The black hole quantum atmosphere

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by  Ramit Dey

The black hole quantum atmosphere
Ramit Dey,∗ Stefano Liberati,† and Daniele Pranzetti‡
SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di Trieste
Ever since the discovery of black hole evaporation, the region of origin of the radiated quanta
hasbeenatopicofdebate. RecentlyitwasarguedbyGiddingsthattheHawkingquantaoriginate
from a region well outside the black hole horizon by calculating the effective radius of a radiating
body via the Stefan–Boltzmann law. In this paper we try to further explore this issue and end
up corroborating this claim, using both a heuristic argument and a detailed study of the stress
energy tensor. We show that the Hawking quanta originate from what might be called a quantum
atmosphere around the black hole with energy density and fluxes of particles peaked at about 4M,
running contrary to the popular belief that these originate from the ultra high energy excitations
very close to the horizon. This long distance origin of Hawking radiation could have a profound
impact on our understanding of the information and transplanckian problems.
7
1 I. INTRODUCTION the underlying geometry 1. While it was debated for a
0
while if Hawking quanta could originate initially, during
2
thestarcollapse,andlaterreleasedoveraverylongtime,
The discovery of Hawking radiation [1] changed our
n it was convincingly argued in [8] that this cannot be the
perspective towards black holes, giving us a deeper in-
a case if an event horizon indeed forms. This leads to the
J sight about the microscopic nature of gravity. At the
same time, within the semi-classical framework, the cur- conclusion that the Hawking quanta are generated in a
2
region outside the horizon. A conclusion corroborated
2 rent understanding of such process still leaves open sev-
by studies of the Hawking modes correlation structure
eral issues. Of course, a well known unresolved problem
where it was shown that mode conversion happens over
] of black hole physics is the information loss paradox [2–
c alongdistancefromthehorizon[9]. Amorerecentclaim
4], i.e. the apparent incompatibility between the com-
q
in this direction, based on calculating the size of the ra-
- plete thermal evaporation of a black hole endowed with
r an event horizon and unitary evolution as prescribed by diatingbodyviatheStefan–Boltzmannlaw,showedthat
g
theHawkingquantaoriginateinanearhorizonquantum
[ quantum mechanics.
For restoring unitarity of Hawking radiation and ad- region,asortofblackhole“atmosphere”[10]. Itisawell
1 known fact that the typical wavelength of the radiated
dressing the information loss problem correctly, it is im-
v quantaiscomparabletothesizeoftheblackhole,soone
portant (among other things) to know from where the
1 mightthinkthatthepointparticledescriptionisnotvery
6 Hawking quanta originate. For example, if one assumes
accurate. However, as measured by a local observer near
1 a near horizon origin of the Hawking radiation, then one
the horizon, the wavelength is highly blue-shifted when
6 way to restore unitarity is by conjecturing some sort of
0 UV-dependent entanglement between partner Hawking traced back from infinity to the horizon, thus validating
. the point particle description.
1 quantawhichwouldenablethelatetimeHawkingfluxto
0 retrivetheinformationintheearlystagesoftheevapora- TheHawkingprocesscanbeexplainedheuristicallyas-
7 tionprocess. Suchscenarioseemstoleadtothesocalled well, for example via a tunnelling mechanism where the
1 “firewall” argument as the conjectured lack of maximal particle tunnels out of the horizon or the anti particle
v: entanglementbetweentheHawkingpairsmakesthenear (propagatingbackwardsintime)tunnelsintothehorizon
i horizonstatesingularandeventuallydemandssomedras- and as a result of this we get the constant Hawking flux
X ticmodificationofthenearhorizongeometry[5]. Onthe at infinity [11]. Alternatively, one popular picture is to
r other hand, if one believes in a longer distance origin of imagine that the strong tidal force near the black hole
a
the Hawking quanta, some effect must be operational at horizon stops the annihilation of the particle and anti-
alargerscaleforrestoringunitarityratherthannearthe particle pairs that are formed spontaneously from the
horizon, avoiding the “firewall”. vacuum. Once the antiparticle is “hidden” within the
black hole horizon, having a negative energy effectively,
A similar open issue is the transplanckian origin of
the other particle can materialise and escape to infinity
Hawking quanta. Hawking’s original calculation indi-
[12, 13].
cates that the quanta originate near the black hole hori-
zoninahighlyblue-shiftedstaterequiringanassumption Inthispaperweshallexplicitlymakeuseofthislatter
on the UV completion of the effective field theory used heuristic picture as well as of a full calculation of the
for the computation and on the lack of back-reaction on stress energy tensor in 1+1 dimensions. We shall see
that both methods seem to agree in suggesting that the
∗[email protected]
†[email protected] 1 See,forinstance,[6,7]forablackholeevaporationanalysiswhere
‡[email protected] theseissuescanbeaddressedinaquantumgravitycontext.
