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Tunneling Glashow-Weinberg-Salam Model Particles from Black Hole Solutions in Rastall Theory.

1. Introduction

General relativity is analogously linked to the thermodynamics and quantum effects which strongly support it [13]. Black holes are the strangest objects in the Universe and they arise in general relativity, a classical theory of gravity, but it is needed to include quantum effects to understand the nature of the black holes properly. After Bekenstein found a relation between the surface area and entropy of a black hole [4], Hawking theoretically showed that black holes with the surface gravity [kappa] radiate at temperature [kappa]/2[pi] [5-7]. On the other hand, Bekenstein-Hawking radiation causes the information loss paradox because of the thermal evaporation. To solve the information paradox, recently soft-hair idea has been proposed by Hawking et al. [8].

Since Bekenstein and Hawking great contribution to the black hole's thermodynamics, the radiation from the black hole gets attention from researchers. There are many different methods to obtain the Bekenstein-Hawking radiation using the quantum field theory or the semiclassical methods. The quantum tunneling method is one of them [9-16]. Nozari and Mehdipour [17] have studied the Hawking radiation as tunneling phenomenon for Schwarzschild BH in noncommutative space-time. Nozari and Saghafi [18] have investigated the tunneling of massless particles for Schwarzschild BH by considering quantum gravity effects. The semiclassical tunneling method by using the Hamilton-Jacobi ansatz with WKB approximation is another way to obtain the Bekenstein-Hawking temperature and the tunneling rate as [GAMMA] [approximately equal to] exp[-2fmS] [19]. Different kinds of particles such as bosons, fermions, and vector particles are used to study the tunneling of the particles from the black holes and wormholes and obtain their Hawking temperature [20-46]. Nozari and Sefidgar [47] have discussed quantum corrections approach to study BH thermodynamics. Nozari and Etemadi [48] have investigated the KMM seminal work in case of a maximal test particles momentum. They showed that, in the presence of both minimal length and maximal momentum, there is no divergence in energy spectrum of a test particle. Moreover, the uncertainty principle is modified as a generalized uncertainty principle (GUP) [49, 50] to work on the effect of the quantum gravity which is applied to different areas. The important contribution of the GUP is to remove the divergences in physics. On the other hand, GUP can be used to modify Proca equation and Klein-Gordon equation to obtain the effects of the GUP on the Hawking temperature and check if it leaves remnants [51-54]. Nozari and Mehdipour [55] have discussed BH remnants and their cosmological constraints. Moreover, GUP is used to modify the thermodynamics of N-dimensional Schwarzschild-Tangherlini black hole, speed of graviton, and the Entropic Force. Feng et al. [56] have studied the difference between the propagation speed of gravitons and the speed of light by using GUP. They have also investigated the modified speed of graviton by considering GUP. Rama [57] has studied the consequences of GUP which leads to varying speed of light and modified dispersion relations, which are likely to have implications for cosmology and black hole physics.

Since Maxwell, it was the dream of theoretical physicist to unify the fundamental forces in the nature in a single equation. Glashow, Weinberg, and Salam unified the theory of weak and electromagnetic interactions as an electroweak interaction in the 1960s. They assumed that the symmetry between the two different interactions would be clear at very large momentum transfers. However, at low energy, there is a mass difference between the photon and the [W.sub.+], [W.sub.-], and [Z.sub.0] bosons which break the symmetry.

This paper is organized as follows: In Section 2, we investigate the Hawking temperature of the black hole solutions surrounded by perfect fluid in Rastall theory using the tunneling of the massive vector particles. For this purpose we solve the equation of the motion of the Glashow-Weinberg-Salam model using the semiclassical WKB approximation with Hamilton-Jacobi method. In Section 3. we use the GUP-corrected Proca equation to investigate the tunneling of massive uncharged vector particles for finding the corrected Hawking temperature of the black hole solutions surrounded by perfect fluid in Rastall theory. In Section 4, we conclude the paper with our results.

