

x

)

xs

x2

󰀃

,x∗=x+xsln(1−

xs

Forlargexs,xs≫1:ηlexact(xs)=ηlas(xs)+O((

1

2

−xs+

2

π

2

)+

1xs

)

1/2

+O(

1

−2ixs)

xs

).

=−2xsln(

2xs

ηlasistheasymptoticformuladerivedbyusintheDWBA(DistortedWaveBornApproximation)scheme[5].

11

ηlasandηlexactareingoodagreement.Forexample,

η0exact(2.3)=−1.2,η0as(2.3)=−1.18η0exact(2.5)=−1.6,η0as(2.5)=−1.60

Wehave

ηl(xs→∞)→−xsln(2xs)→−∞,forfixedlandxs→∞ηl(xs)→−xsln(l+

1

High-orderpartialwavesgiveanimportantcontri-butiontotheelasticscatteringamplitude.Thisfactleadustostudyηl(xs)forl≥2andfixedxs.Wefound:

ηl(xs)=ηlcoul(xs)+α0(xs)+∆l(xs)

where

ηlcoul(xs)=argΓ(l+1−ixs)∆l(xs)=

α2(xs)α3(xs)α1(xs)++23+O()))2

2

2

,l≫xs

(12)

1

2

)

4)

αl(xs)∼(xs)l+1α0(xs)≃

1

2xs2

Ascouldbeexpected,theleadingbehaviourofηl(xs)forlargelisthesameas

thatoftheCoulombphaseshiftηlcoul.

Thedifference

[ηlexact(xs)−ηlcoul(xs)]=xsf(b)

canbewrittenasxstimesafunctionf(b)oftheimpactparameterb=whichisanalyticatb=0.

(l+1xs

,

Thefactthatηldoesnotvanishesintheinfiniteenergylimitisaconsequenceofthestrongattractivecharacteroftheblackholeinteractionatshortdistances(thetermofthetype−1)intheradialwaveequation).4r2

12

III

DIFFERENTIALELASTICCROSSSEC-TION

Thescatteringamplitude,whosesquaredmodulusgivesthedifferentialelas-ticcrosssection,isgivenby

f(θ)=

∞󰀂l=0

(2l+1)

θ4

−

C1(xs)

θ2

+

C2(xs)

xs

C2(xs)=

+

󰀅󰀅

cos(2γ(−))4

4xs21

󰀆

1+2α1sin(2γ(−))+

1+

186

where

+

4xs2

+

363

288

1

α2

xs4

󰀅

γ(±)=argΓ(

1

5

xs,

α1∼

3

4

xs3

Forintermediateangles,|f(θ)|2hasacomplexbehaviourwithpeaksand

dropswhichdisappearatthegeometrical-opticslimit.

13

IVHAWKINGEMISSIONRATES

TheenergyemittedbytheblackholeineachmodeoffrequencykandangularmomentumlisgivenbytheHawking’sformula[1]

dHl(k)=

Pl(k)

π

kdk

LetusrecallourfirstanalyticexpressionforthepartialabsorptionratePl(k)(S´anchez,1976)[5].Thisisaformulaforhighfrequencies(krs≫1)obtainedwithintheDWBAscheme:

Pl(k)

krs≫1

=

1−exp(−4πkrs)

1+exp{(2l+1)π[1−

27k2rs2

2l+1

(exp(4πkrs)+1)

(exp(4πkrs)−1){1+exp[(2l+1)π(1−

27k2rs2

predominatesinHawkingradiation.Forexample,themaximaofHl(k)forl=0,1,2areintheratio

1

.1:

453

Forangularmomentahigherthantwo,Hl(k)isextremelysmall.

ThespectrumoftotalemissionhasonlyonepeakfollowingcloselytheS-waveabsorptioncrosssectionbehaviour.Itsmaximumliesatthesamepointthemaximumofσ0(xsmax=0.23).

Thepeaksofσ1andσ2turnouttohavenoinfluenceonH(k).

Inconclusion,Hawkingemissionisonlyimportantinthefrequencyrange

0≤k≤

1

beentakenintoaccount.Withhisapproximation,it

ispossibletofindfortherealpartofthephaseshiftalinearbehaviourinxs(ηl∼alxs),butinaccuratevaluesforthecoefficientalareobtained.

Inthecontextofamassivefield,UnruhincludedtheCoulombinteraction.However,hisapproximationisnotsufficientlyaccuratetogivetherealpartofthephaseshiftatleastforl=0inthezero-masscase.

15

ThediscrepancybetweenUnruh’sapproximationwiththeexactcalculationcanbeexplainedasfollows:inUnruh’sapproach,forr≫rs,alltermsoforderhigherthan(rs

(ξRl)+k+2

󰀁

2

2k2rs

(r−rs)2

dr

+

1

r2(r−rs)2

−

l(l+1)

4rs

2

rs8

k[r(r−rs)]

1

3

)2ordertermk2rs2

r

VICONCLUSIONS

Accurateandpowerfulcomputationalmethods,basedontheanalyticresolu-tionofthewaveequationintheblackholebackground,developedbythepresentauthorallowtoobtainthetotalabsorptionspectrumoftheBlackHole.Aswellasphaseshiftsandcrosssections(elasticandinelastic)forawiderangeofen-ergyandangularmomentum,theangulardistributionofabsorbedandscatteredwaves,andtheHawkingemissionrates.