2
Hawkingquantaoriginatefromtheblackholeatmosphere allowsfornegativeenergystatesgiventhatinitthenorm
andnotfromaregionveryclosetothehorizon. Insection of the timelike Killing vector, with respect to which we
II, based on the heuristic picture of Hawking radiation compute energy, changes sign.
described above and invoking the uncertainty principle Indeed,ifaSchwinger-likeprocesstakesplacenearthe
and tidal forces, we show that most of the contribution black hole horizon, due to the tidal force of the black
to the radiation spectrum comes from a region far away hole and the peeling of geodesics, the pair can get spa-
from the horizon. In section III we further strengthen tiallyseparatedandonepartnercanentertheblackhole
our claim by a detailed calculation of the renormalized horizon following a timelike or null curve with negative
stress energy tensor, which indicates a similar result. energywhiletheotherparticlecanescapetoinfinityand
contribute to the Hawking flux. In this picture, we are
implicitly assuming that virtual particles in the vicinity
II. A GRAVITATIONAL SCHWINGER EFFECT of a black hole horizon move along geodesics when they
ARGUMENT are just about to go on-shell.
Therefore, the physical scenario we want to envisage
One ingredient of our heuristic argument to identify is that of a particle-antiparticle pair pulled apart by the
a quantum atmosphere outside the black hole horizon, black hole tidal force outside the horizon until they go
where particle creation takes place, is the uncertainty on-shell as one of them reaches the horizon 3 located at
principle. However, the use of the uncertainty principle rs = 2GM/c2 (actually an infinitesimal distance inside
alone, as originally suggested by Parker [14], does not it so that the geodesic motion will drag it further in-
containanyphysicallyrelevantinformationaboutthelo- side) while the other particle is at a radial coordinate
cationofparticleproductionandwhysmallerblackholes distance r = r∗. Once on-shell, the outgoing particle
shouldbehotter. Indeed,theuncertaintyprincipleinthis eventually reaches infinity and contributes to the Hawk-
case provides a rough estimate of the region of particle ing spectrum. In order to do so though, it has to be cre-
productionasinverselyproportionaltotheenergyofthe ated with an energy corresponding to the energy of the
Hawkingquantawhentheyareproduced,butitdoesnot Hawking quanta at a distance r∗ > rs from the center
take into account any dynamical mechanism to estimate of the black hole as measured by a local static observer;
the probability of spontaneous emission. this can be reconstructed by noticing that
Thus one can improve this argument by invoking a
ω
physical process of creation of the Hawking quanta and ω = √∞ , (1)
r g
using the uncertainty principle as a complementary tool 00
to estimate the region of origin of the quanta. In this
where ω is the energy at infinity and we are using the
section, we try to achieve this goal by relying on tidal ∞
(+,−,−,−)signature. Atinfinity,thethermalspectrum
forces.
of Hawking radiation gives
Let us then consider a situation where a virtual pair,
consisting of a particle and anti-particle, pops out of the
k T
vacuum spontaneously for a very short time interval and ω =γ B H , (2)
∞ (cid:126)
then annihilates itself. In the Schwinger effect [15] a
staticelectricfieldisassumedtoactonavirtualelectron-
where the Hawking temperature for a black hole of mass
positron pair until the two partners are torn apart once M reads k T = (cid:126)c3 . Thus, we get
thethresholdenergynecessarytobecomearealelectron- B H 8πGM
positronpairisprovidedbythefield. Energyisconserved
c3
due to the fact that the electric potential energy has op- ω =γ (3)
∞ 8πGM
posite sign for partners with opposite charge. However,
in its gravitational counterpart a priori only vacuum po-
and
larisation can be induced by a static field in the absence
of an horizon. c 1
ω =γ , (4)
In fact, only in the presence of the latter one has both r 4πr (cid:113) r
s 1− s
the characteristic peeling structure of geodesics (diverg- r
ing away from the horizon on both its sides) as well as
the presence of an ergoregion behind it. 2 The presence
of an ergoregion is crucial for energy conservation as it
3 Onecouldalsoconsiderthecasewheretheingoingparticletun-
nels through the horizon and goes on-shell well inside the hori-
zon(ase.g.suggestedbytheresultsof[9]);however,sinceinour
analysisbelowweareinterestedinthetidalforceascomputedin
2 Thisisstrictlytrueonlyfornon-rotatingblackholes,forrotating the outgoing particle rest frame, this should not affect the final
ones the ergoregion lies outside of the horizon allowing for the expressionfortheforce. Thus,fromthepointofviewofanout-
classical phenomenon of superradiance. However, the quantum sidestaticobserver, theworkdonebythegravitationalfieldon
emissionstillrequiresthepeculiarpeelingstructureofgeodesics the pair (in our heuristic derivation) is insensitive to the exact
typicalofthehorizon. locationwheretheingoingparticlebecomesreal.
