CHARGED FERMIONS TUNNELING FROM REGULAR BLACK HOLES
M. Sharif* W. Javed**
Department of Mathematics, University of the Punjab 54590, Lahore, Pakistan
Received February 15, 2012
We study Hawking radiation of charged fermions as a tunneling process from charged regular black holes, i.e., the Bardeen and ABGB black holes. For this purpose, we apply the semiclassical WKB approximation to the general covariant Dirac equation for charged particles and evaluate the tunneling probabilities. We recover the Hawking temperature corresponding to these charged regular black holes. Further, we consider the back-reaction effects of the emitted spin particles from black holes and calculate their corresponding quantum corrections to the radiation spectrum. We find that this radiation spectrum is not purely thermal due to the energy and charge conservation but has some corrections. In the absence of charge, e = 0, our results are consistent with those already present in the literature.
1. INTRODUCTION
Classically, a black hole (BH) is considered to absorb all matter and energy in the surrounding region into it due a strong gravitational field. Bekenstein fl] was the first to discuss the BH thermodynamics. Later, Hawking [2] investigated BH thcrmodynamical properties and proposed [3] that a BH could emit black-body radiation. According to this, a particle antiparticle pair appears near the event horizon of a BH due to vacuum fluctuations. In order to preserve the total energy, one member of the pair with negative energy must fall into the BH while the other escapes with positive energy. In this process, the BH loses mass and it appears to an outside observer that the BH has just emitted a particle. This semiclassical process is called quantum tunneling [4, 5]. In this approach, particles follow classically forbidden trajectories from inside the horizon to infinity, for which the action becomes complex. This means that the tunneling probability for the outgoing particle is governed by the imaginary part of this action. Because a particle can classically only fall inside the horizon, the action for the ingoing particle must be real.
There are two different methods to evaluate the imaginary part of the action. One is the semiclassi-
E-mail: msharif.math'fflpu.edu.pk
**E-mail: wajihajaved84(&yahoo.com
cal Worn /.el Kramers Brillouin (WKB) approximation method, first used in fC, 7], and the other is the radial null-geodesic method [5]. These methods have been used to evaluate the tunneling probabilities of quantum fields passing through an event horizon. Different semiclassical approaches have been adopted to evaluate tunneling of scalar and Dirac particles (charged and uncharged). In Refs. [8, 9], the tunneling of spin-1/2 particles through event horizons of the Rindler spacetime was investigated and Unruh temperature was obtained. In these papers fermion tunneling from the general non-rotating BH as well as the Kerr Newman BH was also discussed and their corresponding Hawking temperatures was recovered.
Fermions tunneling from the Kerr BH were investigated in [10] by applying the WKB approximation to the general covariant Dirac equation, which allowed finding the Hawking temperature for the Kerr BH. Charged fermion tunneling from dilatonic BHs, the rotating Einstein Maxwell dilaton Axion BH, and a rotating Ivaluza Klein BH were studied in [11] and their corresponding Hawking temperatures were recovered. Hawking radiation of spin-1/2 particles from the Reissner Nordstrom BH was investigated in [12] using the Dirac equation for charged particles. The tunneling of scalar and Dirac particles from the Kerr Newman BH was explored in [13] and its Hawking temperature was obtained. The semiclassical fermion tunneling from the Kerr Newman Ivasuya BH was studied
in [14] and the Hawking temperature was obtained. Some work has also been done for three-dimensional spacetimes [15].
Tunneling of charged fermions from accelerating and rotating BHs with electric and magnetic charges have been studied in [16 18] using the WKB approximation. Tunneling probabilities of charged fermions and the corresponding Hawking temperature were found. In recent papers [19], the tunneling probabilities of incoming and outgoing scalar and charged/'uncharged fcrmion particles from accelerating and rotating BHs have been investigated. Recently, we have examined the radiation spectrum of an RN-liko nonconiniutativc BH [20] by quantum tunneling process (radial null geodesic method). Also, we have investigated quantum corrections of regular BHs [21, 22].
In this paper, we use the procedure in [8] to investigate the tunneling probabilities of charged fermions for charged regular BHs, i.e., the regular Bardeen and regular Ayon-Beato Garcia Bronnikov (ABGB) BHs. We recover the corresponding Hawking temperatures for charged massive as well as massless fermions. Also, we explore the radiation spectrum by using the radial null-geodesic method [12]. This paper is organized as follows. In Sec. 2, we review the basic formalism for the pure thermal spectrum of charged fermions using the Dirac equation for charged particles. Section 3 is devoted to the study of fermion tunneling from the regular Bardeen and ABGB BHs. In Sec. 4, we discuss the correction spectrum of charged fermions due to back-reaction effects. Finally, Sec. 5 summarizes the results.
2. REVIEW: TUNNELING OF CHARGED FERMIONS
In this section, we briefly review some basic material used to evaluate the tunneling probabilities of charged fermions. For this purpose, we apply the WKB approximation to the general covariant Dirac equation for charged particles. The line element of a spherically symmetric BH can be written as
charge q is given by [9]
n'1 ( Dfl
tq
A, ) *
m
$ = 0.
