The GUP effect on tunnelling of massive vector bosons from the 2+1 dimensional black hole

In this study, the Generalized Uncertainty Principle (GUP) effect on the Hawking radiation formed by tunneling of a massive vector boson particle from the $2+1$ dimensional New-type Black Hole was investigated. We used modified massive vector boson equation based on the GUP. Then, the Hamilton-Jacobi quantum tunneling approach was used to work out the tunneling probability of the massive vector boson particle and, Hawking temperature of the black hole. Due to the GUP effect, the modified Hawking temperature was found to depend on the black hole properties, the AdS$_{3}$ radius, and on the energy, mass and total angular momentum of the tunneling massive vector boson. In the light of these results, we also observed that modified Hawking temperature increases by the total angular momentum the particle while it decreases by the energy and mass of the particle, and the graviton mass. Also, in the context of the GUP, we see that the Hawking temperature due to the tunnelling massive vector boson is completely different from both that of the spin-0 scalar and the spin-1/2 Dirac particles obtained in the previous study. We also calculate the heat capacity of the black hole using the modified Hawking temperature and then discuss influence of the GUP on the stability of the black hole.


Introduction
Classically, black holes are considered as a spacetime region where the gravitational field is so strong that does not emit any radiation. However, this consideration has changed under the quantum mechanical approach [1][2][3][4][5][6][7][8]. Meanwhile, by using the formulation of quantum field theory on curved spacetime, Hawking proved that a black hole emits thermal a e-mail: gecimganim@gmail.com b e-mail: ysucu@akdeniz.edu.tr radiation, well known the Hawking radiation in the literature. Furthermore, the Hawking's discovery has established a connection between general relativity, quantum field theory and thermodynamics. Hence, this discovery is very important to understand the black hole physics and the quantum gravity [9,10]. Therefore, to compute the Hawking radiation, it has been developed many different methods. One of these methods is the Hamilton-Jacobi method based on the tunnelling probabilities for the classically forbidden trajectory from inside to outside of an horizon. The tunneling probability, Γ , is related to the classical action of the particle, S, as Γ = e − 2 h ImS . In this connection, the Hawking radiation as a quantum tunnelling process of a point elementary particle, such as spin-0 scalar particle described by Klein-Gordon equation, spin-1/2 Dirac particle described by Dirac equation [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and spin-1 particle described by different equations: classically Proca equation [28][29][30][31][32][33][34][35][36][37] and, quantum mechanically, massive vector boson equation [38] that is derived by quantizing the classical Zitterbewegung model [39], was studied. In all of these studies, we see that Hawking radiation of a black hole was completely independent of the internal properties of a tunneled point particle, such as mass, angular (orbital+spin) momentum, energy, charge etc.
Alternative approaches related to quantum gravity predict the presence of a minimal observable length in Planck scale [40][41][42][43][44][45][46][47]. The existence of a minimal length leads to the generalized uncertainty principle (GUP) [41,48,49]. Given the GUP, the Hawking radiation of a black hole is related to the internal properties of a tunnelled particle. This effect is known as "Quantum Gravity Correction" in the literature [50][51][52][53]. To include the quantum gravity effect in the Hawking radiation, the Klein-Gordon equation for spin-0 particle, Dirac equation for spin-1/2 particle, and massive W ± -boson equations derived from the Lagrangian given by the Glashow-Weinberg-Salam model for spin-1 particle are modified in the GUP framework [54][55][56]. Using these modified relativistic wave equations, the modified Hawking temperature of various black holes is calculated by quantum tunnelling process of the spinning particles. The calculations show that the quantum gravity correction term is related not only to the black hole's properties but also to the tunnelled particle's properties, such as mass, angular (orbital+spin) momentum, energy, charge etc. [56][57][58][59][60][61][62][63][64][65][66][67][68]. Hence, it is evident that the studies of the Hawking radiation by quantum tunnelling process of the relativistic quantum mechanical particles with various spin have an important situation in the context of the quantization of black hole. In this motivation, using the GUP relations, we will modified the massive vector boson equation which recently proposed to describe the behaviour of the spin-1 particle in 2 + 1 dimensional quantum electrodynamics (QED 2+1 ) [39].
In the context of the quantization of gravity, recently, many important studies have been carried out by focusing on 2 + 1 dimensional theories as a toy model [69][70][71][72][73]. Especially, the 2+1-dimensional massive gravity provides an active research area, both mathematically and physically, among these theories. For example, the New Massive Gravity is one of them [74]. In this theory, the graviton, which is the quantum particle of gravity in the standard model, gains a mass, topologically, [74][75][76][77][78][79]. Furthermore, the theory has a black hole solution, for example, the new-type black hole [80]. And, the Hawking radiation of the black hole was discussed by using quantum tunneling process of point particles, such as massive spin-0, spin-1/2 and spin-1 particle [26,27,38]. All the point particles are tunnelled in the same way from the black hole because the Hawking radiation is only dependent of the black hole properties. To investigate the effect of the tunnelleing particle properties on the Hawking radiation, in this study, we aim to work out the GUP effect on the Hawking radiation of the 2+1 dimensional Newtype black hole as a quantum tunnelling process of a massive spin-1 particle described by the 2 + 1 dimensional modified massive vector boson wave equation.
This paper is organized as follows. In the following section, we will modified the relativistic quantum mechanical vector boson equation under the GUP. After that, using Kerner-Mann's quantum tunneling method, we calculate the Hawking temperature as a quantum tunnelling process of the massive vector boson from the New-type black hole with the GUP effect. In section 4, we summarize the results.