3
where γ is a numerical factor spanning the energy range
of the quanta giving rise to the radiation thermal spec-
2M
trum. At the peak of the spectrum γ ≈2.82. ar| = λ (6)
r∗ r3 C
This energy is provided by the work done by the grav- ∗
itational field to pull the two partners apart. We can Our aim is to determine the work done on the spon-
compute this work in the static frame outside a black taneously created particle pair by the tidal force in the
hole and compare it with ω(r∗). Using this relation, we static frame outside the black hole. For this we need to
candeterminetheregionfromwhichtheHawkingquanta compute the tidal force as measured by a static observer
originate. Thisistheprocesswenowwanttoimplement. outside the black hole at the instant when the outgoing
Let us clarify that, in a general relativistic frame- partner goes on shell. This can be achieved by consider-
work, the geodesic deviation equation does not describe ing the particle rest frame and the static observer frame
the force acting on a particle moving along a geodesic. as locally two inertial frames: The latter sees the parti-
Rather, it expresses how the spacetime curvature in- cle as moving with outward velocity given by the radial
fluences two nearby geodesics, making them either di- component of the geodesic tangent vector ur = dr/dτ.
vergeorconverge,i.e.iteffectivelymeasurestidaleffects. Oncethisisknown, wecanderivetheradialacceleration
Therefore,wecaninterprettheseeffectsasthepullofthe observed by the static observer by performing a boost
gravitational force on particles and talk about the work with rapidity ζ =tanh−1(ur).
done by the gravitational fieldonly in an heuristic sense. We thus need to determine the instantaneous radial
Nevertheless, in the case considered here where the test componentofthefreefallvelocityoftheoutgoingparticle
particles have a mass much smaller than the black hole when it goes on-shell. This can be computed from the
and we can neglect back-reaction effects, we expect this geodesic equation and it is given by
interpretation ofthe gravitationalfield effects tocapture
(cid:115)
some relevant aspects of black hole physics. With these dr 2M(cid:18) r (cid:19)
ur = = 1− , (7)
assumptions spelled out, let us proceed.
dτ r r
0
In the rest frame of the outgoing particle, one would
see the antiparticle accelerating towards the horizon due wherer comesasanintegrationconstantcorresponding
0
to the tidal force. This radial acceleration in the rest to the coordinate distance at which the particle veloc-
frameoftheparticlecanbecomputedusingthegeodesic ity goes to zero. Since we are interested in the value of
deviation equation, namely the radial component of the geodesic tangent vector at
the instant when the outgoing particle goes on-shell and
becomes an Hawking quantum which eventually reaches
ar|r∗ ≡ DDnτ2r(cid:12)(cid:12)(cid:12)(cid:12) = Rrµνρuµuνnρ|r∗ , (5) ii.nefi.nHitayw, kwiengcaqnuatnaktae tchaen ibnetegcrreaatitoend cwointshtaznetror0ve→loc∞ity,
r∗ only at infinity. Hence, we get
where the r.h.s. is expressed in terms of the Riemann (cid:114)
2M
tensor components, nr denotes the separation between ur| = . (8)
the two radially infalling geodesics followed by the pair r∗ r∗
of particles and uµ = [1,0,0,0] in the rest frame of the We can now boost the acceleration vector aµ =
particle. (0,ar,0,0),wherear givenby(6),withavelocityparam-
The separation between the particle and the anti- eter given by (8), in order to determine the tidal force in
particle when the pair forms spontaneously (i.e. they the static frame ar . We get ar =arcosh(ζ) so that the
st st
go “on-shell”) is given by their Compton wavelength, radial component of the force under this transformation
