Quantum tunnelling for Hawking radiation from a dynamical Black Hole

Abstract

The paper deals with Hawking radiation related to non-static spherically
symmetric black hole. Quantum corrections are incorporated using
Hamilton-Jacobi method beyond semi-classical approximation. It is found that
different order correction terms satisfy identical differential equation as the
semiclassical action and are solved by a typical technique. It has been shown
that with proper choice of the proportionality factors, one loop back reaction
effect in the space time can be obtained. Finally, using the law of black hole
mechanics, a general modified form of the black hole entropy is obtained
considering modified Hawking temperature.

Full text

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arXiv:1106.4375v1 [gr-qc] 22 Jun 2011
Quantum tunnelling for Hawking radiation from a dynamical Black
Hole
Nairwita Mazumder1
Ritabrata Biswas2
Subenoy Chakraborty3
Department of Mathematics, Jadavpur University, Kolkata-700 032, India.
Abstract
The paper deals with Hawking radiation related to non-static spherically symmetric black hole.
Quantum corrections are incorporated using Hamilton-Jacobi method beyond semi-classical ap-
proximation. It is found that different order correction terms satisfy identical differential equation
as the semiclassical action and are solved by a typical technique. It has been shown that with
proper choice of the proportionality factors, one loop back reaction effect in the space time can be
obtained. Finally, using the law of black hole mechanics, a general modified form of the black hole
entropy is obtained considering modified Hawking temperature.
Keywords : Hawking Temperature, Tunnelling, Quantum Correction
Pacs no : 04.70.Dy, 04.60.Kz, 04.62.+v
1 Introduction
The classical idea of a black hole(BH) that nothing can escape from it was ruled out by Hawking [1] in 1974.
Based on quantum field theory, he had shown (in semi classical approximation) that there is a continuous
emission of radiation from a BH identical to blck body radiation [2, 3] having temperature T =
the surface gravity of the BH. Subsequently in the past decades other semiclassical methods were developed for
BH radiation [4, 5, 6, 7]. In fact these studies of black hole radiation can be classified into two groups. The
first approach developed by Parikh and Wilczek [8, 9] is based on the heuristic pictures of visualization of the
source of radiation as tunnelling and is known as radial null geodesic method. The essence of this method is to
calculate the imaginary part of the action for the s-wave emission using the radial null geodesic equation and
then Hawking temperature is obtained by relating it to the Boltzmann factor for emission. The alternative way
of looking into this aspect is known as complex paths method developed by Sriivasan et. al. [10, 11]. In this
approach, the differential equation of the action S(r, t) of a classical scalar particle can be obtained by plugging
the scalar wave function φ(r, t) = exp?−i
temperature is obtained, using the ”Principle of detailed balance”[10, 11, 12].
In this work we consider a general non-static metric for dynamical BH. Hamilton Jacobi (HJ) method is
extended beyond semiclassical approximation to consider all the terms in the expansion of the one particle
action. It is found that the higher order terms (quantum corrections) satisfy identical differential equations as
the semiclassical action and the complicated terms are eliminated considering BH horizon as one way barrier.
We derive the modified Hawking Temperature using both the above approaches and are found to be identical at
the semiclassical level. Finally, modified form of the BH entropy with quantum correction has been evaluated.
κ
2π, κ being
¯ hS(r, t)?into the Klein Gordon(KG) equation in a gravitational back
ground. Then Hamilton-Jacobi method is employed to solve the differential equation for S. Finally, Hawking
1nairwita15@gmail.com
2biswas.ritabrata@gmail.com
3schakraborty@math.jdvu.ac.in
1

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2Method of radial null geodesic : A survey of earlier works
This section deals with a brief survey of the method of radial null geodesic method [8] considering the picture of
Hawking radiation as quantum tunnelling. In a word, the method correlates the imaginary part of the action for
the classically forbidden process of s-wave emission across the horizon with the Boltzmann factor for the black
body radiation at the Hawking temperature. We start with a general class of non-static spherically symmetric
BH metric of the form
ds2= −A(r, t)dt2+
dr2
B(r, t)+ r2dΩ2
2
(1)
where the horizon rh is located at A(rh, t) = 0 = B(rh, t) and the metric has a coordinate singularity at
the horizon. To remove this coordinate singularity we make the following Painleve-type transformation of
coordinates :
?
and as a result the metric (1) transforms to
dt → dt −
1 − B
AB
dr
(2)
ds2= −Adt2+ 2
?
