The black hole quantum atmosphere

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

black

hole

quantum

atmosphere

Ramit Dey

a

,

b

,

Stefano Liberati

a

,

b

,

Daniele Pranzetti

a

,

b

,

∗

a_{SISSA,ViaBonomea265,34136Trieste,Italy}

b_{INFN,SezionediTrieste,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21August2017

Receivedinrevisedform13September 2017

Accepted21September2017 Availableonline29September2017 Editor:M.Cvetiˇc

Eversincethediscoveryofblackholeevaporation,theregionoforiginoftheradiatedquantahasbeen a topicofdebate. Recently it was argued byGiddings that the Hawkingquanta originate from a re-gionwelloutsidetheblackhole horizonbycalculatingtheeffectiveradiusofaradiatingbodyviathe Stefan–Boltzmannlaw.Inthispaperwetrytofurtherexplorethisissueandendupcorroboratingthis claim,usingbothaheuristic argumentandadetailedstudyofthestressenergytensor.Weshowthat theHawkingquantaoriginatefromwhatmightbecalledaquantumatmospherearoundtheblackhole withenergydensityandfluxesofparticlespeakedatabout4MG,runningcontrarytothepopularbelief thattheseoriginatefromtheultrahighenergyexcitationsveryclosetothehorizon.Thislongdistance originofHawkingradiationcouldhaveaprofoundimpactonourunderstandingoftheinformationand transplanckianproblems.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The discovery ofHawking radiation [1] changed our perspec-tive towards black holes, giving us a deeper insight about the microscopicnatureofgravity.At thesametime, withinthe semi-classical framework, the current understanding of such process still leaves open several issues. Of course, a well known unre-solvedproblemofblackholephysicsistheinformationloss para-dox[2–4],i.e. theapparentincompatibility betweenthecomplete thermalevaporationofablackholeendowedwithanevent hori-zonandunitaryevolutionasprescribedbyquantummechanics.

ForrestoringunitarityofHawkingradiationandaddressingthe information loss problemcorrectly, it is important (among other things)toknowfromwheretheHawkingquantaoriginate.For ex-ample,ifoneassumesanearhorizonoriginoftheHawking radia-tion,thenonewaytorestoreunitarityisbyconjecturingsomesort ofUV-dependententanglementbetweenpartner Hawkingquanta which would enable the late time Hawking flux to retrieve the information in the early stages of the evaporation process. Such scenarioseemstoleadtothesocalled“firewall”argumentasthe conjectured lackofmaximal entanglement betweentheHawking pairs makes the near horizon state singular and eventually de-mandssomedrasticmodificationofthenearhorizongeometry[5].

*

Correspondingauthor.

E-mailaddresses:rdey@sissa.it(R. Dey),liberati@sissa.it(S. Liberati), dpranzetti@sissa.it(D. Pranzetti).

Ontheotherhand,ifonebelievesinalongerdistanceoriginofthe Hawkingquanta,someeffectmustbeoperationalatalargerscale for restoringunitarity ratherthan nearthe horizon, avoiding the “firewall”.

A similar open issue is the transplanckian origin of Hawking quanta. Hawking’s original calculation indicates that the quanta originateneartheblackholehorizoninahighlyblue-shiftedstate requiring an assumption on the UV completion of the effective field theory used for the computation and on the lack of back-reactionontheunderlyinggeometry.1 _{Whileitwas debatedfora}

while ifHawking quanta could originate initially, during the star collapse, andlaterreleasedoveraverylongtime,itwas convinc-ingly argued in[8]that thiscannot bethe caseifanevent hori-zon indeedforms.This leads tothe conclusionthat the Hawking quantaaregeneratedinaregionoutsidethehorizon.Aconclusion corroborated by studies of theHawking modescorrelation struc-ture where it was shown that mode conversion happens over a longdistancefromthehorizon[9].Amorerecentclaiminthis di-rection,basedoncalculatingthesizeoftheradiatingbodyviathe Stefan–Boltzmannlaw,showedthat theHawkingquantaoriginate in a near horizon quantum region, a sort of black hole “atmo-sphere” [10]. It is a well knownfact that the typical wavelength oftheradiatedquantaiscomparabletothesizeoftheblackhole,

1 _{See,forinstance,Refs.[6,7]forablackholeevaporationanalysiswherethese} issuescanbeaddressedinaquantumgravitycontext.

https://doi.org/10.1016/j.physletb.2017.09.076

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

soonemightthink thatthe pointparticledescriptionisnot very accurate.However, asmeasuredbyalocalobservernearthe hori-zon,thewavelengthishighlyblue-shiftedwhentracedbackfrom infinityto thehorizon, thus validatingthepoint particle descrip-tion.

TheHawkingprocesscanbeexplainedheuristicallyas-well,for exampleviaatunneling mechanismwheretheparticletunnelsout ofthehorizonortheantiparticle(propagatingbackwardsintime) tunnelsintothehorizonandasaresultofthiswegettheconstant Hawkingfluxatinfinity[11].Alternatively, onepopularpictureis toimaginethat thestrongtidalforceneartheblackholehorizon stopsthe annihilation of the particle andanti-particle pairs that areformedspontaneouslyfromthevacuum.Oncetheantiparticle is“hidden”withintheblackholehorizon,havinganegativeenergy effectively,theotherparticlecanmaterialize andescapetoinfinity

[12,13].

Inthispaperweshallexplicitlymakeuseofthislatterheuristic picture as well as of a full calculation of the stress energy ten-sorin1

+

1 dimensions. Weshallseethat bothmethodsseemto agree in suggesting that the Hawking quanta originate from the black hole atmosphere and not from a region very close to the horizon. In section 2, based on the heuristic picture of Hawking radiationdescribed above and invokingthe uncertainty principle andtidalforces,weshowthatmostofthecontributiontothe ra-diationspectrumcomes fromaregionfarawayfromthehorizon. Insection3 wefurther strengthenour claimby adetailed calcu-lationoftherenormalizedstressenergytensor,whichindicates a similarresult.