namely nρ = [0,nr,0,0] where nr ∼ λC = (cid:126)/mc, and is given by
m(cid:28)M is the particles rest mass (from now on we shall
work in units where (cid:126) = c = 1). So in the end, Eq. (5)
implies that the radial component of the tidal acceler- Fr (cid:12)(cid:12) = marst (cid:12)(cid:12)(cid:12) = mλC 2M ,
ation (as computed in the rest frame of the particle at tidal−st r∗ (1−2M/r)(cid:12)r∗ (1−2M/r∗)2 r∗3
coordinate r ) is given by 4 (9)
∗
where we have rescaled the mass in the rest frame by
the appropriate Lorentz factor, (1−2M/r )−1. Finally,
∗
usingthefactthatλ ∼1/m,themagnitudeoftheforce
C
4 Forcomputationoftheaccelerationintherestframeofthepar- is given by
ticle we need the Riemann tensor in the inertial frame of the
pSachrtwicalrez.schOilndeccoaonrdcinomatpesutaendthtehReniebmoaonsnt ittenussoinrginthtehefreset-aftailcl ||Fr ||= 2M(cid:18)1− rs(cid:19)−23 . (10)
tidal−st r3 r
velocity of the particle as measured in the static frame. A fea- ∗ ∗
ture of the Schwarzschild geometry is that the components of
In analogy with the Schwinger effect, we shall now as-
the Riemann tensor remains invariant under such a boost [16].
Thus,in(5)wehaveRrttr =−2M/r3. sume that the work done by the tidal force to split the
4
Γ
virtual pair can be approximated by the product of the
forcecomputedabovewiththedistanceoverwhichitap- 10
pears to have acted, i.e. the separation of the two Hawk-
ing quanta as they go on-shell as measured by a static 8
observer at r . Given that we have assumed that the in-
∗
going Hawking quantum goes on shell as soon as it can 6
do so, i.e. at horizon crossing, this distance will coincide
4
with the static observer’s proper distance to the horizon
d(r ).
∗ 2
Therefore, the workrequiredbythetidal force tosplit
the pair apart is given by 5
Α
1 2 3 4 5 6
W ∼||Fr ||d(r )= 2M(cid:18)1− rs(cid:19)−23d(r ), FIG. 1: This plot shows the variation of γ with respect to
tidal tidal−st ∗ r3 r ∗ the radial distance from the center of the black hole. The
∗ ∗
(11) red dashed line corresponds to the horizon location at α=1
where d(r ) is given by where the expression for the tidal force work diverges, indi-
∗
cating that the quanta in the far UV tail of the Hawking
(cid:90) r∗√ spectrum originate from very near the horizon.
d(r ) = g dr(cid:48) (12)
∗ rr
rs
  (cid:32) (cid:114) (cid:33)2
(cid:112) 1 1
= rs α(α−1)+ 2logα 1+ 1− α  , outgoing particles distance from the horizon in their rest
frame,givenbyλ ,resultinginashorterproperdistance
C
d(r ) at which they are detected.
∗
and we have defined α≡r /r .
∗ s
Wecanthenequatethisworktothetotalenergyofthe Also, by using Eq. (12) and expressing the rest of
twoHawkingquantabeingcreated,namelyW =2ω . Eq. (11) in terms of α, we can see that the work doable
tidal r
This gives us at fixed α by the tidal forces scales as the inverse of the
mass of the black hole so making evident that smaller
2M(cid:18) 2M(cid:19)−32 γ (cid:18) 2M(cid:19)−12 holes can produce hotter particles at the same relative
r3 1− r d(r∗)= 2πr 1− r . (13) distance from the horizon.
∗ ∗ s ∗
Let us stress again the heuristic nature of our argu-
Finally, from eq. (13) we get
ment. We are considering the instantaneous value of the
tidal force observed by the outgoing partner at a given
2π (cid:18) 1(cid:19)−12 coordinate distance r where it goes on-shell. However,
γ = 1− (14) ∗
α2 α wethenusethisinstantaneousvaluetocomputethework
  (cid:32) (cid:114) (cid:33)2 done by the gravitational field over a distance d(r∗), as
1 1
· 1+ √ logα 1+ 1− . if the force was actually at work with the same constant
2 α2−α α value throughout the whole splitting process.