2. Tunneling of Charged Massive Vector Bosons

In this section, we study the tunneling of the charged massive bosons from the different types of black holes surrounded by the perfect fluids in Rastall theory.

2.1. The Black Hole Surrounded by the Dust Field in Rastall Theory. First we study the line element of the black hole surrounded by the dust field [58]:

[mathematical expression not reproducible], (1)

where M is a mass of black hole, [kappa] and [lambda] are the Rastall geometric parameters, [N.sub.d] is the dust field structure parameter, and Q is a charge of black hole. Now, we can rewrite (1) in the following form:

d[s.sup.2] = -G (r) d[t.sup.2] + B (r) d[r.sup.2] + C (r) d[[theta].sup.2] + D (r) d[[phi].sup.2], (2)

where G(r), B(r), C(r), and D(r) are given below:

[mathematical expression not reproducible]. (3)

The equation of motion for the Glashow-Weinberg-Salam model [59-61] is

[mathematical expression not reproducible], (4)

where [absolute value of g] is a coefficients matrix, m is particles mass, and [[PHI].sup.[mu]v] is antisymmetric tensor, since

[mathematical expression not reproducible], (5)

where [A.sub.[mu]] is the vector potential of the charged black hole and [A.sub.0] and [A.sub.3] are the components of [A.sub.[mu]], e is the charge of the particle, and [[DELTA].sub.[mu]] is covariant derivative. The values of [[PHI].sup.[mu]] and [[PHI].sup.[mu]v] are given by

[mathematical expression not reproducible]. (6)

Using WKB approximation for the wave function ansatz [62], i.e.,

[mathematical expression not reproducible], (7)

to the Lagrangian (4) (where [I.sub.0] and [I.sub.i] correspond to particles action, for i = 1,2,3, ...) and neglecting the higher order terms, we get the following set of equations given below:

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

We can choose [I.sub.0] by using separation of variables technique, i.e.,

[mathematical expression not reproducible], (12)

where [??] is the angular momentum of the BH and E and [??] represent particle energy and angular momentum, respectively. From (8)-(11), we can obtain the following matrix equation:

[??] [([c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3]).sup.T] = 0, (13)

which provides "[??]" as a matrix of order 4 x 4 and its components are given by

[mathematical expression not reproducible], (14)

where [mathematical expression not reproducible], and [??] = [[partial derivative].sub.[phi]][I.sub.0]. For the nontrivial solution [absolute value of [??]] = 0 and solving above equations one can yield

[mathematical expression not reproducible], (15)

where + and - represent the outgoing and incoming particles, respectively, whereas, "X" is the function which can be defined as

[mathematical expression not reproducible] (16)

and [??] is the angular velocity at event horizon.

By integrating (15) around the pole, we get

[mathematical expression not reproducible], (17)

where the surface gravity [kappa]([r.sub.+]) of the charged black hole is given by

[mathematical expression not reproducible]. (18)

The tunneling probability [GAMMA] for outgoing charged vector particles can be obtained by

[mathematical expression not reproducible]. (19)

Now, we can calculate the [[??].sub.H](Im[W.sup.+]) by comparing the [??](Im[W.sup.+]) with the Boltzmann formula [mathematical expression not reproducible], and we get

[mathematical expression not reproducible]. (20)

The result shows that the [??](Im[W.sup.+]) is dependent on r, the vector potential components ([A.sub.0] and [A.sub.3]), energy E, angular momentum [??], and mass of black hole M; [kappa] and [lambda] are the Rastall geometric parameters; and [N.sub.d] and Q are dust field structure parameter and charge of black hole, respectively.

2.2. The Black Hole Surrounded by the Radiation Field. Second example of the line element of black hole surrounded by the radiation field [58] is given below:

[mathematical expression not reproducible], (21)

where M is a mass of black hole, [N.sub.r] is the negative radiation structure parameter, and Q is a charge of black hole.

