Physics Letters B 662 (2008) 62–65

www.elsevier.com/locate/physletb

Quantum tunneling and back reaction

Rabin Banerjee, Bibhas Ranjan Majhi ∗

S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India

Received 9 January 2008; received in revised form 20 February 2008; accepted 23 February 2008

Available online 4 March 2008

Editor: T. Yanagida

Abstract

We give a correction to the tunneling probability by taking into account the back reaction effect to the metric of the black hole spacetime. Wethen show how this gives rise to the modifications in the semiclassical black hole entropy and Hawking temperature. Finally, we reproduce thefamiliar logarithmic correction to the Bekenstein–Hawking area law.© 2008 Elsevier B.V. All rights reserved.

In 1975, Hawking discovered [1] the remarkable fact thatblack holes, previously thought to be completely black regionsof spacetime from which nothing can escape, actually radiatea thermal spectrum of particles and that the temperature of thisradiation depends on the surface gravity of the black hole by therelation TH = K0

2π. This discovery was consistent with an earlier

discovery [2–5] of a connection between black holes and ther-modynamics which revealed that the entropy of a black hole isproportional to the surface area of its horizon, SBH = A

4 . Fromthis, using the second law of thermodynamics dM = TH dSBH,the temperature of the black hole can be calculated. This isactually due to an analogy between the second law of thermo-dynamics and the black hole equation dM = K0

8πdA. These rela-

tions are all based on classical or semiclassical considerations.It is possible to include quantum effects in this discussion

of Hawking radiation. Using the conformal anomaly methodthe modifications to the spacetime metric by the one loop backreaction was computed [6,7]. Later it was shown [8,9] thatthe Bekenstein–Hawking area law was modified, in the lead-ing order, by logarithmic corrections. Similar conclusions werealso obtained by using quantum gravity techniques [10–12].Likewise, corrections to the semiclassical Hawking tempera-ture were derived [7,8].

* Corresponding author.E-mail addresses: rabin@bose.res.in (R. Banerjee), bibhas@bose.res.in

(B.R. Majhi).

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.02.044

A particularly useful and intuitive way to understand theHawking effect is through the tunneling formalism as devel-oped in [13]. The semiclassical Hawking temperature is verysimply and quickly obtained [13,14] in this scheme by ex-ploiting the form of the semiclassical tunneling rate. A naturalquestion that arises in this context is the feasibility of this ap-proach to include quantum corrections. Although there havebeen sporadic attempts [15,16] a systematic, thorough and com-plete analysis is still lacking.

In this Letter we compute the corrections to the semiclas-sical tunneling rate by including the effects of self gravitationand back reaction. The usual expression found in [13], given

in the Maxwell–Boltzmann form e− E

TBH , is modified by a pref-actor. This prefactor leads to a modified Bekenstein–Hawkingentropy. The semiclassical Bekenstein–Hawking area law con-necting the entropy to the horizon area is altered. As obtained inother approaches [9,10,12], the leading correction is found to belogarithmic while the nonleading one is a series in inverse pow-ers of the horizon area (or Bekenstein–Hawking entropy). Wealso compute the appropriate modification to the Hawking tem-perature. Explicit results are given for the Schwarzschild blackhole.

Let us consider a general class of static, spherically symmet-ric spacetime of the form

(1)ds2 = −f (r) dt2 + dr2

g(r)+ r2 dΩ2

R. Banerjee, B.R. Majhi / Physics Letters B 662 (2008) 62–65 63

where the horizon r = rh is given by f (rh) = g(rh) = 0. Themetric has a coordinate singularity at the horizon which is re-moved by transforming to Painleve coordinates [17]. A suitable

choice is dt → dt −√

1−g(r)f (r)g(r)

dr under which the metric takesthe form,

ds2 = −f (r) dt2 + 2f (r)

√1 − g(r)

f (r)g(r)dt dr + dr2 + r2 dΩ2.

(2)

The radial null geodesics (ds2 = dΩ2 = 0) are given by

(3)r ≡ dr

dt=

√f (r)

g(r)

(±1 − √1 − g(r)

)where the positive (negative) sign gives outgoing (incoming)radial geodesics. Now expanding the quantities f (r) and g(r)

about the horizon rh we get

f (r) = f ′(rh)(r − rh) + O((r − rh)

2),(4)g(r) = g′(rh)(r − rh) + O

((r − rh)

2).The surface gravity of the black hole on the horizon is definedby

K(M) = Γ 000

∣∣r=rh

(5)= 1

2

[√1 − g(r)

f (r)g(r)g(r)

df (r)

dr

]∣∣∣∣r=rh

.

