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Scalar field radiation from dilatonic black holes

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Abstract

We study radiation of scalar particles from charged dilaton black holes. The Hamilton-Jacobi method has been used to work out the tunneling probability of outgoing particles from the event horizon of dilaton black holes. For this purpose we use WKB approximation to solve the charged Klein-Gordon equation. The procedure gives Hawking temperature for these black holes as well.
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General Relativity and Gravitation
ISSN 0001-7701
Volume 44
Number 12
Gen Relativ Gravit (2012) 44:3163-3167
DOI 10.1007/s10714-012-1449-x
Scalar field radiation from dilatonic black
holes
H.Gohar & K.Saifullah
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Gen Relativ Gravit (2012) 44:3163–3167
DOI 10.1007/s10714-012-1449-x
RESEARCH ARTICLE
Scalar field radiation from dilatonic black holes
H. Gohar ·K. Saifullah
Received: 9 March 2012 / Accepted: 18 August 2012 / Published online: 7 September 2012
© Springer Science+Business Media, LLC 2012
Abstract We study radiation of scalar particles from charged dilaton black holes.
The Hamilton–Jacobi method has been used to work out the tunneling probability of
outgoing particles from the event horizon of dilaton black holes. For this purpose we
use WKB approximation to solve the charged Klein–Gordon equation. The procedure
gives Hawking temperature for these black holes as well.
Keywords Quantum tunneling ·Scalar particles ·Dilaton black holes
Quantum mechanical effects combined with gravity theories present a picture of black
holes that emit radiations and can evaporate [1,2]. Different techniques have been
developed to study these radiations from a variety of black configurations [318]. Four-
and five-dimensional dilaton black holes have also been studied for Hawking radiations
using different techniques [1922]. In this paper we study emission of scalar particles
from charged dilaton black holes. For this purpose we use the Hamilton–Jacobi method
and apply WKB approximation to the Klein–Gordon equation to calculate the imag-
inary part of the classical action for outgoing trajectories across the horizon. WKB
approximation has widely been used to calculate the tunneling probability of particles
and Hawking temperature of black holes. This approximation is valid in the range
where the size of the particle is much smaller than that of the black hole and thus can
be treated as point-like. After working out the tunneling probability for the classically
forbidden trajectory, we compare this with the Boltzmann factor =exp (βE),for
H. Gohar
National Centre for Physics, Islamabad, Pakistan
K. Saifullah (B
)
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
e-mail: saifullah@qau.edu.pk
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3164 H. Gohar, K. Saifullah
particle of energy E. We obtain Hawking temperature for the black hole also, as βis
the inverse of the horizon temperature [6,7].
Dilaton black hole is a solution of Einstein’s field equations in which charged
dilaton field is coupled with the Maxwell field. Dilaton is a scalar field, which occurs
in low energy limit of the string theory in which the fields like axion and dilaton are
incorporated in Einstein’s action. The four dimensional Langragian in low energy is
given by [19,22]
S=dx4gR+2()+e2aF2,(1)
where a, the coupling parameter, denotes the strength of the coupling of the dilaton
field to the Maxwell field F,gis the determinant of the metric tensor gμν and Ris
the Ricci scalar. The line element for charged and spherically symmetric dilaton black
hole is given by [19,22]
ds2=−e2u(r)dt2+e2u(r)dr2+R2(r)dθ2+sin2θdφ2,(2)
where
e2u(r)=1r+
r1r
r1a2
1+a2,(3)
R(r)=r1r
ra2
1+a2.(4)
Here, and Fare given as
e2=1r
r2a
1+a2,(5)
F=Q
r2dt dr.(6)
The ADM mass Mand electric charge Qof the dilaton black hole are given by
M=r+
2+r
21a2
1+a2,(7)
Q2=r+r
1+a2.(8)
The outer and inner horizons, r+and r, of the dilaton black hole are given by
r±=1+a2
1±a2M±M2(1a2)Q2.(9)
Here ais confined in the interval 0 a1. When a=0, the metric reduces to
the Reissner–Nordström solution. The electric potential of the dilatonic black hole is
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Scalar field radiation from dilatonic black holes 3165
given as
Aμ=Atdt =Q
rdt.(10)
To deal with scalar tunneling we use the charged Klein–Gordon equation for scalar
field, =(t,r
), given by
1
gμiq
¯
hAμggμυ (∂νiq
¯
hAν)m2
2=0,(11)
where qand mare the charge and mass of the scalar particle and ¯
his Planck’s constant.