2. AgravitationalSchwingereffectargument

Oneingredientofourheuristicargumenttoidentifyaquantum atmosphereoutsidetheblackholehorizon,whereparticlecreation takesplace,is the uncertaintyprinciple. However, theuse ofthe uncertaintyprinciplealone,asoriginallysuggestedbyParker[14], doesnotcontainanyphysicallyrelevantinformationaboutthe lo-cationofparticleproduction andwhy smallerblackholes should be hotter. Indeed,the uncertainty principle in thiscase provides arough estimateoftheregion ofparticleproductionasinversely proportionaltothe energyofthe Hawkingquantawhenthey are produced,butitdoesnottakeintoaccountanydynamical mecha-nismtoestimatetheprobabilityofspontaneousemission.

Thus one can improve this argument by invoking a physical process of creation ofthe Hawking quanta and using the uncer-taintyprincipleasacomplementarytooltoestimatetheregionof originofthequanta.Inthissection,wetrytoachievethisgoalby relyingontidalforces.

Letusthenconsiderasituationwhereavirtualpair,consisting of a particle and anti-particle, pops out of the vacuum sponta-neouslyfora veryshorttime interval andthen annihilatesitself. In the Schwinger effect [15] a static electric field is assumed to acton a virtualelectron–positron pairuntil thetwo partnersare tornapart oncethe thresholdenergynecessaryto become areal electron–positronpairisprovidedbythefield.Energyisconserved duetothefactthattheelectricpotentialenergyhasoppositesign for partners with opposite charge. However, in its gravitational counterparta priori onlyvacuumpolarization can be induced by astaticfieldintheabsenceofanhorizon.

Infact,onlyinthepresenceofthelatteronehasboththe char-acteristicpeelingstructure ofgeodesics(divergingaway fromthe horizononbothitssides)aswellasthepresenceofanergoregion behindit.2Thepresenceofanergoregioniscrucialforenergy

con-2 _{Thisis strictlytrue only for non-rotatingblackholes, for rotatingones the} ergoregionliesoutside ofthehorizonallowingfor the classicalphenomenon of

servationasitallowsfornegativeenergystatesgiventhatinitthe normofthetimelikeKillingvector,withrespecttowhichwe com-puteenergy,changessign.

Indeed,if a Schwinger-likeprocess takes place nearthe black hole horizon, due to the tidal force of the black hole and the peelingofgeodesics,thepaircan getspatiallyseparatedandone partner can enter the black holehorizon following a timelike or null curve withnegative energywhile the other particle can es-capetoinfinityandcontributetotheHawkingflux.Inthispicture, weareimplicitlyassumingthatvirtualparticlesinthevicinityofa blackholehorizonmovealonggeodesicswhentheyarejustabout togoon-shell.

Therefore,thephysicalscenariowewanttoenvisageisthatofa particle–antiparticlepairpulledapartbytheblackholetidalforce outsidethehorizonuntiltheygoon-shellasoneofthemreaches thehorizon3_locatedatr

s

=

2G M

/

c2 (actuallyaninfinitesimal dis-tance inside it so that the geodesic motion will drag it further inside)while the other particleis ata radial coordinate distance r

=

r_∗.Onceon-shell,theoutgoingparticleeventually reaches in-finityandcontributestotheHawkingspectrum.Inordertodoso though,it hastobecreatedwithan energycorrespondingto the energyoftheHawkingquantaatadistancer_∗

>

rs fromthe cen-terofthe blackhole asmeasured bya localstatic observer;this canbereconstructedbynoticingthat

ω

r

=

ω

_∞

√

g00

,

(1)

where

ω

∞ is the energy at infinity and we are using the

(

+,

−,

−,

−)

signature.Atinfinity,thethermalspectrumof Hawk-ingradiationgives

ω

_∞

=

γ

kBTH

¯

h

,

(2)

wheretheHawkingtemperatureforablackholeofmass M reads kBTH

=

hc¯

3

8πG M, and

γ

is a numerical factorspanning the energy rangeofthequantagivingrisetotheradiationthermalspectrum. Atthepeakofthespectrum

γ

≈

2

.

82.

Thus,weget

ω

∞

=

γ

c 3 8

π

G M (3) and

ω

r

=

γ

c 4

π

rs 1



1

−

rs r

.

(4)

Thisenergyisprovided bythework doneby thegravitational fieldtopullthetwopartnersapart.Wecancomputethisworkin the staticframe outsidea blackhole andcompareit with

ω

(

r_∗

)

. Using thisrelation,we can determinethe regionfromwhich the Hawking quanta originate. This is the process we now want to implement. Although in the rest of this Section we present the detailed derivation ofthe relation between the outgoing particle energy and the radial distance at which it goes on-shell for the massivecase,ourresultholdsalsoformasslessparticles.We com-mentattheendofthisSectiononhowthesameSchwingereffect

superradiance.However,thequantumemissionstillrequiresthepeculiarpeeling structureofgeodesicstypicalofthehorizon.

3 _{Onecouldalsoconsiderthecasewheretheingoingparticletunnelsthroughthe} horizonandgoeson-shellwellinsidethehorizon(ase.g. suggestedbytheresults of[9]);however,sinceinouranalysisbelowweareinterestedinthetidalforce ascomputed intheoutgoingparticle restframe,thisshouldnotaffectthefinal expressionfortheforce.Thus,fromthepointofviewofanoutsidestaticobserver, theworkdonebythegravitationalfieldonthepair(inourheuristicderivation)is insensitivetotheexactlocationwheretheingoingparticlebecomesreal.

argument can be implemented straightforwardly to the massless case.

Let us clarify that, in a general relativistic framework, the geodesic deviation equation does not describe the force acting on a particle moving along a geodesic. Rather, it expresses how thespacetimecurvature influencestwo nearbygeodesics,making them eitherdiverge orconverge, i.e. it effectivelymeasures tidal effects.Therefore,wecaninterprettheseeffectsasthepullofthe gravitationalforce onparticles andtalk aboutthe work done by thegravitational field onlyin an heuristicsense. Nevertheless,in the case considered here where the test particles have a mass muchsmallerthantheblackholeandwecanneglectback-reaction effects,we expect thisinterpretation ofthegravitationalfield ef-fectstocapturesomerelevantaspectsofblackholephysics.With theseassumptionsspelledout,letusproceed.

