MASSIVE PARTICLE TUNNELING RATE OF KERR-NEWMAN-ANTI-DE SITTER BLACK HOLE BY HAMILTON-JACOBI METHOD

 

M. Ilias Hossain ¹, M. Jakir 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

 

     Equation 1

  

 

Where

, , ,                      Equation 2

 

Here the parameters  and q are the associated with the mass, angular momentum, cosmological radius, and charge parameters of the spacetime respectively in the background of the rotating anti de Sitter space. The spacetime causal structure depend strongly on the singularities of the metric given by the zeros of  as follows

 

                                 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

 

        Equation 4   

 

 

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

 

                                        Equation 10

 

which can be written as

 

                                 Equation 11

 

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 I is the action of the radiating particle and is the emission rate.

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 m and  are the mass of the particle and the inverse metric tensors respectively.

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  on the line element (1), we obtain the new line element of the Kerr-Newman-anti-de Siter black hole as

 

                                              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