A
?1
B− 1
?
dtdr + dr2+ r2dΩ2
2
(3)
This metric (i.e., the choice of coordinates) has some distinct features over the former one namely
(i) the metric is singularity free across the horizon
(ii) at any fixed time we have flat spatial geometry
(iii) both the metric has the same boundary geometry at any fixed radius
The radial null geodesic (characterised by ds2= 0 = dΩ2
2) has the differential equation (using (3))
dr
dt=
?
A
B
?
±1 −
?
1 − B(r, t)
?
(4)
where outgoing or ingoing geodesic is identified by the′+′or′−′sign within the square bracket in equation (4).
In the present case we deal with the outgoing particles through the horizon (i.e.,′+′sign only) and according
to parikh and Wilczek [8] the imaginary part of the action is obtained as
ImS = Im
?rout
rin
prdr = Im
?rout
rin
?pr
0
dp′
rdr = Im
?rout
rin
??H
0
dH′
dr
dt
?
dr
(5)
Note that in the last step of the above derivation we have used the Hamilton’s equation ˙ r =
are canonical pair. Further, it is to be mentioned that in quantum mechanics the action of a tunnelled particle in
a potential barrier having energy larger than the energy of the particle will be imaginary as pr=
For the present non-static BH the mass of the BH is not constant and hence the dH′integration extends over
all values of energy of outgoing particle, from zero to E(t) [13] (say). As we are dealing with tunnelling across
the BH horizon so using taylor series expansion about the horizon rhwe write
dH
dpr
???
rwhere (r, pr)
?2m(E − V ).
A(r, t)|t=∂A(r, t)
∂r
)
????
????
t
(r − rh) + O(r − rh)2???
(r − rh) + O(r − rh)2???
t
(6)
B(r, t)|t=∂B(r, t)
∂r
)
t
t
(7)
So in the neighbourhood of the horizon the geodesic equation (4) can be approximated as
dr
dt≈1
2
?
A′(rh, t)B′(rh, t)(r − rh)(8)
Substituting this value ofdr
dtin the last step of equation (5) we have
ImS =
2πE(t)
?A′(rh, t)B′(rh, t),
2
(9)

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where the choice of contour for r-integration is on the upper-half complex plane to avoid the coordinate singu-
larity at rh. Thus the tunnelling probability is given by
Γ ∼ exp
?
−2
¯ hImS
?
= exp
?
?
−4πE(t)
¯ h√A′B′
?
,
(10)
which in turn equating with the Boltzmann factor exp
?
−E(t)
T
, the expression for the Hawking temperature is
TH=¯ h?A′(rh, t)B′(rh, t)
4π
(11)
From the above expression it is to be noted that TH is time dependent.
Recently, a drawback of the above approach has been noted [14, 15, 16]. It has been shown that Γ ∼
exp?−2
should be modified as exp−
across the ordinary barrier it is immaterial whether the particle goes from the left to the right or the reverse
path. So in that case
?
and there is no problem of canonical invariance.But difficulty arises for black horizon which behaves as
barrier for particles going from inside of the BH to outside but it does not act as a barrier for particles going
from outside to the inside. So relation (12) is no longer valid. Also using the tunnelling the probability as
Γ ∼ exp?−1
back reaction are taken into account. But unfortunately, no general apprach to account for the above effects
are there in the literature − only few results are available for some known BH solutions [17, 18, 19, 20].
¯ hImS?= exp
?
−2
¯ hIm?rout
?
rin
Im?
prdr
?
?
is not canonically invariant and hence is not a proper observable, it
prdr
¯ h
. The closed path goes across the horizon and back. For tunnelling
prdr = 2
?rout
rin
prdr
(12)
¯ hIm?prdr?, there will be a problem of factor two in Hawking temperature [15, 16].