In therest frame ofthe outgoing particle,one would seethe antiparticleacceleratingtowardsthehorizonduetothetidalforce. This radial acceleration in the rest frame of the particle can be computedusingthegeodesicdeviationequation,namely

ar



_r ∗

≡

Dnr D

τ

2





r_∗

=

Rrμνρuμuνnρ



_r∗

,

(5) where the r.h.s. is expressed in terms of the Riemann tensor components,nr _{denotes the separation betweenthe two radially} infalling geodesics followed by the pair of particles and uμ

=

[

1

,

0

,

0

,

0

]

intherestframeoftheparticle.

Theseparationbetweentheparticleandtheanti-particlewhen thepair formsspontaneously (i.e.they go “on-shell”)is givenby their Compton wavelength, namely nρ

= [

0

,

nr

,

0

,

0

]

where nr

∼

λ

C

= ¯

h

/

mc,andm



M istheparticlesrestmass(fromnowonwe shallworkinunitswhereh

_¯

=

c

=

1).Sointheend,Eq.(5)implies that theradial componentof thetidal acceleration(ascomputed intherestframeoftheparticleatcoordinater_∗)isgivenby4

ar

|

r∗

=

2MG

r3

∗

λ

C (6)

Ouraimis todetermine thework done onthespontaneously createdparticlepairbythe tidalforceinthestaticframe outside theblackhole.Forthisweneedtocomputethetidalforceas mea-sured by a static observer outside the black hole at the instant whentheoutgoingpartnergoesonshell.Thiscanbe achievedby considering theparticle restframe and thestatic observerframe aslocallytwoinertialframes:Thelatterseestheparticleas mov-ing with outward velocity given by the radial component ofthe geodesic tangent vector ur

₌

_dr

_/

_d

_τ

_{. Once thisis known, we can} derive the radial acceleration observed by the staticobserver by performingaboostwithrapidity

ζ

=

tanh−1

(

ur

₎

_.

Wethusneedtodeterminetheinstantaneousradialcomponent ofthefree fallvelocity oftheoutgoingparticle whenitgoes on-shell.Thiscan becomputedfromthe geodesicequation anditis givenby ur

=

dr d

τ

=



2MG r



1

−

r r0



,

(7)

where r0 comes asan integration constant corresponding to the

coordinate distance at which the particle velocity goes to zero. Sinceweareinterestedinthevalueoftheradialcomponentofthe

4 _{Forcomputationoftheaccelerationintherestframeoftheparticleweneed} theRiemanntensorintheinertialframeoftheparticle.Onecancomputethe Rie-manntensorinthestaticSchwarzschildcoordinatesand thenboostit usingthe free-fallvelocityoftheparticleasmeasuredinthestaticframe.Afeatureofthe SchwarzschildgeometryisthatthecomponentsoftheRiemanntensorremain in-variantundersuchaboost[16].Thus,in(5)wehaveRrttr= −2MG/r3.

geodesic tangentvectorattheinstant whentheoutgoingparticle goes on-shell and becomes an Hawking quantum which eventu-allyreachesinfinity,wecantaketheintegrationconstantr0

→ ∞

,

i.e. Hawkingquantacan becreatedwithzerovelocity onlyat in-finity.Hence,weget

ur



_r ∗

=



2MG

r_∗

.

(8)

We can now boost the acceleration vector aμ

= (

0

,

ar

,

0

,

0

)

, wherear _{givenby (6),withavelocity parametergivenby (8),in} order todetermine thetidalforce inthe staticframear_st.We get arst

=

arcosh

(ζ )

=

ar

(

1

−

2MG

/

r

)

−1 so that theradial component oftheforceunderthistransformationisgivenby

F_tidalr _−st



_r ∗

=

mar_st

(1

−

2MG/r

)





r_∗

=

m

λ

C

(1

−

2MG/r_∗

)

2 2MG r3 ∗

,

(9)

wherewehaverescaledthemassintherestframebythe appro-priate Lorentz factor,

(

1

−

2MG

/

r_∗

)

−1. Finally,usingthe factthat

λ

C

∼

1

/

m,themagnitudeoftheforceisgivenby

||

Fr_tidal−st

|| =

2MG r3 ∗



1

−

rs r_∗



−3 2

.

(10)

InanalogywiththeSchwingereffect,weshallnowassumethat theworkdonebythetidalforcetosplitthevirtualpaircanbe ap-proximatedbytheproduct oftheforcecomputedabovewiththe distance over which it appears to have acted,i.e. the separation ofthetwoHawkingquantaastheygoon-shellasmeasuredbya staticobserveratr_∗.Giventhatwehaveassumedthattheingoing Hawkingquantumgoesonshellassoonasitcandoso,i.e.at hori-zoncrossing, thisdistancewillcoincidewiththestaticobserver’s properdistancetothehorizond

(

r_∗

)

.

Therefore,theworkrequiredbythetidalforcetosplitthepair apartisgivenby5 Wtidal

∼ ||

Frtidal−st

||

d

(

r∗

)

=

2MG r3 ∗



1

−

rs r_∗



−3 2 d

(

r_∗

) ,

(11) whered

(

r_∗

)

isgivenby d

(

r_∗

)

=

r∗



rs

√

grrdr (12)

=

rs

⎛

⎝

α

(

α

−

1)

+

1 2log

⎡

⎣

α

1

+



1

−

1

α



2

⎤

⎦

⎞

⎠ ,

andwehavedefined

α

≡

r_∗

/

rs.

We can then equate thiswork to thetotal energyof thetwo Hawkingquantabeingcreated,namelyWtidal

=

2

ω

r.Thisgivesus

2MG r3 ∗



1

−

2MG r_∗



−3 2 d

(

r_∗

)

=

γ

2

π

rs



1

−

2MG r_∗



−1 2

.