(2.2)
//,i/ = 0,1,2,3,
where m is the rilciSS of fermion particles, Afl is the 4-po-tential, 'I' is the wave function, and 7,J are the Dirac matrices [14]. The antisymmetric property of the Dirac matrices, i.e.,
[7°, 7"] =
0.
m = 3, a ± /?,
—[y* 5 y
reduces Dirac equation (2.2) to the form
■n'1 ( d, - -$A,_
m
T
$ = 0.
(2.3)
The spinor wave function 'I' has two spin states: spin-up (radially outward, i.e., in positive r-dircction) and spin-down (radially inward, i.e., in negative r-dircction). The solutions for spin-up and spin-down particles arc respectively given by [8]
%(i,r, 0,<P) =
\I/4. 0,0) =
Mj.r.H.o) 0
B(t,r,e,<p) 0
x cxp
0
C(t.r.H.o) 0
D(t,r,e,<p) x cxp
:If(t,r,e,<p)
(2.4)
Th(t,r,e,<i>)
(2.5)
where is the action of the emitted spin-up /'spin-down particles. I11 what follows, we discuss the spin-up case in detail; the spin-down case follows in a similar fashion. Using Eq. (2.4) in Dirac equation (2.3), we obtain the set of equations
ds2 = —F dt2
where
F 1dr2 + r2(I02 + r2 sin2 9 d(f)2, (2.1)
F = 1
M(r)
This metric can be reduced to well-known BHs for special choices of M(r). The Dirac equation with electric
iA
sfF^i
-.¡hi- - BsjTÎT);),.!.
r sin 0
iA
--qA0
30It
= 0.
sfFÏn + m A = 0, (2.6)
(2.7)
IB
-.dtJî —
IB
,qA0
vW
+ m,B = 0
-.4
'-do if
r sin 0
d0it
= 0.
(2.9)
To find the action from the above equations, we use separation of variables in accordance with
It = —Et + W'"(r) + J(0, <t>),
(2.10)
where E and J denote the energy and angular momentum of the emitted particle, and W is an arbitrary function of r. Inserting this value of the action in Eqs. (2.6) (2.9), we also use Taylor's expansion to expand F(r) near the outer horizon r+, neglecting squares and higher powers. Substituting the values of A0(r+) and setting L4 = B and IB = ,4 in the above set of equations, we obtain
B
-E - qA0
y/(r-r+)F'(r+)W
ni A = 0, (2.11)
i
-d0J , . a
r r Sill t>
ditJ
= o,
(2.12)
.4
-E - qAo
s/(r -r+)F'(r+)
y/(r-r+)F'(r+)W
m H 0. (2.13)
7
-do-J , . a
r r sm 0
ditJ
= o,
(2.14)
wliere tlie prime denotes tlie derivativo witli respect to r. Equations (2.12) and (2.14) yield
--'—¡K,l 0
r r sm 0
(2.15)
which implies
J = ox\\iko_
Ci / ese 8 (19 + c-2
(2.16)
where k, c\, and c2 are arbitrary functions of 8 and <f>. This quantity must be same for both outgoing and incoming cases. As a result, it cancels from the formula for the tunneling probability from inside to outside the horizon (which is the ratio of outgoing and incoming modes [8]).
In the massless case (m = 0), Eqs. (2.11) and (2.13) yield the respective solutions
W'(r) = WUr) =
W'(r) = WL(r) =
E + qAo
(r — r+ )F'(r+ ) '
E + qAo (r — r+)F'(r+) '
(2.17)
(2.18)
where W"+ correspond to the outgoing/incoming solutions. The tunneling probability of a particle going from outside to inside the horizon is equal to unity [9]. Also, Eqs. (2.17) and (2.18) lead to
Im = — Im W-.
Hence, the overall tunneling probability of the outgoing particle turns out to be
r =
Probfout] exp[^2(Im M"+)] Probfin] exp[^2(Im W- )]
= oxp[—41m W+],
(2.19)
We can recover the Hawking temperature Th from the relation as
1
F = oxp[—/?£'], 3 =
TH
In the massive case (m ^ 0), Eqs. (2.11) and (2.13) no longer decouple. We eliminate the function W from these two equations by respectively multiplying Eqs. (2.11) and (2.13) with ,4 and B. After some manipulations, it follows that
A ~B
-(E + qAo) ± \J(E + qAo)2 + m2(r - r+)F'(r+)
m
s/{r -r+)F'{r+)
(2.20)
The limit r r+ yields either A/B 0 or A/B —^ —oo, i. e., either .4->0 or B 0. For ,4 —^ 0, we can evaluate the value of m from Eq. (2.13) as
m =
A ~B
-s/(r -r+)F'(r+)W'(r) + -(E + qAo)
s/{r -r+)F'{r+)
(2.21)
Inserting this value in Eq. (2.11) and simplifying, we obtain the same value of M"^(r) as in Eq. (2.17). Similarly, for B —¥ 0, the same expression for W_(r) is found as in (2.18). Consequently, the Hawking temperature turns out to be the same as in the massless case. In the spin-down case, for both massive and massless fcrmions, the Hawking temperature remains the same as for the spin-up case. Thus, both spin-up and spin-down particles are emitted at
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