ThetotalabsorptionspectrumoftheBlackHoleisknownexactly.Theabsorptionspectrumasafunctionofthefrequencyshowsaremarkableoscil-latorybehaviourcharacteristicofadiffractionpattern.Theabsorptioncrosssectionoscillatesarounditsopticalgeometriclimitwithdecreasingamplitudeandalmostconstantperiod.SuchoscillatoryabsorptionpatternisanuniquedistinctivefeatureoftheBlackHole.Absorptionbyordinarybodies,complexrefractionindexoropticalmodelsdonotpresentthesefeatures.

Forordinaryabsorptivebodies,theabsorptiontakesplaceinthewholemediumwhilefortheBlackHoleittakesplaceonlyattheorigin(r=0).FortheBlackHole,theeffectiveHamiltoniandescribingthewave-blackholeinteractionisnon-hermitian,despiteofbeingreal,duetoitssingularityattheorigin(r=0).Thewellknownunitarity(optical)theoremofthepotentialscatteringtheorycanbegeneralizedtotheBlackHolecase,explicitlyrelatingthepresenceofanonzeroabsorptioncrosssectiontotheexistenceofasingularity(r=0)inthespacetime.

All√partialabsorptionamplitudeshaveabsolutemaximaatthefrequence(l+1k=34M

DWBA(DistortedWaveBornApproximation)fortheBlackHoleasitwasimplementedbythepresentauthormorethantwentyyearsago[5]isanaccuratebetterapproximationforhighfrequencies(krs≫1)tocomputetheabsorption(imaginarypart)phaseshiftsandrates,bothforhigh(l≫krs)andlow(l≪krs)angularmomenta.

Approximativeexpressions(whatevertheybe),forveryhighfrequencies,orforlowfrequenciesdonotallowtofindtheremarkableoscillatorybehaviourofthetotalabsorptioncrosssectionasafunctionoffrequency,oftheBlackHole.TheknowledgeofthehighlynontrivialtotalabsorptionspectrumoftheBlackHoleneededthedevelopmentofcomputationalmethods[2]morepowerfulandaccuratethanthecommonlyusedapproximations.

Theangulardistributionofabsorbedandelasticallyscatteredwaveshavebeenalsocomputedwiththesemethods.

TheconceptualgeneralfeaturesoftheBlackHoleAbsorptionspectrumwillsurviveforhigherdimensional(D>4)genericBlackHoles,andincludingchargeandangularmomentum.TheywillbealsopresentforBlackHolebackgroundssolutionsofthelowenergyeffectivefieldequationsofstringtheoriesandDbranes.TheAbsorptionCrossSectionisaclassicalconcept.Itisexactlyknownandunderstoodintermsofclassicalphysics(classicalperturbationsaroundfixedbackgrounds).(Although,ofcourse,itispossibletorederiveandcomputemag-nitudesfromseveraldifferentwaysandtechniques).

Anincreasingamountofpaper[11]hasbeendevotedtothecomputationofabsorptioncrosssections(“greybodyfactors”)ofBlackHoles,whateverD-dimensional,ordinary,D-braneous,stringy,extremalornonextremal.Allthesepapers[11]dealwithapproximativeexpressionsforthepartialwavecrosssec-tions.Inallthesepapers[11]thefundamentalremarkablefeaturesoftheTotalAbsorptionSpectrumoftheBlackHoleareoverlooked.

References

[1]S.Hawking,“ParticleCreationbyBlackHoles”,Comm.Math.Phys.43

(1975)199.[2]N.S´anchez,“AbsorptionandEmissionSpectraforaSchwarzschildBlack

Hole”,Phys.Rev.D18(1978)1030.

18

[3]N.S´anchez,“WaveScatteringTheoryandtheAbsorptionProblemforaBlack

Hole”,Phys.Rev.D16(1977)937.[4]N.S´anchez,“ElasticScatteringofWavesbyaBlackHole”,Phys.Rev.D18

(1978)1798.[5]N.S´anchez,“ScatteringofScalarWavesfromaSchwarzschildBlackHole”,

J.Math.Phys.17(1976)688.[6]N.S´anchez,“SurlaPhysiquedesChampsetlaG´eom´etriedel’Espace-Temps”,

Th`esed’Etat,Paris(1979).[7]J.A.H.Futterman,F.A.HandlerandR.A.Matzner,“ScatteringfromBlackHoles”,CambridgeUniversityPress,Cambridge,U.K.(1988),andreferencestherein.[8]S.Persides,Int.Jour.Math.Phys.48(1976)165;50(1976)229.[9]A.A.Starobinsky,

37(1973)1).

Zh.Eks.Teor.Fiz.

64

(1973)

48.

(Sov.Phys.-JETP

[10]W.G.Unruh,Phys.Rev.D14(1976)3251.[11]Forexample:

S.S.GubserandI.R.Klebanov,Phys.Rev.Lett.77(1996)4491;J.MaldacenaandA.Strominger,

Phys.Rev.D56(1997)4975;

Phys.Rev.D55

(1997)

861

and

S.S.Gubser,Phys.Rev.D56(1997)4984andhep-th/9706100;

M.CivetiˇcandF.Larsen,Phys.Rev.D56(1997)4994andhep-th/9705192;I.R.KlebanovandS.D.Mathur,hep-th/9701187;

S.Das,A.DasguptaandT.Sarkar,Phys.Rev.D55(1997)12;S.P.deAlwisandK.Sato,Phys.Rev.D55(1997)6181;R.Emparan,hep-th/9704204;

H.W.Lee,Y.S.MyungandJ.Y.Kim,hep-th/9708099.

19