MASSIVE PARTICLE TUNNELING RATE OF KERR-NEWMAN-ANTI-DE SITTER BLACK HOLE BY HAMILTON-JACOBI METHOD M. Ilias Hossain 1 1 Professor,
Department of Mathematics, Rajshahi University, Rajshahi - 6205, Bangladesh
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
Where
Here the parameters
Depending on the
black hole parameters, the function
Solving the above
equation, the position of the black hole horizons is given by
and
where and
Taking only the
positive sign which is the event horizon of KNAds black hole as follows
Expanding
which can be
written as
Now if we set
where I is the action
of the radiating particle and 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 where m and 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 I. For the convenience of our research to study the Hawking
radiation, adopting the transformation 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 where
Calculating the
non-null inverse metric tensors from the metric (15) and employing these in Eq.
(13) as follows Equation 17 To solve action Equation 18 where 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 Equation 21 where 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 To get the maximum
value of the integration, neglecting higher order terms above and equal 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 Equation 24 For the maximum
value of integration, neglecting Equation 25 Replacing |