Modified Vector Boson Equation
The covariant form of the massive vector boson (massive spin-1 relativistic quantum particle) equation in a curved spacetime can be written as [39] where the m 0 is mass of the vector boson and the β µ (x), are the 2+1 dimensional spacetime-dependent Kemmer matrices described by the 2+1 dimensional spacetime-dependent Dirac matrices as σ µ (x) = (σ 3 (x), iσ 1 (x), iσ 2 (x)), on the other hand, they are related to the flat spacetime Kemmer matrices, β (i) , as β µ (x) = e µ (i) β (i) , the covariant derivative, ∇ µ , is defined by means of the spin connection coefficients for the spin-1 particle, Σ µ (x), as ∇ µ =∂ µ − Σ µ (x) and Σ µ (x) can be written in terms of the spin-1/2 particle connections, and the Γ µ (x) is defined as [81] where the e µ (i) (x) are triads, the Γ α ν µ are Christoffel symbols and s λ ν (x) is spin operator of the spin-1/2 particle and it is given by and, also, Ψ (x) is a wave function of the spin-1 particle and it has symmetric representation as Ψ (x)=(ψ + , ψ 0 , ψ 0 , ψ − ) T [39] because of the quantization [82][83][84][85][86][87]. Also, in the vector boson equation,h is defined ash = h/2π in terms of the Planck constant, h, and the speed of light in vacuum, c, is taking as c = 1 in throughout of the paper. In the rest frame, this equation has the particle, with the spin up state with positive energy, and antiparticle, with the spin down state with negative energy, solutions and is compatible with the Proca equation in the classical limit. This equation is directly derived as an excited state of the classical zitterbewegung model [39], which the symmetry and integrability properties of this model was discussed in 2+1 dimensions [88]. It is analogues in the 2+1 dimensional spacetimes that of the 3+1 dimensional spin-1 sector of the Duffin-Kemmer-Petiau (DKP) equation [82][83][84][85][86][87]. As discussing the Hawking radiation of the 2 + 1 dimensional New-type and Warped-AdS 3 black holes by the quantum tunneling method of the massive vector boson as a point particle described by the vector boson equation (1) [38], we see that the spin-1 point particle tunnel as that of the scalar and Dirac point particles [26,27].
On the other hand, to investigate the quantum gravity effect on the tunnelling process of the massive vector boson, the massive vector boson wave equation must be rewritten in the framework of the GUP. Using the fact that in the existence of a minimal length at Planck scale, the standard Heisenberg uncertainty principle can be modified as follows [89][90][91][92]; where α = α 0 /M 2 p , the M 2 p is the Planck mass and α 0 is the dimensionless parameter. The Eq.(6) can be derived by using the modified commutation relation given as follows [89][90][91][92]; where x i and p j are the modified position and the modified momentum operators defined by the standard position, x 0i , and the standard momentum, p 0 j =−ih∂ j , satisfying the standard commutation relation x 0i , p 0 j = ihδ i j , as, respectively, and p 2 0 =Σ p 0 j p j 0 [60]. On the other hand, the modified energy relation is given by the following form [55,63,90,91]: where the energy mass shell condition, E 2 = p 2 + m 2 0 , is used. The square of the momentum operator can be expressed after neglecting the higher order terms of the α parameter as follows [92]; Hence, Using the Eq.(8), Eq.(9) and Eq.(10), the modified massive vector boson equation is written as follows; (11) or its explicit form: where the Ψ is the modified wave function of a vector boson.

