A NEW METHOD TO STUDY THE HAWKING RADIATION
OF THE CHARGED BLACK HOLE WITH A GLOBAL MONOPOLE
LIU JING-JING, CHEN DE-YOU, YANG SHU-ZHENG
Institute of Theoretical Physics, China West Normal University, Nanchong, 637002, China
E-mail: liujingjing68@126.com; deyouchen@126.com; szyangcwnu@126.com
Received August 21, 2007
Applying Hamilton-Jacobi method, we investigate the tunneling characteristic of the charged black hole with a global monopole by taking the self-gravitational interaction as well as the energy conservation and charge conservation into account.
The radiation spectrum deviates from the purely thermal one and the tunneling probability is related to the change of Bekenstein-Hawking entropy. The result satisfies an underlying unitary theory and gives a correction to the Hawking radiation.
Key word: self-gravitation; charged particle; tunneling probability.
1. INTRODUCTION
As is well known, black hole is one of the most meaningful prophecies of general relativity theory [1]. The black hole always plays the significant role in physics and astronomy due to the people’s views on the space and time, matters and gravitation. In 1974, Hawking proved the black hole radiates thermal and the radiation spectrum is purely thermal, which plays key role on the cognition and research on black holes [2]. Recently, a semi-classical method to describe the
Hawking radiation as a tunneling process, where a particle moves in dynamical geometry, was proposed by Kraus, Parikh and Wilczek [3–7]. In their works, they pointed out the potential barrier is created by the outgoing particle itself; thereby the cause of the mechanism corresponding to the tunneling potential hill has been resolved. The energy conservation and the unfixed background space-time are taken into account. The radial null geodesics of the emitted particle and to calculate the imaginary part of the particle’s action were found in their studies. Utilizing the WKB approximation and Hamilton equation, they derived the relationship between the tunneling probability and the classical action of the particle. According to this method, quite a few fruits have been achieved and all of their results supported Parikh and Wilczek’s opinion.
However, these researches were focused on the Hawking radiation of massless particles. In 2005, Zhang and Zhao [8, 9] and Yang [10–13] extended the
Rom. Journ. Phys., Vol. 53, Nos. 5– 6 , P. 659–664, Bucharest, 2008

660 Liu Jing-Jing, Chen De-You, Yang Shu-Zheng 2 method to charged particles and made a great deal of success. In the same year, a different method was introduced on this subject by Angheben, Nadalini, Vanzo and Zerbini [14]. But they lost sight of the self-gravitation of the particle so that the derived radiation spectrum is purely thermal.
In this paper, taking the self-gravitation, energy conservation and charge conservation into account, we discuss the tunneling characteristic of the charged black hole with a global monopole by extending Angheben’s method. The derived result is in accordance with Parikh and Wilczek’s result. The remainders of this paper are organized as follows. In the next section, extending Angheben’s method to the sphere symmetry space-time, we obtain the classical action of the particle and the temperature of the black hole when the self-gravitation and unfixed background space-time aren’t considered. Taken the self-gravitation as well as the energy conservation, charge conservation and angular momentum conservation into account, the exact expression of emission spectrum will be shown in Section 3. Some concluding remarks are given in Section 4.
2. HAMILTON-JACOBI EQUATION AND HAWKING RADIATION
According to [15], the line element of the charged black hole with a global monopole is given by ds
2
§
©
1 8 r
2 d sin
2
2 M r
Q 2 r 2
·
¹ dt
2
2
,
§
©
1 8
2 r
M Q 2 r 2
·
¹
1 dr
2
(1) with the electromagnetic potential
A
P
Q r
, 0, 0, 0 ,
¹
(2) where K is the global monopole, M and Q are the mass and charge of the black hole respectively. According to g
00
= 0 and the null super-surface equation g
PQ w
P f w v f 0, we can get the infinite red-shift surface and the horizons of the black hole r
TLS r r
M r M
2
1 8
2
1 8
2
Q
2
, which are coincident with each other. This is helpful for us to discuss the Hawking radiation of the black hole.
The classical action I of the emitted particle tunnels across the event horizon satisfies the relativistic Hamilton-Jacobi equation is written by g
PQ w
P
I qA
P w
Q
I qA
Q u 2 0, (3)