Following the procedure given in Section 2.1 for this line element, we can obtain the surface gravity [kappa]([r.sub.+]) of this charged black hole surrounded by the radiation field in the following form:

[mathematical expression not reproducible], (22)

where [[??].sub.r] = [partial derivative][N.sub.r]/[partial derivative]r. The tunneling rate of particles can be calculated as

[mathematical expression not reproducible] (23)

and the corresponding Hawking temperature at horizon can be obtained as

[mathematical expression not reproducible]. (24)

This temperature depends on radiation structure parameter [N.sub.r], mass M, and black hole charge Q.

2.3. The Black Hole Surrounded by the Quintessence Field. Third example of the line element of the black hole surrounded by the quintessence field [58] is given below:

[mathematical expression not reproducible], (25)

where [N.sub.q] is a quintessence field structure parameter. By following the same process and using the vector potential for this black hole, the surface gravity can be derived as

[mathematical expression not reproducible]. (26)

The corresponding tunneling probability

[mathematical expression not reproducible] (27)

and Hawking temperature

[mathematical expression not reproducible] (28)

are derived and given in the above expressions. The Hawking temperature depends on M, Q, and [N.sub.q], i.e., quintessence field structure parameter, mass, and charge of black hole, respectively.

2.4. The Black Hole Surrounded by the Cosmological Constant Field. Fourth example of the line element of black hole surrounded by the cosmological constant field is given below [58]:

[mathematical expression not reproducible], (29)

where [N.sub.c] is a cosmological constant field structure parameter. For this black hole, the surface gravity at outer horizon is obtained by following the above-mentioned similar procedure; i.e.,

[mathematical expression not reproducible]. (30)

Moreover, the required tunneling probability of particles

[mathematical expression not reproducible] (31)

and their corresponding Hawking temperature is calculated in the following expression:

[mathematical expression not reproducible]. (32)

This temperature depends on [N.sub.c], M, and Q, i.e., cosmological constant field structure parameter, mass, and charge of black hole, respectively.

2.5. The Black Hole Surrounded by the Phantom Field. Last example of the line element of black hole surrounded by the phantom field is [58]

[mathematical expression not reproducible], (33)

where [N.sub.p] is a phantom field structure parameter. For vector potential [A.sub.[mu]] of this black hole, the surface gravity can be derived as

[mathematical expression not reproducible]. (34)

The tunneling probability of particles

[mathematical expression not reproducible] (35)

and the required Hawking temperature of particles can be obtained as given below:

[mathematical expression not reproducible]. (36)

The Hawking temperature depends on M, Q, and [N.sub.p]; these are phantom field structure parameter, mass, and charge of black hole, respectively.

3. GUP-Corrected Proca Equation and the Corrected Hawking Temperature

In this section, we focus on the effect of the GUP on the tunneling of massive uncharged vector particles from the black hole solutions surrounded by perfect fluid in Rastall theory. Firstly we use the GUP-corrected Lagrangian for the massive uncharged vector field [[psi].sub.[mu]] given by [63]

[mathematical expression not reproducible]. (37)

One can derive the equation of the motion for the GUP-corrected Lagrangian of massive uncharged vector field as follows [63]:

[mathematical expression not reproducible], (38)

with

[mathematical expression not reproducible]. (39)

It is noted that we use the Latin indices for the modified tensor [[psi].sub.i[mu]] as follows: i = 1,2, 3; on the other hand, for [[psi].sub.0[mu]], we use the 0 for the time coordinate. Moreover, we note that [beta] = 1/(3[M.sup.2.sub.f]), where [M.sub.f] is the Planck mass and mw stands for the mass of the particle.