Vector boson particle tunnelling from New-type black hole
The New-type black hole is described as [26,80] where f (r) = r 2 + br + c with two constant parameters b and c. The AdS 3 radius, L, is defined as L 2 = 1 2m 2 = 1 2Λ in terms of the m graviton mass or the Λ cosmological constant. And, it has an outer and inner horizon at r ± = 1 2 −b ± √ b 2 − 4c , respectively. Furthermore, the New-type black hole has six different types according to the signatures of the parameters b and c [80] and the each type black hole has different mathematical and physical properties. For example, in case b 2 =4c, the black hole becomes extremal. On the other hand, in the case b=0 and c < 0, it is reduced to the non-rotating BTZ black hole [80].
To calculate the quantum gravity effect on the Hawking temperature of the black hole, firstly, we write the modified massive vector boson equation in the New-type black hole background by using the spin-1 matrices and spin connection coefficients calculated in [38], and secondly, we use the following ansatz for the modified wave function to compute the tunnelling probability of the vector boson [39], where A(t, r, φ ), B(t, r, φ ) and D(t, r, φ ) are functions of the spacetime coordinates and S(t, r, φ ) is the classical action function of the particle trajectory. Then, the modified massive vector boson equation can be reduced to the three coupled differential equations after neglecting the terms withh: These three equations have nontrivial solutions for the coefficients A (t, r, φ ), B (t, r, φ ) and D (t, r, φ ) under the condition of the coefficients matrix determinant having 0. From this condition, we get the Hamilton-Jacobi equation as Thanks to the separation of variables method we can write the S (t, r, φ ) in terms of its variables as follows; where E and j are the energy and total angular momentum of the particle, respectively, and K(r)=K 0 (r)+ αK 1 (r) [64] and the C is a complex constant. Inserting Eqs. (17) into Eqs. (16), we obtain the radial function K (r) as follows: where χ is and, K + (r) and K − (r) represent the radial trajectories of the outgoing and incoming particles on the outer horizon, respectively. Using a contour that has a semicircle around the pole at the outer horizon, the integration in Eqs. (18) are calculated as where Σ is On the other hand, the tunnelling probabilities of particles crossing the outer horizon are given by respectively. Furthermore, the tunneling probability for the classically forbidden trajectory from inside to outside of the black hole horizon is given by [24] Γ = e − 2 h ImS(t,r,φ ) .
Moreover, the total imaginary part of the action is ImS (t, r, φ ) = ImK ± (r) = ImK + (r) − ImK − (r) [93,94]. Hence, using the fact that ImK + (r) = −ImK − (r), the probability of the vector boson tunneling from inside to outside of the outer event horizon of the black hole is calculated as If one expands the classical action in terms of the particle energy, then the modified Hawking temperature is obtained at the linear order. Thus, we can write the probability as where β is the inverse temperature of the outer horizon. According to this, the modified Hawking temperature becomes If we expand the T ′ H in terms of the α powers and neglect the higher order of the α terms, then the modified Hawking temperature of the New-type black hole becomes as follows; where the T H =h (r + −r − ) 4π is the standard Hawking temperature of the black hole. From the T ′ H expression, we see that the modified Hawking temperature is related to not only the mass parameter of the black hole, but also to the AdS 3 radius, L, (and, hence, to the graviton mass) and the properties of the tunnelled massive vector boson, such as angular momentum, energy and mass. On the other hand, in the case of α = 0, the modified Hawking temperature reduces to the standard temperature obtained by quantum tunnelling process of the point particles with spin-0, spin-1/2 and spin-1, respectively [26,38].

Summary and Conclusion
In this study, we investigate the quantum gravity effect on the tunneled massive vector boson from New-type black hole in the context of 2+1 dimensional New Massive Gravity. For this, at first, using the GUP relations, we modify the massive vector bosons equation. Then, using the Kerner-Mann method, the tunnelling probabilities of the massive vector particle are derived, and subsequently, the corrected Hawking temperature of the black hole is calculated. And, we find that the modified Hawking temperature not only depends on the black hole's properties, but also depends on the emitted spin-1 vector boson's mass, energy, total angular momentum. Also, it is worth mention that the modified Hawking temperature depends on the mass of graviton in this context. As can be seen from Eqs. (24), the Hawking temperature increase by the total angular momentum of the tunneled particle while it decreases by the energy and mass of the tunnelled particle and the graviton mass.
-As described previously, the New type black hole is reduced to the static BTZ black hole in the case of b = 0 and c < 0,. Hence, the Hawking temperature of the static BTZ black hole under the quantum gravity effect is where r + = −r − = |c| is used and the T H =h √ |c| 2π is the standard Hawking temperature of the static BTZ black hole in the context of 2+1 dimensional New Massive Gravity theory [26]. In this case, the modified Hawking temperature is higher than standard Hawking temperature when 4E 2 + 9m 2 0 2m 2 |c| < 4 j 2 . On the other hand, when 4E 2 + 9m 2 0 2m 2 |c| > 4 j 2 , the modified Hawking temperature is lower than standard Hawking temperature.
-In the absence of the quantum gravity effect, i.e. α = 0, the modified Hawking temperature is reduced to the standard temperature obtained by quantum tunnelling of the massive spin-0, spin-1/2 and spin-1 point particles [26,38].