Department of Physics, Ranaghat College, Ranaghat, India.
Received Date: 26/04/2016; Accepted Date: 20/07/2016; Published Date: 22/07/2016
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In this paper, we have studied the Hawking radiation of massive spin-1 particles from the black holes in (2+1) dimensions with non-trivial dilaton fields. We consider two special varieties of these black holes one is static charged and other is spinning electrically neutral. By applying the standard method of WKB approximation and Hamilton-Jacobi ansatz we have shown the tunnelling probability and Hawking temperature of massive bosons accordingly. In the certain limit of the dilaton coupling for spinning neutral case we have recovered the Hawking temperature of the BTZ black holes as well.
Hawking black holes, Radiation, Dilaton coupling
Black holes have become the most fascinating objects to study in the context of Einstein’s general theory of relativity over the last century. According to Hawking [1-3] these black holes emit thermal radiation at a constant temperature across the horizon. This temperature is known as Hawking temperature. Several methods have been developed to understand Hawking radiation and calculate Hawking temperature after the original method proposed by Hawking . In recent year’s semi-classical approach [4-29] of deriving Hawking radiation as tunnelling process captured lot of attention. This semi- classical tunnelling method is considered to calculate the imaginary part of the action for the (classically forbidden) process of s-wave emission across the horizon (Kraus and Wilczek [30-32]), which is related to the Boltzmann factor for emission at the Hawking temperature. Applying the WKB approximation the emission and absorption probabilities for the classically forbidden trajectory of the s-wave coming from inside to outside of the horizon is given by, , where I is the classical action of the trajectory of the tunnelling particles. In calculating the Hawking temperature two different methods were developed to evaluate the imaginary part of the action- one was proposed by Parikh and Wilczek  and the other method used by Angheben et.al. , this is an extension of the analysis done by Srinivasan and Padmanabhan .
First method deals with the part of the action that contributes an R imaginary term is , where Pr the emitted null s-wave is. Hence one can calculate imaginary part of the action by using the Hamilton’s equation and the knowledge of null geodesics. This method is known as null geodesic method. The second method is known as the Hamilton-Jacobi method where the imaginary part of the action is derived from the Hamilton-Jacobi equation by choosing suitably the form of the action. This tunnelling method as well applied for the black hole radiation of spin and spin particles and photons at Hawking temperature [24,25,33-36] close to the black hole horizon.
In continuation of using further the tunnelling method to model spin-1 (e.g. bosons like W± and Z) particle emission from black holes S. I. Kruglov  investigated Hawking radiation in (2+1) dimensions. Recently various authors have analysed Hawking radiation of vector bosons in the context of static and rotating black holes [37-44]. Apart from the various black hole solutions with or without cosmological constants here we are interested black holes with nontrivial dilaton fields which are neither asymptotically flat nor anti-de Sitter. We are interested to investigate Hawking radiation and to calculate Hawking temperature of a class of Einste in-Maxwell dilaton black holes with arbitrary dilaton coupling as discovered by Chan and Mann [45-48]. Therefore our purpose is to study the tunnelling of massive vector boson W± through the space- time of a static charged  and spinning charge neutral  dilaton black holes in (2+1) dimensions. In case of static charged black holes we derive the field equation of the charged boson from the Lagrangian given by Glashow-Weinberg-Salam model. After that we use the WKB approximation and then apply the Hamilton- Jacobi ansatz, for solving the radial part of the action by using the determinant of the matrix equals zero, we found the tunnelling rate and the corresponding Hawking temperature. For the latter case of spinning black hole we use Proca equation as the field equation of the massive vector field which is neutral. We again calculate the tunnelling rate in case of spinning dilaton black holes and also recover associated Hawking temperature of this class of black holes. We are mainly interested to study Hawking radiation in the background of these special classes of black holes for generic values of dilaton coupling. We have also obtained two known class of black holes like BTZ  and MSW* in the two special limits of the dilaton coupling and also can reproduce the Hawking temperature for those particular classes of black holes. This paper is organised as follows. In section 2 we consider static charged black holes with nonzero Dilaton filed and investigate the Hawking radiation of charged vector bosons and estimate the Hawking temperature. In section 3 we also study tunnelling of massive vector bosons which are uncharged in the spinning neutral dilaton black hole space- time. We conclude our study in section 4.
We consider the action given in  as follows
with arbitrary couplings a and b. Here is the Ricci scalar, φ is the dilaton field, Λ is the cosmological constant and F is the Maxwell’s field strength. When a=1 and b=1, the equation (1) is an action for low energy string theory expressed in a usual Einstein metric. Now the family of static circularly symmetric black hole solutions to the above action as obtained in  is
Q Is related to the electric charge whereas β, γ are integration constants. k is a real number. Last relations in equation (5) are among the various dilaton couplings, finally related with a single parameter N. As (M,Λ) > 0 the black hole solutions exist only in the domain . When N=1, (2) reduces to the 2+1 dimensional MSW. For N=2 and Q=0, metric (2) represents uncharged BTZ black hole which might be the special case of BTZ black hole given in . In general due to the presence of nontrivial dilaton, black holes are neither asymptotically flat nor anti-de Sitter (de Sitter).