So, although the analogy with the Schwinger effect for
The relation between γ and α, i.e the radial distance
the electron-positron pair production by an electric field
scaled as r /r , is better illustrated in Fig. 1. It is clear
∗ s may be advocated to lend support to our description of
fromtheplotthatthepartoftheHawkingthermalspec-
Hawkingquantaproductionfromaquantumatmosphere
trum around the peak (γ ∼ 2.82), where most of the
that extends well beyond the horizon, we now want to
radiation is concentrated, corresponds to a region which
presentamoresoundanalysisbasedontherenormalized
extendsfaroutsidethehorizon, uptoaround2r (atthe
s stress energy tensor in order to confirm this picture.
peak r ≈4.38M).
∗
The plot above also shows how, in this tidal force
derivation, the quanta with higher velocity (kinetic en-
ergy) are produced closer to the horizon. This is con-
sistent with our analysis since the higher the initial ra-
dial velocity the stronger the Lorentz contraction of the III. STRESS-ENERGY TENSOR
By analyzing the renormalized stress energy tensor
5 Alternatively, we could introduce a 4-vector (cid:96)µ = (0,(cid:96)r,0,0), (RSET)onecanunderstandHawkingradiationinabet-
with ||(cid:96)|| = (cid:112)gµν(cid:96)µ(cid:96)ν = d(r∗), and compute the work as ter way as this is a local object which can help to probe
(cid:12)
Wtidal∼ grrFtridal−st(cid:96)r(cid:12)(cid:12)r∗. Thiswouldgivethesameresult. the physics in the vicinity of the black hole.
5
A. Computation of RSET As mentioned earlier C¯(u,V) is the metric component
ofastaticspacetime,soallthedynamicsofthecollapsing
Let us introduce a set of globally defined affine coor- geometry is captured in the p˙ term of (23). In the above
dinates U,V on I− ,I− respectively. Restricting to analysis, by using another affine null coordinate, we can
left right
the radial and time dimensions, the metric reads differentiatebetweenthestaticcontributiontotheRSET
and that due to the the dynamics associated with the
ds2 =C(U,V)dUdV . (15) collapse [19].
In(1+1)dimensionstherenormalisedstressenergyten-
sor for any massless scalar field in terms of these affine B. RSET for different vacuum states.
null coordinates can be easily computed using the con-
formal anomaly [17, 18]. The components of the RSET
In order to extract physical information from the
computed in some arbitrary vacuum state are given as:
RSET, we want to compute the energy density and the
1 flux experienced by a free falling observer (along con-
(cid:104)TUU(cid:105) = −12πC1/2∂U2C−1/2 stant Kruskal position) long after the collapse has begun
1 (cid:20)C 3(C )2(cid:21) in the two physically relevant states for Hawking radia-
= ,UU − ,U , (16) tion, namely the Hartle–Hawking and the Unruh states.
24π C 2 C2
Therefore, in this section we are going to explicitly eval-
(cid:104)T (cid:105) = − 1 C1/2∂2C−1/2 uate the general expressions for the RSET components
VV 12π V expectation values obtained above in these two cases.
1 (cid:20)C 3(C )2(cid:21) Using (A2) we get the relations
= ,VV − ,V , (17)
24π C 2 C2
p(u)
(cid:104)TUV(cid:105) = 9R6Cπ = 241π∂U∂V lnC, (18) p˙(u) ≡ ∂up(u)=− 2rs , (26)
p(u) p˙(u)
p¨(u) = =− . (27)
where C is the conformal factor introduced in the above 4r2 2r
s s
metric and R is the scalar curvature.