(13)

Finally,fromeq.(13)weget

γ

=

2

π

α

2



1

−

1

α



−1 2 (14)

·

⎛

⎝

1

+

1 2

√

α

2

₋

_α

log

⎡

⎣

α

1

+



1

−

1

α



2

⎤

⎦

⎞

⎠ .

5 _{Alternatively, we could introduce a 4-vector}

μ= (0,r,0,0), with ||||=

gμνμν_{=d(r∗),andcomputetheworkasWtidal∼grrF}r

tidal−str_r

∗.Thiswould

Fig. 1. Thisplotshowsthevariationofγ withrespecttotheradialdistancefrom thecenteroftheblackhole.Thereddashedlinecorrespondstothehorizonlocation atα=1 wheretheexpression forthetidalforcework diverges,indicatingthat thequantainthefarUVtailoftheHawkingspectrumoriginatefromverynear thehorizon.(Forinterpretationofthereferencestocolorinthisfigurelegend,the readerisreferredtothewebversionofthisarticle.)

Therelationbetween

γ

and

α

,i.e. theradialdistancescaledas r_∗

/

rs,is better illustrated in Fig. 1.It is clear fromthe plot that thepartoftheHawkingthermalspectrumaroundthepeak

(

γ

∼

2

.

82

)

,wheremostoftheradiationisconcentrated,correspondsto aregionwhich extendsfaroutsidethehorizon, upto around2rs (atthepeakr_∗

≈

4

.

38MG).

The plotabove also shows how, inthis tidal force derivation, the quanta with higher velocity (kinetic energy) are produced closer to the horizon. This is consistent with our analysis since thehighertheinitialradial velocitythestrongertheLorentz con-tractionoftheoutgoingparticlesdistancefromthehorizonintheir restframe,givenby

λ

C,resultinginashorterproperdistanced

(

r∗

)

atwhichtheyaredetected.

Also, by using Eq.(12) andexpressing the rest ofEq. (11)in terms of

α

,we can see that the work doableat fixed

α

by the tidalforcesscalesastheinverseofthemassoftheblackholeso makingevidentthat smallerholescanproducehotterparticlesat thesamerelativedistancefromthehorizon.

IntheSchwingereffectargumentwedescribed inthisSection wehave considered thecaseofa massivetest particle. However, inthephysicalcontextofa4D Schwarzschildblackhole,mostof theradiationisemitted bymassless particles.Ageneralization of ourargumenttothemassless casecan beachievedina straight-forwardmanner.In fact,despite thelackof arestmass frameof oneofthetwopartners,onecanalways studytheSchwinger-like effectin a localinertial frame inthe vicinity of thehorizonand compute the radial acceleration (5) considering two radially in-fallingnullgeodesicswith4-velocityuμ

= [

1

,

1

,

0

,

0

]

insuchgiven frame; due to the symmetries of the Riemann tensor, this leads tothesameexpression (6)butwiththeCompton wavelength

λ

C replaced by the massless particle de Brogliewavelength

λ

B. The acceleration as measured by a static observer outside the black holethen proceeds along the samelines asin the massive case, since the boost between the two frames, locally both inertial, is the same as in the massive case; we can thus compute the ra-dialcomponentofthetidalforce asin(9),whereonther.h.s.we replacethecombinationm

λ

C withE

λ

B,E beingthemassless par-ticleenergy,asmeasuredbythestaticobserver,whichisrelatedto thedeBrogliewavelengthby thestandard relation E

=

1

/λ

B (re-callthat we switched tounits whereh

_¯

=

c

=

1). Inthis way,we recovertheexpression(10),whichisindependentofthetest par-ticlemass.Therefore,theplotinFig. 1applies alsotothecaseof radiationbeingemittedbymasslessparticles.

Letus stress againthe heuristic nature of our argument. We areconsideringtheinstantaneousvalueofthetidalforceobserved by the outgoingpartner at a givencoordinate distancer_∗ where itgoeson-shell.However,wethenusethisinstantaneousvalueto

compute thework doneby thegravitationalfield overa distance d

(

r_∗

)

,asiftheforcewas actuallyatworkwiththesameconstant value throughoutthe whole splittingprocess. A similarapproach wasalsousedin[17]togiveanestimateofthewavelengthofthe Hawkingquantaasproducedbythegravitationaltidalforce.

So, although the analogy with the Schwinger effect for the electron–positron pairproductionby an electricfield maybe ad-vocatedtolendsupporttoourdescriptionofHawkingquanta pro-ductionfromaquantumatmospherethatextendswellbeyondthe horizon,wenowwanttopresentamoresoundanalysisbasedon therenormalizedstressenergytensorinordertoconfirmthis pic-ture.

3. Stress-energytensor

By analyzing the renormalized stress energy tensor (RSET) in the2-dimensionalcase,one canunderstandHawkingradiationin abetterwayasthisisalocalobjectwhichcanhelptoprobethe physicsinthevicinityoftheblackhole.ThederivationoftheRSET componentshasbeenconsidered inmanyplaces intheliterature

[18–22],herewebuildonthesepreviousresultsandcomputethe energy density and flux as seen by an observer which has zero radialvelocity(thusgivingrisetonokinematicaleffects)andzero accelerationatthehorizon.

3.1. ComputationofRSET

Following[23],let usintroducea setofglobally definedaffine coordinates U , V on

I

_left−,

I

_right− respectively. Restricting to the radialandtimedimensions,themetricreads

ds2

=

C

(

U

,

V

)

dU dV

.

(15)

In

(

1

+

1

)

dimensionstherenormalized stressenergytensorforany massless scalarfield intermsoftheseaffinenullcoordinatescan be easily computed using the conformal anomaly [18–20,24,25]. ThecomponentsoftheRSETcomputedinsomearbitraryvacuum statearegivenas:



TU U

 = −

1 12

π

C 1/2

_∂

2 UC−1/2

=

1 24

π



C,U U C

−

3 2

(

C,U

)

2 C2



,

(16)



TV V

 = −

1 12

π

C 1/2

_∂

2 VC−1/2

=

1 24

π



C,V V C

−

3 2

(

C,V

)

2 C2



,

(17)



TU V

 =

RC 96

π

=

1 24

π

∂

U

∂

Vln C

,

(18)

where C is the conformal factor introduced in the above metric andR isthescalarcurvature.