3 Hawking radiation of the charged black hole 661 where u and q are the mass and charge of the emitted particle respectively.
Ordering g PQ
2 M Q
2
1 8 and substituting the inverse metric tensors r r
2
obtained from the line element (1) into Eq. (3), we have
1 f r
I qA t
2 f r w r
I
2 r
1
2 w
T
I
2 r 2
1 sin 2 T w
M
I
2 u
2
0.
(4)
Considering the symmetry of the black hole, we carry on the following separation variable as
I t R r H , (5) in which Z is the energy of the emitted particle. R ( r ), H ( T ) and ) ( M ) are the generalized momentums. Substituting Eq. (5) into Eq. (4), we get w R r w r
1 f r ©
Z qQ ·
2 r f r r
^
1
2 w
T
H T
2 r
1
2 sin 2 T
¼
2 u
2
,
`
(6) from which we learn that the imaginary part of the emitted particle’s action is only produced from the pole at the event horizon. According to [16], the proper spatial distance is introduced and defined by the spatial metrics as d f r
1 dr
2 r d T r
2 sin
2 d
2
.
(7)
Limiting to the s-wave contribution that contains the bulk of particle emission yields
V ³ dr f r
.
(8)
Employing the near-horizon approximation f r f r r r } , we can get
V
2 f
, r r r r ,
1 f r
2 f
, r r where f
, r
2 M r
2
R V
2 f
, r r u ³ d
V
V u
©
Z qQ r ¹
2
2 Q
2 r
3
. From Eqs. (6)–(10), we obtain f r r
^
1
2 w
T
H T
2 r
2
1 sin
2 T
2 u
2
.
`
(9)
(10)
(11)

662 Liu Jing-Jing, Chen De-You, Yang Shu-Zheng
Where the solution is singular at V = 0 which corresponds to the event horizon. Now it is very convenient to obtain the result from Eq. (11). Utilizing the Feynman prescription and regularizing the singularity by using the contour, we can obtain the integral result. So the action I of the emitted particle is
I
2 S i f
, r r
§
©
Z qQ r
·
¹
(real part), and the temperature over the surface of the black hole is
(12)
T f
, r
4 r
S 2
1
§
2 r
S
2
2 Q
2 r
3
·
¸
¹
.
(13)
In this Section, the action I of the particle has been obtained under the condition which the self-gravitational interaction and the unfixed background space-time weren’t taken into accounted. In the following, we will correct the action and explore the tunneling probability by incorporating these.
4
3. TUNNELING PROBABILITY
Considering energy conservation and charge conservation as well as the self-gravitational interaction of the emitted particle, we fix the ADM mass and charge of the total space-time and allow these of the black hole to fluctuate.
When the particle with energy Z and charge q tunnels out, the mass M and charge Q of the black hole should be replaced by M – Z and Q – q . So the correct action I should be dI
M r c c
S ir c
Q q c
2
¬
ª
« r c 2 d
Q q c r c dq c
º
»
¼
, (14) where r c
M M Z c
2
1 8
2
1 8
2
Q q c
2
.
(15)
Substituting Eq. (15) into (14), the imaginary part of the emitted particle can be expressed as
Im I S
Z , q
³
0,0
M r c c r c
Q q c
2
¬
ª
« d Z r c 2
Q q c r c dq c
º
»
¼
.
(16)

5 Hawking radiation of the charged black hole 663
Finishing the integral, we have
Im I
­ª
S «
2
¯
¬
M Z
«
¬
ª
«
M M
2
1 8
2
M Z
2
1 8
2
1 8
2
2
Q
2
»
»
º
»
2
¾
¼ ¿
½
°
Q
1
2
' S
BH q
,
2
»
¼
º
»
2
(17) in which ' S
BH
S
BH
M Z , Q q S
BH
, is the change of Bekenstein-
Hawking entropy. Therefore the relationship between the imaginary part of the emitted particle and the tunneling probability can be expressed as
* ~ exp 2 Im I exp ' S
BH
.
(18)
The result shows the actual radiation spectrum deviates from the purely thermal one, and the tunneling probability is related to the change of
Bekenstein-Hawking entropy, which gives a correction to the Hawking radiation of the black hole. When K = 0 and Q = 0, the line element (1) is reduced to that of the static Schwarzschild black hole. Now the tunneling probability can be expressed as
* ~ exp 2 Im I exp
ª
SZ M
Z
2
º
¼ exp ' S
BH
, which is full accordant with Parikh and Wilczek’s result.
(19)
4. CONCLUSIONS
In this paper, taking the self-gravitation of emitted particle as well as the conservation of energy and charge, the tunneling characteristic of charged particle via the sphere symmetry black hole with a global monopole is discussed by extending the work of Angheben et al.
The result shows that the derived radiation spectrum deviates from the purely thermal one and the tunneling probability is related to the change of Bekenstein-Hawking entropy. Such result satisfies an underlying unitary theory and also gives a correct correction to
Hawking radiation.
Acknowledgment.
The work is supported by the Sichuan province science and technology department foundation for fundamental research (Grant No.05JY029-092).

664 Liu Jing-Jing, Chen De-You, Yang Shu-Zheng
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