Quantum corrected non-thermal radiation spectrum from the tunnelling mechanism

0.17 MB·English

by  Subenoy Chakraborty

Quantum corrected non-thermal radiation
spectrum from the tunnelling mechanism
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1Subenoy Chakraborty, 2Subhajit Saha and 3,4,5Christian Corda
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May 29, 2015
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8 1,2Department of Mathematics, Jadavpur University, Kolkata
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700032, West Bengal, India

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c 3Dipartimento di Scienze, Sezione di Fisica, Scuola Superiore di
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Studi Universitari e Ricerca "Santa Rita", via San Nicola snc,
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r 81049, San Pietro Infine (CE) Italy
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4Austro-Ukrainian Institute for Science and Technology, Institut for
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Theoretish Wiedner Hauptstrasse 8-10/136, A-1040, Wien, Austria
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6 5International Institute for Applicable Mathematics & Information
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Sciences (IIAMIS), Hyderabad (India) & Udine (Italy)
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. Email addresses: [email protected],
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[email protected], [email protected]
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: Abstract
v
i Tunnelling mechanism is today considered a popular and widely used
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method in describing Hawking radiation. However, in relation to black
r
a hole (BH) emission, this mechanism is mostly used to obtain the Hawk-
ing temperatureby comparing theprobability of emission of an outgoing
particle with the Boltzmann factor. On the other hand, Banerjee and
Majhireformulatedthetunnellingframeworkderivingablackbodyspec-
trum through the density matrix for the outgoing modes for both the
Bose-Einstein distribution and theFermi-Dirac distribution. In contrast,
Parikh and Wilczek introduced a correction term performing an exact
calculation of the action for a tunnelling spherically symmetric particle
and, as a result, the probability of emission of an outgoing particle cor-
responds to a non-strictly thermal radiation spectrum. Recently, one of
us (C. Corda) introduced a BH effective state and was able to obtain a
non-strictly black body spectrum from the tunnelling mechanism corre-
sponding to the probability of emission of an outgoing particle found by
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ParikhandWilczek. Thepresentworkintroducesthequantumcorrected
effective temperature and the corresponding quantum corrected effective
metriciswrittenusingHawking’speriodicityarguments. Thus,weobtain
further corrections to the non-strictly thermal BH radiation spectrum as
thefinaldistributionstakeintoaccountboththeBHdynamicalgeometry
during the emission of the particle and the quantum corrections to the
semiclassical Hawking temperature.
Keywords: Quantum Tunnelling, Quantum corrected effective tem-
perature, BH information puzzle
PACS Numbers: 04.70.Dy, 04.70.-s.
ConsideringHawkingradiation[1]inthetunnellingapproach[2]-[7],[25]-[27]
theparticlecreationmechanismcausedbythevacuumfluctuationsneartheBH
horizon works as follows. A virtual particle pair is created just inside the hori-
zonandthe virtualparticlewithpositiveenergycantunnel outthe BHhorizon
asarealparticle. Otherwise,thevirtualparticlepairiscreatedjustoutsidethe
horizon and the negative energy particle can tunnel inwards. Thus, for both
the possibilities, the particle with negative energy is absorbed by the BH and
as a result the mass of the BH decreases. The flow of positive energy particles
towards infinity is considered as Hawking radiation. Earlier, this approachwas
limited to obtain only the Hawking temperature through a comparison of the
probabilityofemissionofanoutgoingparticlewiththeBoltzmannfactorrather
than the actual radiation spectrum with the correspondent distributions. This
problemwasformallyaddressedbyBanerjeeandMajhi[7]. Byanovelformula-
tion of the tunnelling formalism, they were able to directly reproduce the black
bodyspectrumforeitherbosonsorfermionsfromaBHwithstandardHawking
temperature. However,considering contributions beyond semiclassicalapproxi-
mation in the tunnelling process, Parikhand Wilczek [2, 3] found a probability
of emission compatible with a non-thermal spectrum of the radiationfrom BH.