Let us now consider the Lagrangian of W-bosons which is treated as massive vector boson here, interact with the background electromagnetic field is given by ,
Where and ∇μ is the covariant geometric derivative. Aμ Is the vector potential associated with the electromagnetic field Fμν (Where Fν=∇μ Aν−∇ν Aμ) of the black hole? The charge carries by the W+ boson is denoted by e and the mass is m. Equation of motion of the W-boson should be derived from the above Lagrangian and write down in the following form
In order to study the tunnelling of W+ bosons as the spin-1 particle, one should solve the following equation
The solution of equation (8) can be represented in the form
Where μ=0,1,2 and Cμ represents some arbitrary functions. According to the WKB approximation, one can write down the action as
The determinant of the metric in equation (2)
and the nonzero components of the metric are
Using equations (8)-(12) we can get three sets of equations in the leading order of ħ2
We assume that the action S (t,r,θ) can be written in the form
Where E and J are the energy and the angular momentum of the W+ boson respectively. Whereas K is a complex constant. Inserting equation (16) into equations (13)-(15), one can obtain a matrix equation (C0,C1,C2)T = 0 (here T denotes the transition to the transposed vector) where is a 3×3 matrix whose elements are
To obtain the nontrivial solution of the Equations (13-15), we should have the determinant of the Matrix equals to zero. So by solving det =0 we get
Solving this equation for the radial function W(r) we have the following integration
Integrating around the pole at the event horizon r+ we obtain
W+ Denotes the radial function of outgoing particles and W(r) of the ingoing particles. Then Following  we can obtain the tunnelling probability of W+ bosons
We can find the Hawking temperature by simply comparing equations (21) with the Boltzmann factor , where
Using Equations (3), (22), one can recover the Hawking temperature for a static charged dilaton black hole as derived in 
Where and. here the location of the outer (event) horizon r+ is a function of the black hole parameters M,Q,Λ and dilaton coupling N. We can show for uncharged case (i.e. Q=0) Hawking temperature becomes
Following our earlier discussion the Hawking temperature derived in equation (23) reduces to the Hawking temperature for charged MSW black hole for N=1 which is exactly matched with the result presented in . Furthermore for vanishing Q we can get from equation (24) for uncharged static BTZ black hole , for uncharged MSW black hole the temperature would be , which only depends on the cosmological constant Λ.
We mention below a class of spinning black hole solutions parameterised by the mass M, angular momentum J and dilaton coupling N which metric is given by  as
The mass M and angular momentum J are related with the parameters δ and ω as follows
The solution of the equation (33) after using the form of the metric given in equation (25) is considered in the form
Again μ=0,1,2 With the help of the WKB approximation, one can represent the action similar to equation (10) as
However we further choose the action in the form
Where E and j are the energy and angular momentum of the spin-1 particles, respectively, and K is a (complex) constant. After substituting Equation (34), (35), (36) and (37) into equation (33) and keeping only the leading order term in ħ we obtain a 3×3 matrix (matrix is) equation: (C0,C1,C2)T = 0. Thus, one can read the non-zero components of as follows
The system of linear equations (C0,C1,C2)T = 0 possesses nontrivial solutions if det =0. Then from equation (38) we obtain
From equation (39) we get
The above equation immediately gives the function W (r) as
Where is angular velocity at the outer event horizon located at r+? We mention that W+ denotes the radial function of the outgoing particles and W− of the ingoing particles. Integrating around the pole at the outer horizon, we obtain
As previously we calculate the tunnelling probability of the massive vector particle is
We have found in equation (42), the imaginary part of the function W (r) is
Here k is defined as the surface gravity of the black hole horizon and the Hawking temperature would be
We have put the expression for the location of the outer event horizon r+ by setting the function h(r+)=0 from the metric (25) as
We get the Hawking temperature in equation (45) exactly same as the standard Hawking temperature derived in . If the angular momentum J (or spinning parameter ω ) approaches zero, the metric (25) reach the metric of the non-spinning dilaton black hole discussed in the previous section and the Hawking temperature (45) nicely go back to the temperature of the uncharged non-spinning black hole shown in equation (24). Here we also present the expression for the Hawking temperature for two limits of the dilaton coupling parameter N. The first one is the spinning generalization of the MSW black hole in 2+1 dimensions which temperature can be obtained from equation (45) by simply putting the coupling parameter N=1 as follows:
Secondly for N=2 the metric (25) after suitable coordinate transformation reduces to the metric of spinning BTZ black hole and the Hawking temperature of BTZ black hole is produced immediately as obtained in  is
Here p+=p(r+) and p-=p(r-) are the location of outer and inner event horizon respectively for the BTZ black hole.
In this study we have presented the semi-classical tunnelling processes to understand Hawking radiation of massive vector particles either charged or uncharged, tunnelling through the event horizon of the particular class of dilaton black holes in 2+1 dimensions. We have computed first the probability of tunnelling of charged spin-1 (W+ bosons) particles through the event horizon of static charged black hole which has non-trivial dilaton field. Further we have calculated Hawking temperature at horizon of this charged black hole for arbitrary coupling parameter N. This results exactly matched with the Hawking temperature obtained by the other method in references  for arbitrary coupling N, and in  for N=1. In the later part of the paper we have mainly concentrated on tunneling methods by using Proca equations of massive spin-1 field to find the tunneling probability of outgoing particles and the Hawking temperature at the outer event horizon of the neutral spinning dilaton black holes. Again we have derived Hawking temperature for general dilaton coupling. By tunning that coupling at certain values we have been able to show the radiation temperature of a particular class of black holes like BTZ which is consistent with the result derived in  in the context of Hawking radiatiom. It would also be interesting to study further the tunneling probability of other type of spin particles across the horizon of the black holes considered here.
I would like to thank R. Uniyal for providing some references.