For computing the first term of (23) we can write
Now let us also introduce a null coordinate u affine on
Ir+ight such that C¯1/2∂2C¯−1/2 = 3C¯−2(cid:0)∂ C¯(cid:1)2− 1C¯−1∂2C¯. (28)
u 4 u 2 u
U =p(u); (19)
Using the metric conformal factor C from (A1) we get
from this we get
∂U =p˙−1∂u. (20) ∂uC¯ = ∂u[p˙(u)C]=p¨C+p˙∂uC
(cid:18) 1 r2−r2(cid:19)
In terms of the set (u,V), the metric reads = p˙(u) − + s C
2r 2r2r
s s
ds2 =C¯(u,V)dudV , (21) r
= − s C¯, (29)
2r2
with
and
C¯(u,V)=p˙(u)C(U,V). (22) 1 (cid:18)C¯(cid:19) r2 1r f(r)C¯
∂2C¯ =− r ∂ = s C¯− s . (30)
Assuming that the observer is always outside the col- u 2 s u r2 4r4 2 r3
lapsing star, C¯(u,V) would be the metric component of
Using the above relation in (28) we have
astaticspacetime. Intermsofthisnewlydefinednullco-
oarsdinate, a simple computation shows that TUU is given C¯1/2∂u2C¯−1/2 = 43C¯−2(cid:20)4rrs24C¯2(cid:21)
(cid:104)TUU(cid:105)=−1p˙2−π2 (cid:104)C¯1/2∂u2C¯−1/2−p˙1/2∂u2p˙−1/2(cid:105) . (23) − 21C¯−1(cid:20)4rrs24C¯− 21rsfr(3r)C¯(cid:21)
Now T will have only a static contribution if V =v 3 r2 r 3M2 M
VV = − s + s − + , (31)
but if the affine null coordinate on I+ is defined as 16r4 4r3 4 r4 2r3
left
V =q(v) (24) where f(r) is given in (A8) and we used rs =2M in the
last step. For the second term on the r.h.s. of (23), we
and we define C(cid:48)(U,v)=q˙(v)C(U,V), T is given as have
VV
(cid:104)TVV(cid:105)=−1q˙2−π2 (cid:104)C(cid:48)1/2∂v2C(cid:48)−1/2−q˙1/2∂v2q˙−1/2(cid:105) . (25) p˙1/2∂u2p˙−1/2 =−p˙12/2∂u(cid:18)p˙3p¨/2(cid:19)= (8M1 )2 . (32)
6
We are now ready to compute explicitly the expectation using f(r) = (cid:0)1− 2M(cid:1), as follows from the metric of a
r
value of the different RSET components for the Hartle– black hole in static Schwarzschild coordinates. We have
Hawking (|H(cid:105)) and Unruh (|U(cid:105)) states.
1 (cid:20)3M2 M(cid:21)
We can start by observing that for the T and T
UU UV (cid:104)U|T |U(cid:105)= − , (42)
components, the expectation values are the same in the vv 24π 2r4 r3
two vacuum states [18]. Therefore, in the following we
simply denote and from (40) we get
(cid:104)T (cid:105)≡(cid:104)H|T |H(cid:105)=(cid:104)U|T |U(cid:105), (33) 1 M2(cid:20)3M2 M(cid:21)
UU UU UU (cid:104)U|T |U(cid:105)= − . (43)
(cid:104)T (cid:105)≡(cid:104)H|T |H(cid:105)=(cid:104)U|T |U(cid:105). (34) VV 6π V2 2r4 r3
UV UV UV
By means of (31), (32), (cid:104)T (cid:105) is given by
UU
C. Energy density
p˙−2(cid:20)3M2 M 1 (cid:21)
(cid:104)T (cid:105) = − + (35)
UU 24π 2 r4 r3 32M2
Wenowhavealltheingredientstoextractphysicalin-
V2 (cid:20) 4M 12M2(cid:21) formationfromtheRSET.Letusfirstanalyzetheenergy
= (768πM2)−1 e−r/M 1+ + .
4r2 r r2 density as measured in the frame of an observer moving
along fixed position in Kruskal coordinates.
To compute (cid:104)TUV(cid:105) we use (18), from which LetusconsideranobserveratagivenKruskalposition
with 2-velocity vµ =C−1/2(1,0) (in [T,X] coordinates).
1 1
(cid:104)T (cid:105) = ∂ ∂ lnC = (p˙q˙)−1∂ ∂ lnC The energy density, ρ, measured by this observer for the
UV 24π U V 24π u v
Unruh state is given by
1
= − (p˙q˙)−1C∂2C. (36)
96π r ρ=(cid:104)U|T |U(cid:105)vµvν =C−1(cid:104)U|T |U(cid:105)
µν TT
Using C(t,r) from (A1) andthe exact values of q(u) and =C−1(cid:104)U|T +T +2T |U(cid:105). (44)
VV UU UV
p(v), we get
Using(35),(37),(43)wecancomputetheenergydensity
(cid:104)T (cid:105)=− M2 e−r/2M. (37) exactly and we plot it in FIG. 2 (where α≡r/rs).