This non precisely thermal character of the spectrum is important to resolve
the information loss paradox of BH evaporation [8] because arguments that in-
formation is lost during hole’s evaporation partially rely on the assumption of
strict thermal behavior of the radiationspectrum [3, 8]. Interesting approaches
to resolve the BH information puzzle have been recently proposed in [9, 10]
Thebasicdifferencebetweentheworks[2,3]andthework[7]isconsideration
or non-consideration of the energy conservation. As a result, there will be a
dynamical [2, 3] or static [7] BH geometry. In fact, due to conservation of
energy, in [2, 3] the BH horizon contracts during the radiation process which
deviatesfromtheperfectblackbodyspectrum. Thisnon-thermalspectrumhas
profound implications for realizing the underlying quantum gravity theory. In
thelanguageofthetunnellingmechanism,atrajectoryinimaginaryorcomplex
time joins two separated classical turning points [3]. The key point is that the
forbiddenregiontraversedby the emitting particlehasa finite size [3]fromr=
r to r =r (r is the radius of the horizon of the BH initially and
initial final initial
r is the radiusof the horizonofthe BH after particleemission). This finite
final
sizeimpliesadiscretenatureofthetunnellingmechanism,whichischaracterized
by the physical state before the emission of the particle and that after the
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emission of the particle. As a result, the radiation spectrum is also discrete
[10, 11]. Consequently, particle emission can be interpreted like a quantum
transition of frequency ω between the two discrete states [10, 11]. It is the
particle itself which generates a tunnel through the horizon [3, 10, 11] having
finite size. In thermal spectrum, the tunnelling points have zero separation, so
there is no clear trajectory because there is no barrier [3, 10, 11].
In Planck units (G = c = k = ~ = 1 = 1 ), the strictly thermal
B 4πǫ0
tunnelling probability is given by [1, 2, 3]
ω
Γ∼exp − , (1)
T
(cid:18) H(cid:19)
where T = 1 is the Hawking temperature and ω is the energy-frequencyof
H 8πM
the emitted radiation. However,considering contributions beyond semiclassical
approximation and taking into account the conservation of energy, Parikh and
Wilczek reformulate the tunnelling probability as [2, 3]
ω ω
Γ∼exp − 1− . (2)
T 2M
(cid:20) H (cid:16) (cid:17)(cid:21)
This non-thermal spectrum enables the introduction of an intriguing way to
consider the BH dynamical geometry through the BH effective state. In fact,
one introduces the effective temperature as [10]-[13]
2M 1
T (ω)≡ T = , (3)
E H
2M −ω 4π(2M −ω)
whichpermits to rewrite the probabilityof emission(2) inBoltzmann-Hawking
form as [10]-[13]
ω
Γ∼exp[−β (ω)ω]=exp(− ), (4)
E
T (ω)
E
where the effective Boltzmann factor takes the form [10]-[13]
1
β (ω)≡ . (5)
E
T (ω)
E
One interpretes the effective temperature as the temperature of a black body
emiting the same total amount of radiation [10]-[13]. Hence Hawking tem-
perature is replaced by the effective temperature in the expression
for the probability of emission. It should be noted that this notion
of effective temperature has already been introduced in the litera-
ture for the Schwarzschild BH [12, 13], for the Kerr BH [19] and
for the Reissner-Nordstrom BH [20]. Further, the ratio TE(ω) = 2M
TH 2M−ω
characterize the deviationofthe radiationspectrum ofa BHfrom the
strictlythermalfeature [10, 11,12,13]. Also, thetunnellingapproach
of Parikh and Wilczek shows the probability of emission of Hawking
quanta [see Eq. (2)] is non-thermal in nature (i.e., BH does not emit
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like a perfect black body). Moreover, due to perfect black body char-
acter of Bose-Einstein and Fermi-Dirac distributions, it is natural to
have deviations from these distributions in case of the above effec-
tive temperature. Thus in analogy to BH, the effective temperature
of a body (say star) can be defined as the temperature of a black
body that would emit the same total amount of electromagnetic radi-
ation [10, 14]. So one can consider this effective temperature and the
bolometricluminosityas the two fundamental physical parameters to
identify a star on the Hertzsprung-Russel diagram. It is worthy to
mention here that both the above two physical parameters however
depend on the chemical composition of the star [10, 11, 12, 13, 14].