Therefore using (4) K and r can be approximately expressed as

(6)K(M) � 1

2

√f ′(rh)g′(rh)

and

r � 1

2

√f ′(rh)g′(rh)(r − rh) + O

((r − rh)

2)(7)=K(M)(r − rh) + O

((r − rh)

2)where in the last step (6) has been used.

The imaginary part of the action for an s-wave outgoing pos-itive energy particle which crosses the horizon outwards fromrin to rout can be expressed as

ImS = Im

rout∫rin

pr dr = Im

rout∫rin

pr∫0

dp′r dr

(8)= Im

rout∫rin

H∫0

dH ′

rdr

where in the last step we multiply and divide the integrand bythe two sides of Hamilton’s equation r = dH

dpr|r . Now taking into

account the self-gravitation effects [18], the above integrationcan be expressed as

(9)ImS = Im

rout∫rin

M−ω∫M

dH ′

rdr = − Im

rout∫rin

ω∫0

dω′

rdr

where we used the fact that the Hamiltonian H = M − ω, withM being the original mass of the black hole. Here r can beapproximated by (7) as follows:

(10)r � (r − rh)K(M − ω) + O((r − rh)

2)where rh is the modified Schwarzschild radius and K(M − ω)

is the modified horizon surface gravity. This modification oc-curs due to two effects; self-gravitation which requires the re-placement M → M − ω so that K(M) → K(M − ω) and backreaction to be discussed below. Taking only the first order termof r , the integration in (9) can be written as

(11)ImS = − Im

rout∫rin

ω∫0

dω′

(r − rh)K(M − ω′)dr.

But now the integration over r can be done by deforming thecontour. Ensuring that the positive energy solutions decay intime (i.e. into the lower half of ω′ plane and rin > rout) we haveafter r integration,1

(12)ImS = π

ω∫0

dω′

K(M − ω′).

A derivation of (12), to the leading order in ω, followingsimilar techniques has been presented in [19,20]. In fact if weexpand 1

K(M−ω′) retaining terms linear in ω′, we immediatelyfind,

ImS = π

ω∫0

dω′[

1

K(M)+O(ω′)

]

= πω

K(M)+O

(ω2)

(13)= 2πω√f ′(rh)g′(rh)

+O(ω2)

which is the expression given in [19,20]. In getting the finalform the value of K(M) from (6) has been used.

Now the modified surface gravity due to one loop back reac-tion effects is given by [6,7],

(14)K(M) =K0(M)

(1 + α

M2

)

where K0 is the classical surface gravity at the horizon of theblack hole. Such a form is dictated by simple scaling arguments.As is well known, a loop expansion is equivalent to an expan-sion in powers of the Planck constant h. Since, in natural units,√

h = Mp , the one loop correction has a form given by α

M2 . Theconstant α is related to the trace anomaly coefficient taking intoaccount the degrees of freedom of the fields [7,8]. Its explicitform is given by [8],

1 A similar treatment can be done by taking the contour in the upper halfplane but then one has to replace M → M + ω [18].

64 R. Banerjee, B.R. Majhi / Physics Letters B 662 (2008) 62–65

(15)α = 1

360π

(−N0 − 7

4N 1

2+ 13N1 + 233

4N 3

2− 212N2

)

where Ns denotes the number of fields with spin ‘s’.For the classical Schwarzschild spacetime

(16)f (r) = g(r) = 1 − 2M

r; rH = 2M

and so by Eq. (6) the value of K0(M) is

(17)K0(M) = f ′(rH = 2M)

2= 1

4M.

Substituting these in (12) and then integrating over ω′ we have

(18)ImS = 4πω

(M − ω

2

)+ 2πα ln

[(M − ω)2 + α

M2 + α

].

Now according to the WKB-approximation method the tunnel-ing probability is given by

(19)Γ ∼ e−2 ImS .

So the modified tunneling probability due to back reaction ef-fects is

(20)Γ ∼[

1 − 2ω(M − ω2 )

M2 + α

]−4πα

e−8πω(M− ω2 ).