Now let usalsointroducea nullcoordinateu affine on

I

_right+ suchthat

U

=

p

(

u

)

;

(19)

fromthisweget

∂

U

= ˙

p−1

∂

u

.

(20)

Intermsoftheset

(

u

,

V

)

,themetricreads

ds2

= ¯

C

(

u

,

V

)

dudV

,

(21)

¯

C

(

u

,

V

)

= ˙

p

(

u

)

C

(

U

,

V

) .

(22)

Assuming that the observer is always outside the collapsing star,C

¯

(

u

,

V

)

wouldbethemetriccomponentofastaticspacetime. Intermsofthisnewlydefinednullcoordinate,asimple computa-tionshowsthat TU U isgivenas



TU U

 = −

˙

p−2 12

π



¯

C1/2

∂

_u2C

¯

−1/2

− ˙

p1/2

∂

_u2p

˙

−1/2



.

(23)

Now TV V will have only a staticcontribution if V

=

v but if theaffinenullcoordinateon

I

_left+ isdefinedas

V

=

q

(

v

)

(24)

andwedefineC

(

U

,

v

)

= ˙

q

(

v

)

C

(

U

,

V

)

,TV V isgivenas



TV V

 = −

˙

q−2

12

π



C1/2

∂

2_vC −1/2

− ˙

q1/2

∂

_v2q

˙

−1/2



.

(25)

As mentioned earlier C

¯

(

u

,

V

)

is the metric component of a static spacetime,so all the dynamics of the collapsing geometry iscapturedinthe p term

˙

of(23).In theaboveanalysis, by using another affine null coordinate, we can differentiate between the staticcontributionto theRSET andtheone dueto thedynamics associatedwiththecollapse[23].

3.2. RSETfordifferentvacuumstates

Capturing the dependenceatdifferent radii ofthe RSET com-ponents wouldrequireaknowledge ofthefull p

(

u

)

atanyvalue of u, i.e.to specify a collapse history. However, thiswould lead to the inclusion of transient effects which are not relevant for thepresentdiscussion. Forthisreason,we shallhererely onthe fact that, well afterthe collapse has settle down, the blackhole geometryisformallyindistinguishablefromthatofaneternal con-figuration[26,27] (wherethe formof p

(

u

)

issimplyfixed bythe geometry,see(A.2)).

So,inordertoextract physicalinformationfromthe RSET,we shallcompute theenergydensityandtheflux experiencedby an observerat constant Kruskal positionlong after the collapse has takenplacein thetwo physicallyrelevant statesforHawking ra-diationinthe eternalblackholecase, namelytheUnruhandthe Hartle–Hawkingstates.Weshallstart inthisSection by explicitly evaluating the general expressions for the RSET components ex-pectationvalues.

Using(A.2),wegettherelations

˙

p

(

u

)

≡ ∂

up

(

u

)

= −

p

(

u

)

2rs

,

(26)

¨

p

(

u

)

=

p

(

u

)

4r2s

= −

p

˙

(

u

)

2rs

.

(27)

Forcomputingthefirsttermof(23)wecanwrite

¯

C1/2

∂

_u2C

¯

−1/2

=

3 4C

¯

−2



_∂

uC

¯



2

−

1 2C

¯

−1

_∂

2 uC

¯

.

(28)

UsingthemetricconformalfactorC from(A.1)weget

∂

uC

¯

= ∂

u

[ ˙

p

(

u

)

C

] = ¨

pC

+ ˙

p

∂

uC

= ˙

p

(

u

)



−

1 2rs

+

r2

−

r2_s 2r2_r s



C

= −

rs 2r2C

¯

,

(29) and

∂

u2C

¯

= −

1 2rs

∂

u

 ¯

C r2



=

r2s 4r4C

¯

−

1 2 rsf

(

r

) ¯

C r3

.

(30)

Usingtheaboverelationin(28)wehave

¯

C1/2

∂

u2C

¯

−1/2

=

3 4C

¯

−2



r2 s 4r4C

¯

2



−

1 2C

¯

−1



r2 s 4r4C

¯

−

1 2 rsf

(

r

) ¯

C r3



= −

3 16 r2_s r4

+

rs 4r3

−

3 4 M2G2 r4

+

MG 2r3

,

(31)

where f

(

r

)

isgivenin(A.8)andweusedrs

=

2MG inthelaststep. Forthesecondtermonther.h.s.of(23),wehave

˙

p1/2

∂

_u2p

˙

−1/2

= −

p

˙

1/2 2

∂

u



_¨

p

˙

p3/2



=

1

(8MG)

2

.

(32)

We arenow readytocompute explicitlythe expectationvalueof the differentRSETcomponents fortheHartle–Hawking (

|

H



) and Unruh(

|

U



)states.

We can start by observingthat forthe TU U andTU V compo-nents, the expectation values are the same in the two vacuum states[20].Therefore,inthefollowingwesimplydenote



TU U

 ≡ 

H

|

TU U

|

H

 = 

U

|

TU U

|

U

 ,

(33)



TU V

 ≡ 

H

|

TU V

|

H

 = 

U

|

TU V

|

U

 .

(34) Bymeansof(31),(32),



TU U



isgivenby



TU U

 =

˙

p−2 24

π



3 2 M2_G2 r4

−

MG r3

+

1 32M2_G2



= (

768

π

M2G2

)

−1V 2 4r2e −r/MG

·



1

+

4MG r

+

12M2G2 r2



.

(35)

Tocompute



TU V



weuse(18),fromwhich



TU V

 =

1 24

π

∂

U

∂

Vln C

=

1 24

π

(

p

˙

q

˙

)

−1

_∂

u

∂

vln C

= −

1 96

π

(

p

˙

q

˙

)

−1_C

_∂

2 rC

.