Further, in analogywith the effective temperature, one can define the effec-
tive mass and the effective horizon radius as [10]-[13]
ω
M =M − and r =2M =2M −ω. (6)
E E E
2
Note that these effective quantities are nothing but the average value of the
corresponding quantities before (initial) and after (final) the particle emission
(i.e., M = M, M = M −ω; r = 2M and r = 2M ). Accordingly, T is
i f i i f f E
the inverse of the average value of the inverses of the initial and final Hawking
temperatures [10]-[13]. Hence, there is a discrete character (in time) of the
Hawking temperature. Thus, the effective temperature may be interpreted as
the Hawking temperature during the emission of the particle [10]-[13].
Following[11]onecanuseHawking’speriodicityargument[11,23,24]toobtain
the effective Schwarzschild line element
2M dr2
ds2 =−(1− E)dt2+ +r2(sin2θdϕ2+dθ2), (7)
E r 1− 2ME
r
which takes into account the BH dynamical geometry during the emission of
the particle.
Recently, one of us (C. Corda) introduced the above discussed BH effec-
tive state [10]-[13] and was able to obtain a non-strictly black body spectrum
from the tunnelling mechanism corresponding to the probability of emission of
an outgoing particle found by Parikh and Wilczek [11]. The final non-strictly
thermal distributions which take into account the BH dynamical geometry are
[10, 11]
<n> = 1
boson exp[4π(2M−ω)ω]−1
(8)
<n> = 1 .
fermion exp[4π(2M−ω)ω]+1
Now, we further modify the effective temperature by incorporating the quan-
tumcorrectionstothesemiclassicalHawkingtemperaturediscussedin[4]. Asa
result, the quantum physics of BHs will be further modified. Banerjee and Ma-
jhi [4] have formulated the quantum corrected Hawking temperature using the
Hamilton-Jacobi method [15] beyond semiclassical approximation. According
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to them [4], the quantum corrected Hawking temperature (termed as modified
Hawking temperature) is given by
β −1
T(m) = 1+Σ i T , (9)
H iM2i H
(cid:20) (cid:21)
where the β are dimensionless constant parameters. However, if these param-
i
eters are chosen as powers of a single parameter α, then in compact form [4]
α
T(m) = 1− T . (10)
H M2 H
(cid:16) (cid:17)
This modified Hawking temperature is very similar in form to the temperature
correctionin the context of one-loop back reaction effects [16, 17] in the space-
time with α related to the trace anomaly [18]. Further, using conformal field
theory, if one considers one-loop quantum correction to the surface gravity for
SchwarzschildBH then α has the expression [4]
1 7 233
α=− −N − N +13N + N −212N , (11)
360π 0 4 21 1 4 23 2
(cid:18) (cid:19)
where N denotes the number of field with spin s. Also considering two-loop
s
backreactioneffects in the spacetime,the quantumcorrectedHawkingtemper-
ature becomes [4]
α γ
T(m) = 1− − T , (12)
H M2 M4 H
where second loop contributionshare related to thie dimensionless parameter γ.
Thus, it is possible to incorporate higher loop quantum corrections by proper
choices of the β . It should be noted that these correction terms dominate at
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large distances [21].