To apply WKB approximation in lowest order we choose the scalar field of the form
(t,r) =ei
I(t,r,θ,φ)+I1(t,r,θ,φ)+O().(12)
Here Iis the action for the outgoing trajectory. Substituting Eq. (12)in(11)inthe
lowest order in , dividing by the exponential term and multiplying by 2, yields
0=gtt(∂tIqA
t)2+grr(∂rI)2+gθθ (∂θI)2+gφφ(∂φI)2+m2.(13)
We note that tand φare the only Killing fields for the spacetime at hand (Eq. (2)).
So we can assume the following separation of variables for the action
I=−Et +W(r)+Jφ+K,(14)
where E,Jand Kare constants; Eand Jrepresent the energy and the angular
momentum of the emitted particle and Kcan be complex also. It is pertinent to mention
here that we are only considering radial trajectories. This is because in the tunneling
approach of Hawking radiation the particles are locally considered to follow θ=
constant geodesics. Further, in the case of spin-1/2 particles (like fermions) we deal
with spin-up and spin-down cases separately and correspondingly get two equations
(see, for example, Refs. [11,16,17]). For scalars this is not the case and we have only
one equation. Using Eq. (14)inEq.(13), and solving for W(r)for θ=θ,gives
W(r)
(E+qA
t)21r+
r
1r
r
12a2
1+a2
sin θ(J2)2+1r
r1a2
1+a2(m2)
1r+
r1r
r1a2
1+a2
dr.
With the obvious substitution we can write the above integral as
W(r)f(r)
rr+
dr.(15)
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3166 H. Gohar, K. Saifullah
Thus we have a simple pole at r=r+. We evaluate this integral around the pole at the
outer horizon by using the residue theory for semi circles. This yields
W±(r)πir+(E+qA
t)
1r
r+1a2
1+a2
.(16)
Here ‘+’ and ‘’ represent the outgoing and incoming trajectories, respectively. The
above equations implies that
ImW+=πr+(E+qA
t)
1r
r+1a2
1+a2
.(17)
The tunneling probabilities of crossing the horizon from inside to outside and outside
to inside are given by [6,7]
Pemissi on exp 2
¯
hImI=exp 2
¯
h(ImW++ImK),(18)
Pabsorption exp 2
¯
hImI=exp 2
¯
h(ImW+ImK).(19)
An incoming particle will definitely cross the horizon and fall into the black hole,
therefore, to normalize the probability of the incoming particle we must have
ImK =−ImW,(20)
in Eqs. (18) and (19). From Eq. (16) we note that
W+=−W.(21)
This means that the probability of a particle tunneling from inside to outside the
horizon is given by
=exp 4
¯
hImW+.(22)
By putting the value of ImW+from Eq. (17) into Eq. (22), the tunneling probability
comes out to be
=exp
4πr+(E+qA
t)
¯
h1r
r+1a2
1+a2
.(23)
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Scalar field radiation from dilatonic black holes 3167
By comparing with the Boltzmann factor this gives us Hawking temperature as
TH=1
4πr+1r
r+1a2
1+a2
,(24)
where r+and rare the outer and inner horizons of the black hole. This formula is
consistent with the previous literature [19,20].
Acknowledgments We are thankful for the referee’s comments which helped us improve our manuscript.
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