MASSIVE PARTICLE TUNNELING RATE OF KERR-NEWMAN-ANTI-DE SITTER BLACK HOLE BY HAMILTON-JACOBI METHOD M. Ilias Hossain
1. INTRODUCTION Recently, a semiclassical tunnelling process applied to
find the Hawking radiation of the static Schwarzschild and Reissner-Nordstr¨om
black holes by Parikh and Wilczek Parikh
and Wilczek (2000), Parikh
(2002), Parikh (2004) and their result shows
that the radiation spectrum is not pure thermal but satisfies the unitary
principle and support the result of information conservation. In their process,
the tunneling potential barrier is produced by the self-gravitation interaction
and the position of the horizons before and after the particle’s emission.
Following this method, several researchers studied the Hawking radiation of
various spacetime Hemming and Keski-Vakkuri (2001), Arzano et al. (2005), Medved (2002), Medved (2002), Medved and Vagenas (2005), Medved and Vagenas
(2005), Vagenas (2002), Vagenas (2003), Shankaranarayanan et al. (2002), Angheben et al. (2005) by using Painleav´e or dragging or tortoise
or Eddington-Finkelstein coordinate transformations and these radiations are
limited to uncharged massless particle only. In this article, we
use the Parikh and Wilczek’s opinion Parikh
and Wilczek (2000), Parikh
(2002), Parikh (2004) and employing standard Hamilton-Jacobi method to investigate the Hawking
non-thermal and purely thermal tunneling rates of the Kerr-Newman-anti-de
Sitter (KNAdS) black hole for massive particle. In
order to carry-over this article, KNAdS black hole spacetime is
described as follows. The Kerr-Newman
anti-de Sitter (KNAdS) black hole which is the KAdS black hole generalized with
a charge parameter, described by the metric Equation 1 Where
Here the parameters
and Equation 3 Depending on the
black hole parameters, the function with has
four distinct roots. For the KNAdS black hole case we are interested to find
the real root of , namely the real root corresponds to the radius of the black hole’s
outer event horizon, while the other real root represents the radius of the inner cauchy
horizon and as the
cosmological horizon. Equation (3) can be written as
Solving the above
equation, the position of the black hole horizons is given by Equation 5 and
Equation 6 where , . Equation 7 and is the another cosmological horizon. With the
black hole horizon can be approximated as Equation 8 Taking only the
positive sign which is the event horizon of KNAds black hole as follows Equation 9 Expanding in
terms of black hole parameters with negative cosmological constant under the
condition, we obtain
which can be
written as
Now if we set then with and hence the event horizon of KNAds black
hole is less than Kerr-Newmann Chen
and Yang (2007) event horizon As the event horizon of KNAds black hole
coincides with the outer infinite red-shift surface, we apply the geometrical
optical limit and the “s-wave” approximation. Using the semiclassical WKB
method Massar
and Parentani (2000), the tunneling
probability is found to be related to the imaginary part of the action of the
following form , Equation 12 where The later section describes near the event horizon the new
line element of KAdS black hole. In
section 3 and 4, we derived the Hawking non thermal and thermal radiation respectively.
In section 5, we present our results and discussion. Finally, in section 6, we present our concluding
remarks. 2. The HJ Method for KNAdS Spacetime The Hamilton-Jacobi
method was applied extensively to the non-thermal radiation in 1990s and
attracted people’s attention Srinivasan and Padmanabhan
(1999), Angheben et al. (2005), Kerner and Mann (2007). In 2005, applying semiclassical tunneling
method, Angheben, Nadalini, Vanzo and Zerbini Angheben et al. (2005) developed Hamilton-Jacobi method Shankaranarayanan et al. (2001), Shankaranarayanan et al. (2002), Shankaranarayanan (2003), Srinivasan and Padmanabhan
(1999), Padmanabhan (2004) ignoring the self-gravitational effect of the
emitted scalar particles. Here we now consider the method of Chen et al. Chen and Yang (2007), Chen et al. (2008) to calculate the imaginary part of the action from the relativistic
Hamilton-Jacobi equation. The action of the radiating particle I satisfies the
relativistic Hamilton-Jacobi equation Equation 13 where In this method, we
avoid the exploration of the equation of motion in the Painlev´e coordinates
systems for calculate the imaginary part of the action Equation 14 The position of
black hole horizon of the metric given by Eq. (14) is same as given in Eq.
(11). Therefore, the line element near the event horizon rewritten as , Equation 15 where and are
defined as follows and Equation 16 Calculating the
non-null inverse metric tensors from the metric (15) and employing these in Eq.
(13) as follows Equation 17 To solve action , we consider the properties of the black hole
spacetime and carry out the separation of variables as
Equation 18 where is the
energy of the emitted particle, and are the
generalized momentums, and s the
angular momentum of the particle with respect to -axis. Inserting Eq. (18) into Eq. (17) to
seek a solution of the following form
Equation 19 where the angular
velocity of the particle at the event horizon is
Equation 20 We treat the
emitted particle as an ellipsoid shell of energy to
tunnel across the event horizon. Finishing the above integral by using the
Cauchy’s integral formula, we obtain
Equation 21 where sign
comes from the square root. Therefore, the imaginary part of the action corresponding to the outgoing particle is
obtained by times
the residue of the integrand
Equation 22 Using Eqs. (11) and
(20) into Eq. (22), we get the imaginary part of the true action of the
radiation particle as where
. where and . To get the maximum
value of the integration, neglecting higher order terms above and equal in the
denominator, we then get
Equation 23 3. Non-thermal Tunneling Rate Since the emitted
particle can be treated as a shell of energy ω, Eqs. (22) and (23) should
be modified when the particle’s self-gravitational interaction is incorporated.
Taking into account the energy conservation as well as
angular momentum, the mass parameter and the angular momentum in these
equations will be replaced with and when
the particle with energy ω and angular momentum j tunnels out of the event
horizon. We fix the ADM mass, charge and angular
momentum of the total spacetime and in presence of comological constant KNAdS
spacetime is dynamic and allow mass and angular momentum of the black hole to
fluctuate. Then the imaginary part of the true action can be calculated from
Eq. (23) in the following integral
Equation 24 For the maximum
value of integration, neglecting . Equation (24) becomes
Equation 25 Replacing and by and respectively, we obtain |