UV 12πr4
Ρ
On the other hand, the dependence of (cid:104)T (cid:105) on the
VV
stateinwhichwearecomputingtheexpectationvalueis
0.06
important. For the Hartle–Hawking state (eternal black
hole scenario, non-singular vacuum state in both past 0.05
and future horizons) in Kruskal coordinates the modes
are given by e−iωU,e−iωV, where we defined V as 0.04
0.03
V ≡q(v)=2r ev/2rs. (38)
s
0.02
Using this definition of V we can proceed in a similar
way as for the computation of (cid:104)TUU(cid:105). From (25), we 0.01
obtain
Α
q˙−2(cid:20)3M2 M 1 (cid:21) 2 4 6 8
(cid:104)H|T |H(cid:105)= − + (39)
VV 24π 2 r4 r3 32M2
FIG. 2: Plot of the energy density as a function of the the
U2 (cid:20) 4M 12M2(cid:21) radial distance from the centre of the black hole in Unruh
= (768πM2)−14r2e−Mr 1+ r + r2 . state.
FortheUnruhstateinKruskalcoordinates,themodes We see that the energy density blows up at the hori-
are given by e−iωU,e−iωv and there is no regularization zon(r =2M)sincewearecomputingtheenergydensity
condition imposed in the past horizon. The expectation as observed by a free falling observer in the Unruh state
value of the T component can be obtained from the whichiswellknowntobeilldefinedonthepasthorizon.
VV
relation The significant thing for us is the peak in the distribu-
tion of ρ that is obtained outside the horizon which is at
(cid:104)U|TVV|U(cid:105)=16M2q˙−2(cid:104)U|Tvv|U(cid:105), (40) r ≈4.32M.Quiteinagreementwithourheuristicpredic-
tionbasedonthegravitationalanalogueoftheSchwinger
where (cid:104)U|T |U(cid:105) can be computed from
vv effect.
1 Togetanon-singularenergydensityforthefreefalling
(cid:104)U|Tvv|U(cid:105)=−12πf(r)1/2∂v2f(r)−1/2 (41) observer we should consider the Hartle–Hawking state.
7
This is given by UsingthesecondrelationwegetA=0andfromthefirst
relation we get B =C−1/2. Therefore, zν =C−1/2[0,1].
ρ = (cid:104)H|Tµν|H(cid:105)vµvν =C−1(cid:104)H|TTT|U(cid:105) Using these expressions for vµ,zν, we get
= C−1(cid:104)H|T +T +2T |H(cid:105). (45)
VV UU UV
F =C−1(cid:104)U|T |U(cid:105)=C−1(cid:104)U|[T −T ]|U(cid:105). (48)
TX VV UU
Usingtheexpectationvaluesgivenin(35),(37),(39),we
can plot the energy density (45) with respect to radial Plugging in the expectation values (35), (43) found
distance parametrized by α. This is shown in FIG. 3, above, we can plot the flux as a function of α. This is
where we see a similar nature of the distribution with a showninFIG.4. Weseethatthefluxhasamaximumat
peak outside the horizon; however, as expected, in this
case the energy density is regular everywhere. Remark- F
ably,thepeakislocatedatr ≈4.37M,incloseagreement
0.025
with our heuristic findings.
0.020
Ρ
0.015
0.06
0.010
0.04
0.005
0.02 Α
2 4 6 8
FIG. 4: This plot shows the variation of the flux of Hawking
Α
2 4 6 8 radiation with respect to the radial distance as measured by
an observer in the Unruh state.
FIG. 3: Plot of the variation of energy density computed
r =4.32M and most of the contribution to the Hawking
in Hartle–Hawking state with respect to the radial distance
radiation comes from a region between the horizon and
from the centre of the black hole at fixed time measured in
r ≈ 6M. Let us notice that in the Unruh state the flux
the static frame. Notice that close to the horizon the energy
divergesatr ,againbecauseofthedivergenceofT on
density is negative. s UU
the past horizon.
Theseresultsstronglysupportourpreviousclaimthat
the radiation density is maximized in a region outside
IV. SUMMARY AND DISCUSSION
thehorizon. Wenowshowthatasimilarbehaviorwitha
peak away from the horizon is exhibited also by the flux
part of the RSET. It has been widely believed that Hawking radiation
originates from the excitations close to the horizon and
thiseventuallysuggestedsomedrasticmodificationofthe
D. Flux states in the near horizon regime as a resolution to the
information loss paradox [5, 20–22]. One of the primary
reasons for such an argument is based on the way Hawk-
The flux of the Hawking radiation in the Unruh vac-
uum is given by 6 ing did his original calculation, tracing back the modes
all the way from future infinity to the past null infinity
F =(cid:104)U|Tµν|U(cid:105)vµzν, (46) through the collapsing matter so that one has a vacuum
state at the horizon for a free-falling observers.