The exponential factor of the tunneling probability was previ-ously obtained by Parikh and Wilczek [13]. The factor beforethe exponential is actually due the effect of back reaction. Itwill eventually give the correction to the Bekenstein–Hawkingentropy and the Hawking temperature as will be shown below.

It is known [13,21,22] that change in the Bekenstein–Hawking entropy due to the tunneling through the horizon isrelated to ImS by the following relation,

(21)�Sbh = −2 ImS.

Therefore the corrected change in Bekenstein–Hawking en-tropy is

�Sbh = −8πω

(M − ω

2

)− 4πα ln

[(M − ω)2 + α

](22)+ 4πα ln

(M2 + α

).

Next using the stability criterion d(�Sbh)dω

= 0 for the black hole,one obtains the following condition

(23)(ω − M)3 = 0

which gives the only solution as ω = M . Substituting this valueof ω in (22) we will have the change in entropy of the blackhole from its initial state to final state.

(24)Sfinal − Sinitial = −4πM2 + 4πα ln

(M2

α+ 1

).

So the Bekenstein–Hawking entropy of the black hole withmass M is

(25)Sbh = 4πM2 − 4πα ln

(M2

α+ 1

).

Ignoring back reaction (i.e. α = 0) we just reproduce the usualsemiclassical area law [2–5] for the Bekenstein–Hawking en-tropy,

(26)SBH = 4πM2 = A

4where A is the area of the black hole horizon given by

(27)A = 4πr2H = 16πM2.

Substituting (26) in (25) and expanding the logarithm, we ob-tain the final form,

Sbh = A

4− 8πα lnM

− 64π2α2[

1

A− 16πα

2A2+ (16πα)2

3A3− · · ·

]+ const (independent of M)

= SBH − 4πα lnSBH

− 16π2α2

SBH

[1 − (4πα)

2SBH+ (4πα)2

3(SBH)2− · · ·

](28)+ const (independent of M).

The well-known logarithmic correction [8–12] appears in theleading term. Quantum gravity calculations lead to a prefac-tor − 1

2 for the lnSBH term which would correspond to choosingα = 1

8π. Also, the nonleading corrections are found to be ex-

pressed as a series in inverse powers of A (or SBH), exactly ashappens in quantum gravity inspired analysis [10,12].

Now using the second law of thermodynamics

(29)Th dSbh = dM

we can find the corrected form of the Hawking temperature Th

due to back reaction. This is obtained from (25)

(30)1

Th

= dSbh

dM= 8πM

(M2

M2 + α

).

Therefore the corrected Hawking temperature is given by

(31)Th = TH

(1 + α

M2

)

where TH = 18πM

is the semiclassical Hawking temperatureand the other term is the correction due to the back reaction.A similar expression was obtained previously in [8] by the con-formal anomaly method.

It is also possible to obtain the corrected Hawking tempera-ture (31) in the standard tunneling method to leading order [13]where this temperature is read off from the coefficient of ‘ω’in the exponential of the probability amplitude (20). Recastingthis amplitude as,

(32)Γ ∼ e−8πω(M− ω

2 )−4πα ln(1− 2ω(M− ω2 )

M2+α)

and retaining terms up to leading order in ω, we obtain

Γ ∼ e−8πMω+4πα( 2Mω

M2+α)

(33)= e−( 8πM3

M2+α)ω = e

− ωTh .

R. Banerjee, B.R. Majhi / Physics Letters B 662 (2008) 62–65 65

The inverse Hawking temperature, identified with the coeffi-cient of ‘ω’, reproduces (31).

It is also observed that the usual (semiclassical) identifi-cation between the surface gravity and Hawking temperatureTH = K0(M)

2πpersists even after including the back reaction.

From (14) and (31) we easily infer that Th = K(M)2π

.To conclude, we have considered self-gravitation and (one

loop) back reaction effects in tunneling formalism for Hawkingradiation. The modified tunneling rate was computed. From thismodification, corrections to the semiclassical expressions forentropy and Hawking temperature were obtained. Also, the log-arithmic corrections to the semiclassical Bekenstein–Hawkingarea law was reproduced. Although our analysis was presentedfor the Schwarzschild black hole, it is general enough to includeother examples as well.

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