ArticlePDF Available

Scalar field radiation from dilatonic black holes



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.
1 23
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
H.Gohar & K.Saifullah
1 23
Your article is protected by copyright and
all rights are held exclusively by Springer
Science+Business Media, LLC. This e-offprint
is for personal use only and shall not be self-
archived in electronic repositories. If you
wish to self-archive your work, please use the
accepted author’s version for posting to your
own website or your institution’s repository.
You may further deposit the accepted author’s
version on a funder’s repository at a funder’s
request, provided it is not made publicly
available until 12 months after publication.
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
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
Author's personal copy
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]
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]
Here, and Fare given as
r2dt dr.(6)
The ADM mass Mand electric charge Qof the dilaton black hole are given by
The outer and inner horizons, r+and r, of the dilaton black hole are given by
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
Author's personal copy
Scalar field radiation from dilatonic black holes 3165
given as
Aμ=Atdt =Q
To deal with scalar tunneling we use the charged Klein–Gordon equation for scalar
field, =(t,r
), given by
hAμggμυ (∂νiq
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
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
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
sin θ(J2)2+1r
With the obvious substitution we can write the above integral as
Author's personal copy
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
Here ‘+’ and ‘’ represent the outgoing and incoming trajectories, respectively. The
above equations implies that
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
Pabsorption exp 2
hImI=exp 2
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
This means that the probability of a particle tunneling from inside to outside the
horizon is given by
=exp 4
By putting the value of ImW+from Eq. (17) into Eq. (22), the tunneling probability
comes out to be
Author's personal copy
Scalar field radiation from dilatonic black holes 3167
By comparing with the Boltzmann factor this gives us Hawking temperature as
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.
1. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975)
2. Hawking, S.W.: Black hole explosions? Nature 248, 30 (1974)
3. Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation.
Phys. Rev. D 15, 2738 (1977)
4. Iso, S., Umetsu, H., Wilczek, F.: Anomalies, Hawking radiations and regularity in rotating black holes.
Phys. Rev. D 74, 044017 (2006)
5. Parikh, M.K., Wilczek, F.: Hawking radiation as tunnelling. Phys. Rev. Lett. 85, 5042 (2000)
6. Srinivasan, K., Padmanabhan, T.: Particle production and complex path analysis. Phys. Rev. D 60,
24007 (1999)
7. Shankaranarayanan, S., Padmanabhan, T., Srinivasan, K.: Hawking radiation in different coordinate
settings: complex paths approach. Class. Quantum Grav. 19, 2671 (2002)
8. Akhmedova, V., Pilling, T., de Gill, A., Singleton, D.: Temporal contribution to gravitational WKB-like
calculations. Phys. Lett. B 666, 269 (2008)
9. Akhmedov, E.T., Akhmedova, V., Singleton, D.: Hawking temperature in the tunneling picture. Phys.
Lett. B 642, 124 (2006)
10. Kerner, R., Mann, R.B.: Tunnelling, temperature and Taub-NUT black holes. Phys. Rev. D 73, 104010
11. Kerner, R., Mann, R.B.: Charged Fermions tunnelling from Kerr-Newman black holes. Phys. Lett. B
665, 277 (2008)
12. Vagenas, E.C., Das, S.: Gravitational anomalies, Hawking radiation, and spherically symmetric black
holes. JHEP 10, 025 (2006)
13. Yale, A.: Exact Hawking radiation of scalars, fermions, and bosons using the tunneling method without
back-reaction. Phys. Lett. B 697, 398 (2011)
14. Ding, C., Jing, J.: Deformation of contour and Hawking temperature. Class. Quantum Grav.27, 035004
15. Gillani, U.A., Rehman, M., Saifullah, K.: Hawking radiation of scalar partcles from accelerating and
rotating black holes. JCAP 06, 016 (2011)