(36)

UsingC

(

t

,

r

)

from(A.1)andtheexactvaluesofq

(

u

)

andp

(

v

)

,we get



TU V

 = −

M2G2

12

π

r4e−

r/2MG

_.

₍₃₇₎

On the other hand,the dependence of



TV V



on the state in which we are computing the expectationvalue is important.For theHartle–Hawkingstate(eternalblackholescenario,non-singular vacuumstateinbothpastandfuturehorizons)inKruskal coordi-nates themodesaregivenby e−iωU_,e−iωV_{,wherewe definedV} as

V

≡

q

(

v

)

=

2rsev/2rs

.

(38)

Using thisdefinitionofV we canproceed inasimilar wayas forthecomputationof



TU U



.From(25),weobtain



H

|

TV V

|

H

 =

˙

q−2 24

π



3 2 MG2 r4

−

MG r3

+

1 32MG2



= (

768

π

M2G2

)

−1U 2 4r2e− r MG

·



1

+

4MG r

+

12M2_G2 r2



.

(39)

Fig. 2. Plotoftheenergydensityatagiventimeasafunctionofthe radialdistance fromthecenter oftheblackholeinUnruhstateatagiveninstantoftime.

FortheUnruhstateinKruskalcoordinates,themodesaregiven bye−iωU,e−iωv andthere isnoregularizationcondition imposed inthepasthorizon.The expectationvalueofthe TV V component canbeobtainedfromtherelation



U

|

TV V

|

U

 =

16MG2q

˙

−2



U

|

Tv v

|

U

 ,

(40) where



U

|

Tv v

|

U



canbecomputedfrom



U

|

Tv v

|

U

 = −

1 12

π

f

(

r

)

1/2

_∂

2

vf

(

r

)

−1/2 (41) using f

(

r

)

=



1

−

2MG_r



,asfollowsfromthemetricofablackhole instaticSchwarzschildcoordinates.Wehave



U

|

Tv v

|

U

 =

1 24

π



3M2_G2 2r4

−

MG r3



,

(42)

andfrom(40)weget



U

|

TV V

|

U

 =

1 6

π

M2G2 V2



3M2G2 2r4

−

MG r3



.

(43) 3.3.Energydensity

Wenowhavealltheingredientstoextractphysicalinformation fromtheRSET.Letusfirstanalyzetheenergydensityasmeasured intheframeofanobservermovingalongfixedpositioninKruskal coordinates.

Letus consider an observer at a given Kruskalposition with 2-velocity vμ

=

C−1/2

₍

₁

_,

₀

₎

_(in

_[

_T

_,

_X

_]

_{coordinates).}6 _{The energy}

density,

ρ

,measuredbythisobserverfortheUnruhstateisgiven by

ρ

= 

U

|

Tμν

|

U



vμvν

=

C−1



U

|

TT T

|

U



=

C−1



U

|

TV V

+

TU U

+

2TU V

|

U

 .

(44) Using(35),(37),(43)we cancompute theenergydensityexactly andweplotitinFig. 2(where

α

≡

r

/

rs).

Theenergydensity(44)blowsupatthehorizon

(

r

=

2M

)

since wearecomputingtheenergydensityasobservedbyafreefalling (inKruskalcoordinates)observerintheUnruhstatewhichiswell knowntobeilldefinedonthepasthorizon.Suchdivergencearises

6 _{Thischoiceoftrajectoryisnotgeodesic;howevertheaccelerationthatthe} ob-serverexperiencesisirrelevantcomparedtotheHawkingtemperatureandonecan showeasilythattheaccelerationvanishesatthehorizon.Onemightthinkthata freefallingobserverwouldhavebeenabetterchoice.However,theproblemwith suchchoicewouldbethenon-zeroradialvelocityofthefreefallingobserveratthe horizon,aswellasnearthehorizon.Inthatcase,itwouldthenbedifficultto sep-arateouttheHawkingradiationcontributionfromotherkinematicaleffects[28].

Fig. 3. NearhorizonbehavioroftheenergydensityintheUnruhstateatdifferent times.ThefirstplotcorrespondstothesameinstantoftimeastheplotinFig. 2; thesecondonetoacloseinstantafter.

fromthe1

/

V2_{terminthecomponent(43)whenV}

₌

_0,i.e.atthe

past horizon.The horizonlocation conditionin Schwarzschild ra-dialcoordinate,

α

=

1,cannotdistinguishbetweenpastandfuture horizons andthus the divergent contribution wouldenter in the plotabove ofthe energydensityexpression (44)whenevaluated at

α

=

1. However, a free falling observer at the future horizon wouldnotseethisdivergence,whichisjustanartifactofKruskal coordinates.7Thisisawellknownfactalreadypointedin[19].For thisreason, wehaveremoved the point

α

=

1 in theplotshown inFig. 2.

Near the horizonthe energy densitybecomes negative; these negative values are attained closerto the horizonas the energy densityismeasuredatlatertimes.Weshowthisnearhorizon be-havior in the two plots in Fig. 3, where the first is evaluated at thesametimeastheplotinFig. 2andthesecond oneataclose instantafter(asimilarbehaviorwas foundalsoin[29]);the neg-ative divergent behavior of the energy density at the horizon is clearfromtheplots.

However, letusremarkagainthatthisdivergenceis just ficti-tiousforanobservercrossing thefuturehorizonU

=

0 atagiven valueofV

>

0 anditisaninevitablefeatureofplottingtheenergy densityintheUnruhstateasafunctionofr forafixedinstantof timet.