where vµ is the velocity of the observer and zν is the
The other disturbing feature about this argument is,
contravariant component of the normal to the observer.
when the modes are traced back they become highly
Let us consider a static observer at fixed distance in a
blueshifted near the horizon and we are not well aware
Kruskal frame with vµ = C−1/2[1,0] and indicate the
of the laws of physics in such high transplanckian do-
normal vector as zν = [A,B]. The latter has to satisfy
main. Some resolutions to the above problem has been
the following conditions
proposed several time in the literature [23–25] but they
g zµzν =−1, zµv =0. (47) all demand some challenging modification to our present
µν µ
knowledge of gravitation or quantum field theory.
Let us stress, however, that the UV departures from
Lorentz invariance through the introduction of a funda-
6 IntheHartle–Hawkingvacuumthefluxvanishesduetothether- mentalcutoffpostulatedin[26,27]arerelevantonlyvery
malequilibriumofthestate. close to the horizon for large black holes (in units of the
8
Lorentzbreakingscale). Hence,evencontemplatingsuch logical behavior (other than a true curvature singularity,
scenario,ouranalysisinsectionIIIwouldbebasicallyun- like in the center of a black hole). For this purpose the
changed and unaffected away from the horizon, as also Kruskal coordinate frame is an appropriate choice. The
stressed in the similar analysis carried out in [28]. Kruskal metric is given as
In this paper we have shown evidence that the Hawk-
ing quanta originate from a region which is far outside
r
the horizon, which can be called a black hole atmo- ds2 = se−r/rsdUdV , (A1)
r
sphere. More precisely, from the plots of the energy den-
sityandthefluxintheUnruhstatewegetamaximumat
where r is the radius of the event horizon. For this
s
r ≈4.32M,fortheenergydensityintheHartle–Hawking
coordinate system we have
state the peak is at r ≈ 4.37M. This is strikingly close
to our previous finding for an origin at about r ≈4.38M U =p(u)=−2r e−u/2rs, (A2)
s
for the peak of the thermal spectrum using the heuris-
V =q(v)=2r ev/2rs. (A3)
tic argument based on tidal forces. By large this is also s
The affine null coordinate u,v in terms of radial dis-
in agreement with some previous claims using various
tance from the centre of the black hole, “r”, and time,
other methods, such as calculating the effective radius
“t”, as measured by a static observer is given as
of a radiating body using the Stefan–Boltzmann law or
computingtheeffectiveTolmantemperature[10,29,30],
(cid:20) (cid:18) (cid:19)(cid:21)
r
as well as in close correspondence with the results of the u=t−r =t− r+r ln −1 , (A4)
∗ s r
studyofthestress-energytensorintheBoulwarevacuum s
(cid:20) (cid:18) (cid:19)(cid:21)
of [31]. r
v =t+r =t+ r+r ln −1 . (A5)
If the radiation has a long distance origin then we ∗ s r
s
might not need to worry about the transplanckian issue
at the horizon. Moreover, concerning the fundamental also
issue of unitarity of black hole evaporation, this result
∂r 1 1
suggests to consider some effect operational at this new ∂ = ∗∂ =− ∂ =− f(r)∂ , (A6)
scale in order to eventually restore unitarity of Hawk- u ∂u r∗ 2 r∗ 2 r
ing radiation. A possible scenario is the one of non- ∂ = ∂r∗∂ = 1∂ = 1f(r)∂ . (A7)
violent nonlocality advocated in [32, 33]. We hope that v ∂v r∗ 2 r∗ 2 r
the present contribution will stimulate further investiga-
where we used
tions in these directions.
dr (cid:18) r (cid:19)−1
∗ =[f(r)]−1 = 1− s . (A8)
V. ACKNOWLEDGMENT dr r
We can also define a set of time like and radial coordi-
We thank Renaud Parentani, Sebastiano Sonego and
nates (T,X) as
Matt Visser for illuminating discussions. We also ac-
knowledge the John Templeton Foundation for the sup-
1 1
porting grant #51876. T = (V +U),X = (V −U). (A9)
2 2
Using this metric (A1) is given as
Appendix A: Kruskal frame.
r
ds2 = se−r/rs(dT2−dX2). (A10)
WewanttoexaminethecomponentsoftheRSETina r
globallywelldefinedcoordinatesystemfreeofanypatho-
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