16. Gillani, U.A., Saifullah, K.: Tunneling of Dirac particles from accelerating and rotating Black holes.
Phys. Lett. B 699, 15 (2011)
17. Ahmed, J., Saifullah, K.: Hawking temperature of rotating charged black strings from tunneling. JCAP
11, 023 (2011)
18. Gohar, H., Saifullah, K.: Emission of scalar particles from cylindrical black holes. arXiv:1109.5836
19. Liu, C.Z., Zhang, J.Y., Zhao, Z.: Charged particle’s tunneling from a dilaton black hole. Phy. Lett. B
639, 670 (2006)
20. Chen, D.Y., Jiang, Q.Q., Zu, X.T.: Fermions tunnelling from the charged dilatonic black holes. Class.
Quantum Grav. 25, 205022 (2008)
21. Becar, R., González, P.A.: Hawking radiation for scalar and Dirac fields in five dimensional dilatonic
black hole via anomalies. arXiv:1104.0356
22. Gibbons, G.W., Maeda, K.: Black holes and membranes in higher dimensional theories with dilaton
fields. Nucl. Phys. B 298, 741 (1988)
Author's personal copy
... In this way Wick rotation Method (Gibbons and Hawking, 1977a;1977b), Anomaly method (Iso et al., 2006) and methods of dimensional reduction (Umetsu, 2010) has been widely used by different authors to investigate the Hawking radiations from different black holes. Similarly Hawking radiation as quantum tunneling effect at event horizons of the black holes is discussed by different authors and quantum tunneling method (Parikh and Wilczek, 2000;Padmanabhan, 2004;Srinivasan and Padmanabhan, 1999;Shankaranarayanan et al., 2002;Kraus and Wilczek, 2009;1995;Kerner and Mann, 2006;2007;Rehman and Saifullah, 2011;Gillani and Saifullah, 2011;Ahmed and Saifullah, 2011;Gohar and Saifullah 2012a, 2012bJan and Gohar, 2013) is robust method to discuss these radiation. We can apply this method to almost all type of black holes in different gravity theories from higher dimensions to lower dimensions (Ejaz et al., 2013;Vagenas, 2002;Medved and Vagenas, 2005;Matsuno and Umetsu, 2011;Kim, 2011;Hod, 2011;Wu and Peng, 2011). ...
... Recently, Cai et al. [32] investigated Hawking radiation of an apparent horizon in a Friedmann-Robertson-Walker universe using the Hamilton-Jacobi method. Using the same method, Gohar and Saifullah [33] investigated scalar field radiation from dilatonic black holes. In this paper, a five-dimensional Lovelock black hole is considered to investigate Hawking radiation by including the influence of the ultraviolet correction to the black hole. ...
Full-text available
We investigate Hawking radiation from a five-dimensional Lovelock black hole using the Hamilton–Jacobi method. The behavior of the rate of radiation is plotted for various values of the ultraviolet correction parameter and the cosmological constant. The results show that, owing to the ultraviolet correction and the presence of dark energy represented by the cosmological constant, the black hole radiates at a slower rate in comparison to the case without ultraviolet correction or cosmological constant. Moreover, the presence of the cosmological constant makes the effect of the ultraviolet correction on the black hole radiation negligible. © 2015, Higher Education Press and Springer-Verlag Berlin Heidelberg.
... Similarly, Kerner and Mann (2006 extended this work to large number of black holes and they showed that tunnelling method is very powerful method, which can be applied to variety of black holes. Following the work of Kerner and Mann, recently, this method is applied to higher dimensional black holes (Berti et al. 2011;Hod 2011;Wu and Peng 2011), black holes in String theory (Kim 2011), black strings Saifullah 2011a, 2011b;Saifullah 2012a, 2013), accelerating and rotating black holes Rehman and Saifullah 2011), dilaton black holes (Gohar and Saifullah 2012b;Chen et al. 2008b), three dimensional black holes (Li and Ren 2008;Ejaz et al. 2013) and black holes with NUT parameter (Kerner and Mann 2006;Sharif and Javed 2012). In these papers, authors have discussed the emission of fermions and scalar particles from black holes. ...
Full-text available
We study the quantum tunneling of scalars from charged accelerating and rotating black hole with NUT parameter. For this purpose we use the charged Klein-Gordon equation. We apply WKB approximation and the Hamilton-Jacobi method to solve charged the Klein-Gordon equation. We find the tunneling probability of outgoing charged scalars from the event horizon of this black hole, and hence the Hawking temperature for this black hole.