Onewaytoavoidthismisleading behavioroftheenergy den-sity plot at the horizon could be to show it as a function of U forgiven V

=

const

>

0; this wouldindeedremove the singular-ityfromtheplotsince thepoint

α

=

1 wouldnowcorrespond to U

=

0,i.e.tothefuturehorizonwheretheUnruhstateisregular. However, fromsuch plotit wouldbe verydifficult toextrapolate theinformationabouthowtheenergydensityisdistributedinthe

7 _{Letusstressthatalsothecalculationin[23]oftheRSETcomponentsinthe} col-lapsescenarioshowsthatatthewhiteholehorizontheUnruhstatewillnecessarily besingular.Thiscanbeeasilyrealized byapplyingtimereversaltothe subdomi-nanttermsinthedynamicalcontribution(32)derivedin[23](seeEq.(52)there), whichthenshowsanexponentiallygrowingfluxatthewhitehorizonwhichvery rapidlywouldcreateadivergenceintheTU U componentoftheRSETsoonafter

Fig. 4. PlotofthevariationofenergydensitycomputedinHartle–Hawkingstate withrespecttotheradialdistancefromthecenter oftheblackholeatfixedtime measuredinthestaticframe.Noticethatclosetothehorizontheenergydensity isnegativealsointhiscase,butitremainsfiniteatthehorizonduetothe non-divergentbehavioroftheTV Vcomponent(39)intheHartle–Hawkingvacuum.

r coordinateforfixed timet,sincefixing V andlettingU run im-ply that differentvaluesof U correspondto differentvalues ofr and t.

ThesignificantaspectoftheplotinFig. 2forusisthepeakin thedistributionof

ρ

thatisobtainedoutsidethehorizonwhichis atr

≈

4

.

32MG. Quite inagreement withour heuristic prediction based on the gravitationalanalogue of the Schwinger effect. Let us point out that, although we have shown the plot at a given instantofKillingtime,thebehavioroftheenergydensityremains thesameatanytime,inparticularthepresenceofthepeakatthe samelocation persists;theonlydifferenceisthatthevalueofthe energy densityincreases since it accumulates, giventhat we are nottakingintoaccounttheeffectofback-reaction.

Togetanon-singularenergydensityplotforthefreefalling ob-serverweshouldconsidertheHartle–Hawkingstate.Thisisgiven by

ρ

= H|T

μν

|Hv

μvν

=

C−1

H|T

T T

|U

=

C−1



H

|

TV V

+

TU U

+

2TU V

|

H

 .

(45) Using the expectation values given in (35), (37), (39), we can plot the energy density (45) with respect to radial distance parametrizedby

α

.Thisis showninFig. 4,wherewe seea sim-ilar nature of the distribution with a peak outside the horizon; however,asexpected,inthiscasetheenergydensityisregular ev-erywhere.Remarkably,thepeakislocatedatr

≈

4

.

37MG,inclose agreementwithourheuristicfindings.

Theseresultsstrongly supportour previous claimthat the ra-diationdensityismaximizedinaregionoutsidethe horizon.We nowshowthatasimilarbehaviorwithapeakawayfromthe hori-zonisexhibitedalsobythefluxpartoftheRSET.

3.4. Flux

ThefluxoftheHawkingradiationintheUnruhvacuumisgiven by[30]8

F

= −

U

|

Tμν

|

U



vμzν

,

(46)

where vμ is the velocity of the observer and zν is the con-travariant componentof the normalto the observer. Letus con-sider a static observerat fixed distance in a Kruskalframe with vμ

=

C−1/2

_[

₁

_,

₀

_]

_{andindicatethenormalvectoraszν}

_{= [}

_A

_,

_B

_]

_.The

latterhastosatisfythefollowingconditions

8 _{IntheHartle–Hawkingvacuumthefluxvanishesduetothethermalequilibrium} ofthestate.

Fig. 5. ThisplotshowsthevariationofthefluxofHawkingradiationwithrespectto theradialdistanceasmeasuredbyanobserverintheUnruhstateatagiveninstant oftime.

gμνzμzν

= −

1,zμvμ

=

0

.

(47)

Usingthesecondrelationweget A

=

0 andfromthefirstrelation weget B

=

C−1/2_{.Therefore,zν}

₌

_C−1/2

_[

₀

_,

₁

_]

_.

Usingtheseexpressionsforvμ,zν ,weget

F

= −

C−1



U

|

TT X

|

U

 =

C−1



U

|[−

TV V

+

TU U

]|

U

 .

(48) Plugging inthe expectationvalues(35), (43)found above, we canplotthefluxasafunctionof

α

.ThisisshowninFig. 5.Alsoin thiscasetheplotofthefluxwouldreceiveafictitious(forafree fallingobserveratthefuturehorizon)divergentcontributionfrom the component(43),andwe havethus removed thepoint

α

=

1 from the plot, thus avoiding the divergence at the past horizon V

=

0.We seethat the fluxhas amaximum atr

=

4

.

32MG and mostof thecontribution tothe Hawkingradiation comesfrom a regionbetweenthehorizonandr

≈

6MG.

Letusremarkthat ourfindingsareinlinewiththeanalysisof the2-dimensionalRSETdonein[10]whereitwasshownthatthe ingoing andoutgoingnullcomponentsofthestress tensorwould builduptotheirasymptoticvaluesinaregionoutsidethehorizon. Inouranalysiswehavebeenmorepreciseinconfirmingthisresult bychoosinganobserverandexplicitlycomputingthevaluesofthe energy densityand theflux outside the horizonasmeasured by theobserver.

4. Summaryanddiscussion

It hasbeen widely believedthat Hawkingradiation originates fromtheexcitationscloseto thehorizonandthiseventually sug-gestedsomedrasticmodificationofthestatesinthenearhorizon regime asa resolutiontothe informationlossparadox [5,31–33]. Oneoftheprimaryreasonsforsuchan argumentisbasedonthe wayHawkingdidhis originalcalculation, tracingback themodes all the way fromfuture infinity to thepast null infinity through thecollapsingmattersothat onehasavacuumstate atthe hori-zonforafree-fallingobservers.

The otherdisturbingfeatureaboutthisargumentis,whenthe modes are tracedback they become highly blueshifted near the horizonandwe arenotwellawareofthelawsofphysicsinsuch hightransplanckian domain.Someresolutions tothe above prob-lem have been proposed several times in the literature [34–36]

buttheyalldemandsomechallengingmodificationtoourpresent knowledgeofgravitationorquantumfieldtheory.