3.1. The Black Hole Surrounded by the Dust Field in Rastall Theory. The metric is given by

d[s.sup.2] = -G (r) (r) d[t.sup.2] + B (r) d[r.sup.2] + C (r) d[[theta].sup.2] + D(r) d[[phi].sup.2], (40)

where G(r), B(r), C(r), and D(r) are given below:

G(r) = 1 - 2M/r + [Q.sup.2]/[r.sup.2] - [N.sub.d]/[r.sup.(1-6[kappa][lambda])/1-3[kappa][lambda])], (41)

C(r) = [r.sup.2], B(r) = 1/G(r) = 1/1 - 2M/r + [Q.sup.2]/[r.sup.2] - [N.sub.d]/[r.sup.(1-6[kappa][lambda])/1-3[kappa][lambda])], (42)

D (r) = [r.sup.2][sin.sup.2][theta].

Using the WKB method, we define the [[psi].sub.[mu]] as follows:

[[PSI].sub.[mu]] = [c.sub.[mu]] (t, r, [theta], [phi]) exp [1/h I (t, r, [theta], [phi])], (43)

where I is defined as

I (t, r, [theta], [phi]) = [I.sub.0] (t, r, [theta], [phi]) + h[I.sub.1] (t, r, [theta], [phi]) + [h.sup.2][I.sub.2] (t, r, [theta], [phi]) + .... (44)

We use (43,44), and the metric (40) into (38), and then we only consider the lowest order terms in h to calculate the equations with the corresponding coefficients [c.sub.[mu]]:

[mathematical expression not reproducible], (45)

[mathematical expression not reproducible], (46)

[mathematical expression not reproducible], (47)

[mathematical expression not reproducible], (48)

where the [A.sub.[mu]]s are defined as

[mathematical expression not reproducible]. (49)

Using the semiclassical Hamilton-Jacobi method with WKB ansatz, we separate the variables as follows:

[I.sub.0] = -Et + R (r) + [THETA] ([theta],[phi]) + k. (50)

Note that the energy of the radiated particle is defined with E. Afterwards, we obtain a matrix equation as follows:

[universal] [([c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3]).sup.T] = 0, (51)

where [universal] is a 4x4 matrix, the elements of which are

[mathematical expression not reproducible], (52)

where R' = [[partial derivative].sub.r]R, [J.sub.[theta]] = [[partial derivative].sub.[theta]][THETA], and [J.sub.[phi]] = [[partial derivative].sub.[phi][THETA].

It is noted that, for the condition of det [universal] = 0, we find the nontrivial solution of (51). First, we consider only the lowest order terms of [beta] and then calculate the det [universal] = 0. Our main aim is to obtain the radial part of the equation so that we integrate it using the complex integral method around the event horizon as follows:

[mathematical expression not reproducible], (53)

where

[mathematical expression not reproducible], (54)

[T.sub.2] = -G (r) [m.sup.2][r.sup.2] + [r.sup.2] [(E).sup.2] - G (r) ([J.sup.2.sub.[theta]] + [J.sup.2.sub.[phi]]/D (r)). (55)

We rewrite the metric close to the event horizon to obtain solution for the integral:

G ([r.sub.h]) [approximately equal to] [[DELTA],.sub.r] ([r.sub.h])/r[h.sup.2] (r - [r.sub.h]). (56)

Afterwards we manage to obtain the solution of the integral (53) for the radial part as follows:

Im[R.sub.[+ or -]] (r) = [+ or -]i[pi] [r.sup.2.sub.h]/[[DELTA],.sub.r] ([r.sub.h]) (E) x (1 + [beta][XI]), (57)

where [XI] = 6[m.sup.2] + (6/[r.sup.2.sub.h])([J.sup.2.sub.[theta]] + [J.sup.2.sub.[phi]]cs[c.sup.2][theta]).