Using the above mentioned modified Hawking temperature, the modified
form of the Boltzmann factor is
1 1 β
β(m) = = = H . (13)
TH(m) TH 1− Mα2 − Mγ4 1− Mα2 − Mγ4
Thus, the (quantum corrected)(cid:0)modified BH m(cid:1)ass(cid:0)has the express(cid:1)ion
M
M(m) = . (14)
1− α − γ
M2 M4
In case of emitted radiation from th(cid:0)e BH, the mo(cid:1)dified Hawking temperature
(with quantum correction) becomes
1
T(m) = . (15)
H 8πM(m)
As a result, following [23], one can again use Hawking’s periodicity argument
[11, 23, 24] to obtain the modified Schwarzschild like line element, which takes
the form [10, 11]
2M(m) dr2
(ds )2 =−(1− )dt2+ +r2(sin2θdϕ2+dθ2) (16)
m r 1− 2M(m)
r
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with modified surface gravity
1 1 1− α − γ
κ(m) = = = M2 M4 . (17)
4M(m) 2r(m) 4M
(cid:0) (cid:1)
Eq. (16) enables the replacement M → M(m) and T → T(m) in eqs. (3),
H H
(5)and (6). In other words, one can define the (quantum corrected) modified
effective temperature
2M(m) 1
Tm(ω)≡ T(m) = , (18)
E 2M(m)−ω H 4π(2M(m)−ω)
the (quantum corrected) modified effective Boltzmann factor
1
β(m)(ω)≡ (19)
E Tm(ω)
E
andthe(quantumcorrected)modifiedeffectivemass andeffectivehorizon radius
ω
M(m) =M(m)− and r(m) =2M(m) =2M(m)−ω. (20)
E 2 E E
A clarification is needed concerning the definition (18) [28]. Eqs. (14)-(17)
give the quantum corrections using Hamilton-Jacobi method beyond semiclas-
sical approximation. Here we considered the contributions of the non-thermal
spectrum by reformulation of tunnelling probability choosing Eq. (16) as the
modified Schwarzschild line element. It should be noted that a full cal-
culation involving the action of a particle on the BH spacetime also
leads to this result. SoEq. (3)nowbecomeseq. (18). Following[11,23,24],
one uses again Hawking’s periodicity argument. Then, the euclidean form of
the metric will be given by
2 2
2 dτ r
ds(m) =x2 + dx2+r2(sin2θdϕ2+dθ2),
h E i "4M(m) 1− 2Mω(m) # rE(m)!
(21)
(cid:0) (cid:1)
which is regular at x = 0 and r = r(m). τ is treated as an angular variable
E
with period β(m)(ω) [11, 23, 24]. Replacing the quantity β ℏi in [23] with
E i iM2i
the quantity − ω , if one follows step by step the detailed analysis in [23] at
2M(m) P
theendoneeasilygetsthe(quantumcorrected)modified effective Schwarzschild
line element
2 2M(m) dr2
ds(m) =−(1− E )dt2+ +r2(sin2θdϕ2+dθ2). (22)
E r 2M(m)
1− E
h i r
One also easily shows that r(m) in eq. (21) is the same as in eq. (20). Thus,the
E
line element (22) takes into account both the BH dynamical geometry during
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the emission of the particle and the quantum corrections to the semiclassical
Hawking temperature.
Starting from the standard Schwarzschildline element, i.e. [7, 11]
2M dr2
ds2 =−(1− )dt2+ +r2(sin2θdϕ2+dθ2), (23)
r 1− 2M
r
the analysis in [7] permitted to write down the (normalized) physical states of
the system for bosons and fermions as [7]
|Ψ>boson=(1−exp(−8πMω))12 nexp(−4πnMω)|n(oLu)t >⊗|n(oRut) >
|Ψ>fermion=(1+exp(−8πMω))−21P nexp(−4πnMω)|n(oLu)t >⊗|n(oRut) >.
(24)
P
Hereafter we focus the analysis only on bosons. In fact, for fermions the
analysis is identical [7]. The density matrix operator of the system is [7]
ρˆ ≡Ψ> <Ψ|
boson boson boson
=(1−exp(−8πMω)) exp[−4π(n+m)Mω]|n(L) >⊗|n(R) ><m(R)|⊗<m(L)|.