Let us stress, however, that the UV departures from Lorentz invariance through theintroduction ofa fundamental cutoff pos-tulated in [37,38] arerelevant only veryclose to thehorizonfor large black holes (in units ofthe Lorentz breaking scale). Hence, evencontemplatingsuchscenario,ouranalysisinsection 3would

bebasically unchangedandunaffected awayfromthehorizon,as alsostressedinthesimilaranalysiscarriedoutin[39].

InthispaperwehaveshownevidencethattheHawkingquanta originatefromaregionwhichisfaroutsidethehorizon,whichcan becalledablackholeatmosphere.Moreprecisely,fromtheplotsof theenergydensityandthefluxintheUnruhstatewegeta maxi-mumatr

≈

4

.

32MG,fortheenergydensityintheHartle–Hawking state the peak is at r

≈

4

.

37MG. This is strikingly close to our previous finding foran origin atabout r

≈

4

.

38MG forthe peak of the thermal spectrum using the heuristic argument based on tidalforces.Bylargethisisalsoinagreementwithsome previous claimsusingvariousothermethods,suchascalculatingthe effec-tiveradiusofaradiatingbodyusingtheStefan–Boltzmannlawor computing the effective Tolman temperature [10,40–42], as well asin close correspondence with the results of the study of the nullcomponentofthe stress-energytensorintheUnruh vacuum of[43].

Giventhepresenceofaquantumatmospherewherethe Hawk-ingquantaaregeneratedandwhichextendswellbeyondtheblack hole horizon, as originally suggested in [10], it would be inter-esting toinvestigate how its effectiveradius is affected by going to higher dimensions. Applying the Stefan–Boltzmann radiation lawargument proposed in [10] for the

(

3

+

1

)

-dimensional case to

(

D

+

1

)

-dimensional Schwarzschild black holes, it was found in[40] that the effective radiusgets squeezed towards the black holehorizonasthenumberofspatialdimensionsincreases.

GiventhatthereisnoderivationoftheRSETcomponentsin di-mensionshigherthan

(

1

+

1

)

,wecannotapplytheargument pre-sentedinSection 3to confirm thisresult. However,the heuristic derivationthat wepresentedinSection2couldbeeasily general-izedforanyarbitrarynumber ofdimensions. Withoutpresenting a complete derivation, we can understand in a qualitative way how the quantum atmosphere can be effected by going to ar-bitrary

(

D

+

1

)

higher dimensions by considering the fact that theHawkingtemperaturescales asTB H

=

(₈D_π−_MG2)h¯,where D isthe numberof spatial dimensions. It can be shown that this dimen-sionalscaling ofthe temperature, along withthe modificationof theSchwarzschildmetricforanarbitraryD,wouldyield,forgiven r

=

r_∗,ahighervalue of

ω

r (1)as D increases.Atthesametime, itcanbeshownfromdimensionalargumentsthattheworkdone bythetidalforce mustdecreaseinvalue forthesamegivenr as D increases. Thisimpliesthat, fora fixed D

>

3, thepeak ofthe Hawkingradiationspectrumcorrespondstoanhighervalueof en-ergythanintheD

=

3 caseand,inorderforthegravitationalfield tobe able to provideenough work to reach such amountof en-ergy, the outgoingpartnerscomprising the bulk ofthe spectrum atinfinitymust goon-shellcloser tothehorizon. OurSchwinger effectargumentthusconfirmsinaqualitativewaytherelation ob-tainedin[40]forthedecreaseoftheeffectiveradiusintheregime D

1.

Ifthe radiationhas a longdistance originthen we might not needtoworryaboutthetransplanckianissueatthehorizon. More-over,concerning the fundamental issueof unitarityof blackhole evaporation, this result suggests to consider some effect opera-tionalatthisnewscaleinordertoeventually restoreunitarityof Hawkingradiation. A possible scenario is theone ofnon-violent nonlocalityadvocated in[44,45];seealsotheproposalof[46,47]. We hope that the presentcontribution will stimulatefurther in-vestigationsinthesedirections.

Acknowledgments

WethankRenaudParentani,SebastianoSonegoandMattVisser forilluminatingdiscussions. We alsoacknowledge theJohn Tem-pletonFoundationforthesupportinggrant#51876.

Appendix A. Kruskalframe

WewanttoexaminethecomponentsoftheRSETinaglobally well defined coordinatesystem free ofany pathologicalbehavior (other than a true curvature singularity, like in the center of a black hole). For this purposethe Kruskal coordinate frame is an appropriatechoice.TheKruskalmetricisgivenas

ds2

=

rs re

−r/rs_{dU dV}

_,

_(A.1)

where rs is the radius of the eventhorizon. For this coordinate systemwehave

U

=

p

(

u

)

= −

2rse−u/2rs

,

(A.2)

V

=

q

(

v

)

=

2rsev/2rs

.

(A.3)

Theaffinenullcoordinateu, v intermsofradialdistancefrom thecenter ofthe blackhole,“r”,andtime,“t”,asmeasured bya staticobserverisgivenas

u

=

t

−

r_∗

=

t

−



r

+

rsln



r rs

−

1



,

(A.4) v

=

t

+

r_∗

=

t

+



r

+

rsln



r rs

−

1



,

(A.5) also

∂

u

=

∂

r_∗

∂

u

∂

r∗

= −

1 2

∂

r∗

= −

1 2f

(

r

)∂

r

,

(A.6)

∂

v

=

∂

r_∗

∂

v

∂

r∗

=

1 2

∂

r∗

=

1 2f

(

r

)∂

r

,

(A.7) whereweused dr_∗ dr

= [

f

(

r

)

]

−1

₌



1

−

rs r



−1

.

(A.8)

Wecanalsodefineasetoftimelikeandradialcoordinates

(

T

,

X

)

as

T

=

1

2

(

V

+

U

),

X

=

1

2

(

V

−

U

).

(A.9)

Usingthismetric(A.1)isgivenas

ds2

=

rs re

−r/rs

₍

_dT2

−

_{d X}2

_{) .}

_(A.10)

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