Further the above analysis of tunnelling approach remain incomplete unless effects of self gravitation and
Finally, it is worthy to mention that so far the above tunnelling approach is purely semiclassical in nature
and quantum corrections are not included. Also this method is applicable for Painleve type coordinates only −
one can not use the original metric co-ordinates to avoid horizon singularity. lastly, the tunnelling approach is
not applicable for massive particles [21].
3 Hamilton-Jacobi Method : Quantum Corrections
We shall now follow the alternative approach as mentioned in the introduction,i.e., the HJ method to evaluate
the imaginary part of the action and hence the Hawking temperature. We shall analyze beyond semiclassical
approximation by incorporating possible quantum corrections. As this method is not affected by the coordinate
singularity at the horizon so we shall use the general BH metric (1) for convenience.
In the back ground of the gravitational field described by the metric (1), massless scalar particles obey the
Klein Gordan equation
¯ h2
√−g∂µ
For spherically symmetric BH as we are only considering radial trajectories so we shall consider (t, r) -sector
in the space time given by equation (1) ,i.e., we concentrate to two-dimensional BH problems. Using (1) the
above Klein -Gordan equation becomes
−
?gµν√−g∂ν
?ψ = 0(13)
∂2ψ
∂t2−
1
2AB
∂
∂t(AB)∂ψ
∂t−1
2
∂
∂r(AB)∂ψ
∂r− AB∂2ψ
∂r2= 0 (14)
Using the standard ansatz for the semiclassical wave function namely
ψ(r, t) = exp
?
−i
¯ hS(r, t)
?
(15)
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the differential equation for the action S is
?∂S
∂t
?2
− AB
?∂S
∂r
?2
+ i¯ h
?∂2S
∂t2− AB∂2S
∂r2−
1
2AB
∂(AB)
∂t
∂S
∂t−1
2
∂(AB)
∂r
∂S
∂r
?
= 0(16)
To solve this partial differential equation we expand the action S in powers of Planck’s constant ¯ h as
S(r,t) = S0(r,t) + Σ¯ hkSk(r,t)(17)
with κ, a positive integer. Note that in the above expansion terms of the order of Planck’s constant and its
higher powers are considered as quantum corrections over the semiclassical action S0. Now substituting the
ansatz (17) for S into (16) and equating different powers of ¯ h on the both sides we obtain the following set of
partial differential equations :
¯ h0
:
?∂S0
∂S0
∂t
∂t
?2
− AB
?∂S0
∂r
?2
= 0(18)
¯ h1
:
∂S1
∂t
− AB∂S0
∂r
∂S1
∂r
+i
2
?∂2S0
?∂S1
∂t2− AB∂2S0
∂r2−
1
2AB
∂(AB)
∂t
∂S0
∂t
−1
2
∂(AB)
∂r
∂S0
∂r
?
= 0(19)
¯ h2
:
?∂S1
∂t
?2
+ 2∂S0
∂t
∂S2
∂t
− AB
∂r
?2
− 2AB∂S0
∂r
∂S2
∂r
+ i
?∂2S1
∂t2− AB∂2S1
∂r2−
1
2AB
∂(AB)
∂t
∂S1
∂t
−1
2
∂(AB)
∂r
∂S1
∂r
?
= 0(20)
and so on.
Apparantly, different order partial differential equations are very complicated but fortunately there will be
lots of simplifications if in the partial differential equation corresponding to ¯ hk, all previous partial differential
equations are used and finally we obtain identical partial differental equation, namely
¯ hk
:
∂Sk
∂t
= ±
?
A(r, t)B(r, t)∂Sk
∂r,
(21)
for k = 0,1,2.......
Thus quantum corrections satisfy same differential equation as the semiclassical action S0.
solutions will be very similar. To solve S0it is to be noted that due to non-static BHs the metric coefficients
are functions of r and t and hence standard Hamilton-Jacobi method can not be applied, some generalization
is needed. We start with a general ansatz [13]
Hence the
S0(r, t) =
?t
0
ω0(t′)dt + D0(r, t)(22)
Here ω0(t) behaves as the energy of the emitted particle and the justification of the choice of the integral is that
outgoing particle should have time dependent continuum energy.
