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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 [3–18]. Four-

and ﬁve-dimensional dilaton black holes have also been studied for Hawking radiations

using different techniques [19–22]. 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=dx4√−g−R+2(∇)+e−2aF2,(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+e−2u(r)dr2+R2(r)dθ2+sin2θdφ2,(2)

where

e2u(r)=1−r+

r1−r−

r1−a2

1+a2,(3)

R(r)=r1−r−

ra2

1+a2.(4)

Here, and Fare given as

e2=1−r−

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−

21−a2

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−(1−a2)Q2.(9)

Here ais conﬁned in the interval 0 ≤a≤1. 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,θ,φ

), given by

1

√−g∂μ−iq

¯

hAμ√−ggμυ (∂ν−iq

¯

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,θ,φ) =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(∂tI−qA

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,θ)+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,θ)for θ=θ◦,gives

W(r)=±

(E+qA

t)2−1−r+

r⎛

⎝1−r

−

r

1−2a2

1+a2

sin θ◦(J2)2+1−r−

r1−a2

1+a2(m2)⎞

⎠

1−r+

r1−r−

r1−a2

1+a2

dr.

With the obvious substitution we can write the above integral as

W(r)=±f(r)

r−r+

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)

1−r−

r+1−a2

1+a2

.(16)

Here ‘+’ and ‘−’ represent the outgoing and incoming trajectories, respectively. The

above equations implies that

ImW+=πr+(E+qA

t)

1−r−

r+1−a2

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)

¯

h1−r−

r+1−a2

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+1−r−

r+1−a2

1+a2

,(24)

where r+and r−are 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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