n,m out out out out
(25)
P
If one traces out the ingoing modes, the density matrix for the outgoing(right)
modes reads [7]
ρˆ(R) =(1−exp(−8πMω)) exp(−8πnMω)|n(R) ><n(R)|. (26)
boson out out
n
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This implies that the averagenumber of particles detected at infinity is [7]
1
<n> =tr nˆρˆ(R) = , (27)
boson boson exp(8πMω)−1
h i
where the trace has been taken over all the eigenstates and the final result has
beenobtainedthroughabitofalgebra,see[7]fordetails. Theresultofeq. (27)
is the well known Bose-Einstein distribution. A similar analysis works also for
fermions [7], and one easily gets the well known Fermi-Dirac distribution
1
<n> = , (28)
fermion
exp(8πMω)+1
Both the distributions correspond to a black body spectrum with the Hawking
temperature T = 1 . On the other hand, if one follows step by step the
H 8πM
analysis in [7], but starting from the (quantum corrected) modified effective
Schwarzschild line element (22) at the end obtains the correct physical states
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for boson and fermions as
1
|Ψ> = 1−exp −8πM(m)ω 2 exp −4πnM(m)ω |n(L) >⊗|n(R) >
boson E n E out out
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)
P
−1
|Ψ> = 1+exp −8πM(m)ω 2 exp −4πnM(m)ω |n(L) >⊗|n(R) >
fermion E n E out out
(cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (29)
P
and the correct distributions as
<n> = 1 = 1 = 1
boson exp 8πME(m)ω −1 exp[4π(2M(m)−ω)ω]−1 exp 4π 2 M −ω ω −1
(cid:16) (cid:17) " (1−Mα2−Mγ4) ! #
<n> = 1 = 1 = 1 ,
fermion exp 8πME(m)ω +1 exp[4π(2M(m)−ω)ω]+1 exp 4π 2 M −ω ω +1
(cid:16) (cid:17) " (1−Mα2−Mγ4) ! #
(30)
which are not thermal because they take into account both the BH dynamical
geometryduringtheemissionoftheparticleandthequantumcorrectionstothe
semiclassicalHawkingtemperature. Wenotethatsettingα=γ =0ineqs. (30)
we find the results in [11], i.e. eqs. (8). In fact, in [11] only the BH dynamical
geometry was taken into account. Here, we further improved the analysis by
taking into account also the quantum corrections to the semiclassical Hawking
temperature.
Concluding remarks
The present work deals with the quantum correction of non-thermal radia-
tion spectrum in the framework of tunnelling mechanism. Staring from the
Schwarzschild BH, at first the quantum corrections are considered. As a re-
sult, the Hawking temperature and Schwarzschild mass are modified (see Eqs.
(14) and (15)). So one obtains the modified Schwarzschild line element (see
Eq. (16)). Then we consider the non-thermal radiation spectrum of this mod-
ified Schwarzschild BH by the reformulation of the tunnelling probability. The
resulting quantum corrected effective Schwarzschild metric is rewritten using
Hawking’s periodicity arguments. Also we have shown the correct distri-
butions of bosons and fermions using the above quantum corrections
to the semiclassical Hawking temperature. Thus due to quantum
correction at the semiclassical level, BH parameters (and its radia-
tion spectrum), namely, its mass, temperature, surface gravity, and
Boltzmann factors are modified and as a result, we have quantum
corrected effective Schwarzschild metric. Moreover, the one-loopcor-
rection which comes from interaction between graviton and particles
ofvarious species(characterized) inEq. (11)occurred at the horizon.
Hence, the quantum effects lead to a redefinition of surface gravity
and other parameters. But it should be noted that the BH’s gravita-
tional potential may not only be characterized by this modified mass
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far away from the horizon. Therefore, the modified metric [in Eq.
(16) or Eq. (22)] can only be trusted for its near-horizon geometry,
butnowhereelse–andtheeffective metricforarbitrary distancecould
be elaborated in a perturbative way [29, 30] and the potential is not
Coloumb-like in general.
Acknowledgments
TheauthorsS.C.andS.S.arethankfultoIUCAA, Pune, Indiafortheirwarm
hospitality and research facilities as the work has been done there during a
visit. Also S. C. acknowledges the UGC-DRS Programme in the Department
of Mathematics, Jadavpur University. The author S. S. is thankful to UGC-
BSR Programme of Jadavpur University for awarding research fellowship. The
authors thank the unknown referees for important advices which permitted to
improve this work.
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