... In order to solve the above equation, we assume an ansatz for the solution in a form similar to eqs. (2.12) as [49][50][51], ...
In the framework of the three dimensional New Massive Gravity theory introduced by Bergshoeff, Hohm and Townsend, we analyze the behavior of relativistic spin-1/2 and spin-0 particles in the New-type Black Hole backgroud, solution of the New Massive Gravity.We solve Dirac equation for spin-1/2 and Klein-Gordon equation for spin-0. Using Hamilton-Jacobi method, we discuss tunnelling probability and Hawking temperature of the spin-1/2 and spin-0 particles for the black hole. We observe that the tunnelling probability and Hawking temperature are same for the spin-1/2 and spin-0.
... The first one was used by Parikh and Wilczek [11], which followed from the work of Kraus and Wilczek, and the second one, which is the extension of the complex path analysis [7,8] has been used by different authors. Recently these radiations have been studied for charged black holes in string theory [20], squashed Kaluza–Klein black hole [21], black holes in Einstein-Yang-Mills theory [22], slowly rotating Kerr-Newman black hole [23] and dilatonic black holes242526. These methods have also been applied to higher dimensional black holes272829. ...
Full-text available
Using the Hamilton-Jacobi method of quantum tunneling and complex path integration, we study Hawking radiation of scalar particles from rotating black strings. We discuss tunneling of both charged and uncharged scalar particles from the event horizons. For this purpose, we use the Klein-Gordon equation and find the tunneling probability of outging scalar particles. The procedure gives Hawking temperature for rotating charged black strings as well.
Full-text available
In this paper, we start from the Lagrangian analysis on the action to naturally produce the geodesic equation of the massive particle via tunneling. Then, basing on the new definition for the geodesic equation, we revisit the Hawking radiation of the charged massive particle via tunneling from the Reissner-Nordström-de Sitter black hole with a global monopole. In our treatment, the geodesic equation of the charged massive particle is defined uniformly with that of the massless particle, which overcomes the shortcomings of its previous definition, and is more suitable for the tunneling mechanism. It is noteworthy that, the highlight of our work is a new and important development of the Parikh-Wilczek’s tunneling method.
Full-text available
Since Parikh and Wilczek proposed a semiclassical tunneling method to investigate the Hawking radiation of static and spherically symmetric black holes, the method has been extensively developed to study various black holes. However, in almost all of the subsequent papers, there exists a important shortcoming that the geodesic equation of the massive particle is defined inconsistently with that of the massless particle. In this paper, we propose a new idea to reinvestigate the tunneling radiation from the event horizon of the Reissner-Nordström black hole. In our treatment, by starting from the Lagrangian analysis on the action, we redefine the geodesic equation of the massive and massless particle via tunneling from the event horizon of the Reissner-Nordström black hole, which overcomes the shortcoming mentioned above. The highlight of our work is a new and important development for the Parikh-Wilczek’s semiclassical tunneling method.
In the work [J. Y. Zhang and Z. Zhao, Massive particles's black hole tunneling and de Sitter tunneling, Nucl. Phys. B 725 (2005) 173.], the Hawking radiation of the massive particle via tunneling from the de Sitter cosmological horizon has been first described in the tunneling framework. However, the geodesic equation of the massive particle was unnaturally and awkwardly defined there by investigating the relation between the group and phase velocity. In this paper, we start from the Lagrangian analysis on the action to naturally produce the geodesic equation of the tunneling massive particle. Then, based on the new definition for the geodesic equation, we revisit the Hawking radiation of the massive particle via tunneling from the de Sitter cosmological horizon. It is noteworthy that, the highlight of our work is a new and important development of the Parikh-Wilczek's tunneling method, which can make it more physical.
Full-text available
We study quantum tunneling of scalar particles from black strings. For this purpose we apply WKB approximation and Hamilton-Jacobi method to solve the Klein-Gordon equation for outgoing trajectories. We find the tunneling probability of outgoing charged and uncharged scalars from the event horizon of black strings, and hence the Hawking temperature for these black configurations.
QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ~ 10-33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ~ 1017 s which is very long compared to the Planck time ~ 10-43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ~ 10-6 (Msolar/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (Msolar/M)-3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs.
It is shown that the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant. An observer in these models will have an event horizon whose area can be interpreted as the entropy or lack of information of the observer about the regions which he cannot see. Associated with the event horizon is a surface gravity κ which enters a classical "first law of event horizons" in a manner similar to that in which temperature occurs in the first law of thermodynamics. It is shown that this similarity is more than an analogy: An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon. If the observer absorbs some of this radiation, he will gain energy and entropy at the expense of the region beyond his ken and the event horizon will shrink. The derivation of these results involves abandoning the idea that particles should be defined in an observer-independent manner. They also suggest that one has to use something like the Everett-Wheeler interpretation of quantum mechanics because the back reaction and hence the spacetime metric itself appear to be observer-dependent, if one assumes, as seems reasonable, that the detection of a particle is accompanied by a change in the gravitational field.
We consider scale invariant theories which couple gravity to Maxwell fields and antisymmetric tensor fields with a dilaton field. We exhibit in a unified way solutions representing black hole, space-time membrane, vortex and cosmological solutions. Their physical properties depend sensitively on the coupling constant of the dilaton field, there being critical value separating qualitatively different types of behaviour, e.g. the temperature of a charged black hole in the extreme limit. It is also shown that compactification into the 4-dimensional Minkowski space in terms of a membrane solution is possible in 10-dimensional supergravity model. Present address.
A physical account of the processes of black hole explosions is presented. Black holes form when the degeneracy pressure in a neutron star can no longer balance gravitational forces because the mass of the star is too large. Although black holes absorb surrounding matter through the action of a gravitational field, quantum fluctuations have been theoretically demonstrated to occur in the vacuum, and feature a thermal character. The temperature field decreases outwards, in accordance with the nonuniformity of the gravitational field, but does allow thermal radiation, i.e., Hawking radiation, to escape the black hole. The time scale for the radiation shortens as the mass of the black hole decreases, until a time scale is reached which is short enough for the process to be called an explosion. Observations of electron-positron Hawking radiation are suggested to offer proof of a black hole explosion.
In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature \frachk2pk » 10 - 6 ( \fracM\odot M )° K\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.
Hawking radiation viewed as a semi-classical tunneling process of charged particles from the event horizon of the Garfinkle–Horne dilaton black hole is investigated by taking into account not only energy conservation but also electric charge conservation. Our results show that when the effect of the emitted massive charged particle's self-gravitation is incorporated, the tunneling rate is related to the change of the black hole's Bekenstein–Hawking entropy and the emission spectrum deviates from the purely thermal spectrum.
We consider the tunnelling of charged spin 1/2 fermions from a Kerr–Newman black hole and demonstrate that the expected Hawking temperature is recovered. We discuss certain technical subtleties related to the obtention of this result.
Recently, it has been shown that the radiation arising from quantum fields placed in a gravitational background (e.g. Hawking radiation) can be derived using a quasi-classical calculation. Here we show that this method has a previously overlooked temporal contribution to the quasi-classical amplitude. The source of this temporal contribution lies in different character of time in general relativity versus quantum mechanics. Only when one takes into account this temporal contribution does one obtain the canonical temperature for the radiation. Although in this Letter the specific example of radiation in de Sitter space–time is used, the temporal contribution is a general contribution to the radiation given off by any gravitational background where the time coordinate changes its signature upon crossing a horizon. Thus, the quasi-classical method for gravitational backgrounds contains subtleties not found in the usual quantum mechanical tunneling problem.
We examine Hawking radiation from a Schwarzschild black hole in several reference frames using the quasi-classical tunneling picture. It is shown that when one uses, , rather than, , for the tunneling probability/decay rate one obtains twice the original Hawking temperature. The former expression for Γ is argued to be correct since is invariant under canonical transformations, while is not. Thus, either the tunneling methods of calculating Hawking radiation are suspect or the Hawking temperature is twice that originally calculated.