Now substituting the above ansatz for S0(r, t) into equation (18) and using the radial null geodesic in the
usual metric from (1) namely
dr
dt= ±√AB
we have,
∂D0
∂r∂t
i.e.,
dD0
dr
(23)
+∂D0
dt
dr= ∓ω0(t)dt
dr
= ∓ω0(t)
?r
√AB
D0= ∓ω0(t)
0
dr
√AB
(24)
Hence the complete semiclassical action takes the form
S0(r, t) =
?t
0
ω0(t′)dt′∓ ω0(t)
?r
0
dr
√AB
(25)
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Here the′−′(or′+′) sign corresponds to outgoing (or ingoing) particle. As the solution (25) contains an
arbitrary time dependent function ω0(t) so a general solution for Skcan be written as
Sk(r, t) =
?t
0
ωk(t′)dt′∓ ω0(t)
?r
0
dr
√AB
,k = 1, 2, 3,......
(26)
Thus from equation (15) using the solutions (25) and (26) into equation (17) the wave functions for outgoing
and incoming scalar particle can be expressed as
ψout(r, t) = exp
?
−i
¯ h
???t
0
ω0(t′)dt′+ Σk¯ hk
?t
0
ωk(t′)dt′
?
−
?
ω0(t) + Σk¯ hkωk(t)
??r
0
dr
√AB
??
(27)
and
ψin(r, t) = exp
?
−i
¯ h
???t
0
ω0(t′)dt′+ Σk¯ hk
?t
0
ωk(t′)dt′
?
+
?
ω0(t) + Σk¯ hkωk(t)
??r
0
dr
√AB
??
(28)
respectively. Due to tunnelling across the horizon there will be a change of sign of the metric coefficients in the
(r, t)-part of the metric and as a result function of t coordinate has an imaginary part which will contribute to
the probabilities. So we write
Pin= |ψin(r, t)|2= exp
?2Im
¯ h
???t
0
ω0(t′)dt′+ Σk¯ hk
?t
0
ωk(t′)dt′
?
+
?
ω0(t) + Σk¯ hkωk(t)
??r
0
dr
√AB
??
(29)
and
Pout= |ψout(r, t)|2= exp
?2Im
¯ h
???t
0
ω0(t′)dt′+ Σk¯ hk
?t
0
ωk(t′)dt′
?
−
?
ω0(t) + Σk¯ hkωk(t)
??r
0
dr
√AB
??
(30)
To have some simplification we shall now use the physical fact that all incoming particles certainly cross the
horizon, i.e., Pin= 1.So from equation (29)
Im
??t
0
ω0(t′)dt′+ Σk¯ hk
?t
0
ωk(t′)dt′
?
= −Im
?
ω0(t) + Σk¯ hkωk(t)
??r
0
dr
√AB
(31)
and hence Poutsimplifies to
Pout= exp
?
−4
¯ h
?
ω0(t) + Σk¯ hkωk(t)
?
Im
?r
0
dr
√AB
?
(32)
Then from the principle of ”detailed balance” [10, 11, 12] we write
Pout= exp
?
−ω0(t)
Th
?
Pin= exp
?
−ω0(t)
Th
?
(33)
So comparing (32) and (33), the temperature of the BH is given by
Th=¯ h
4
?
1 + Σk¯ hkωk(t)
ω0(t)
?−1?
Im
?r
0
dr
√AB
?−1
(34)
where
TH=¯ h
4
?
Im
?r
0
dr
√AB
?−1
(35)
is the usual Hawking temperature of the BH. Thus due to quantum corrections the temperature of the BH
is modified from the Hawking temperature and both the temperatures are functions of′t′and′r′. Note that
equation(35) is the standard expression for semiclassical Hawking temperature and it is valid for non-spherical
metric also. However for spherical metric, one can use the Taylor series expansions (6) and (7) near the horizon
and obtain THas given in equation (11) by performing the contour integration. The ambiguity of factor of two
(as mentioned earlier) in the Hawking temperature does not arise here.
5