It is quite clear that [XI] > 0. We note that [R.sub.+] represents the radial function for the outgoing particles and [R.sub.-] is for the ingoing particles. Thus, the tunneling rate of W bosons near the event horizon is

[mathematical expression not reproducible]. (58)

If we set h = 1, we find the corrected Hawking temperature as follows:

[T.sub.e-H] = [[DELTA],.sub.r] ([r.sub.h])/4[pi][r.sup.2.sub.h] (1 + [beta][XI]) = [T.sub.0] (1 - [beta][XI]), (59)

Where [mathematical expression not reproducible] is the original Hawking temperature of a corresponding black hole. We find the corrected Hawking temperature with the effect of quantum gravity. In addition, the Hawking temperature is increased if one uses the quantum gravity effects, but then these effects are canceled in some point and black hole remnants occur.

3.2. The Black Hole Surrounded by the Radiation Field. Following the procedure given in Section 2.1 for this line element, we obtain the corrected Hawking temperature with the effect of quantum gravity for this charged black hole surrounded by the radiation field in the following form:

[T.sub.e-H] = [[DELTA],.sub.r] ([r.sub.h])/4[pi][r.sup.2.sub.h] (1 + [beta][XI]) = [T.sub.0] (1 - [beta][XI]), (60)

where original Hawking temperature is

[mathematical expression not reproducible]. (61)

This temperature depends on radiation structure parameter [N.sub.r], mass M, and black hole charge Q. Moreover, the quantum effects explicitly counteract the temperature increases during evaporation, which will cancel it out at some point. Naturally, black hole remnants will be left.

3.3. The Black Hole Surrounded by the Quintessence Field. By following the same process, we calculate the corrected Hawking temperature under the effect of quantum gravity as follows:

[T.sub.e-H] = [[DELTA],.sub.r] ([r.sub.h])/4[pi][r.sup.2.sub.h] (1 + [beta][XI]) = [T.sub.0] (1 - [beta][XI]), (62)

with the original Hawking temperature

[mathematical expression not reproducible], (63)

derived and given in above expressions. The Hawking temperature depends on M, Q, and [N.sub.q], i.e., quintessence field structure parameter, mass, and charge of black hole, respectively. Naturally, black hole remnants will be left.

3.4. The Black Hole Surrounded by the Cosmological Constant Field. One can repeat the process just for this black hole to calculate the corresponding corrected Hawking temperature with the quantum gravity effects as follows:

[T.sub.e-H] = [[DELTA],.sub.r] ([r.sub.h])/4[pi][r.sup.2.sub.h] (1 + [beta][XI]) = [T.sub.0] (1 - [beta][XI]), (64)

with

[mathematical expression not reproducible]. (65)

This temperature depends on [N.sub.c], M, and Q, i.e., cosmological constant field structure parameter, mass, and charge of black hole, respectively. Again here, remnants are left.

3.5. The Black Hole Surrounded by the Phantom Field. Last example is the line element of black hole surrounded by the phantom field. Now we again repeat the same process to obtain the following corrected Hawking temperature:

[T.sub.e-H] = [[DELTA],.sub.r] ([r.sub.h])/4[pi][r.sup.2.sub.h] (1 + [beta][XI]) = [T.sub.0] (1 - [beta][XI]), (66)

with

[mathematical expression not reproducible]. (67)

The Hawking temperature depends on M, Q, and [N.sub.p]; these are phantom field structure parameter, mass, and charge of black hole, respectively.

4. Conclusions

In this research paper, we have successfully analyzed the GUP-corrected Hawking temperature of [W.sup.[+ or -]] boson vector particles using the equation of motion for the Glashow-Weinberg-Salam model. First of all, we analyzed the modified Hamilton-Jacobi equation by resolving the modified Lagrangian equation utilized for the magnetized particles in the space-time. We have analyzed the GUP effect on the radiation of black holes surrounded by perfect fluid in Rastall theory.

As the original Hawing radiation, the Hawking temperature [T.sub.H] of the black holes is associated with its mass M and charged Q. However, these results indicated that the effect of quantum gravity is counted and the behavior of the tunneling boson vector particle on the event horizon is observed from the original event. The Hawking temperature [T.sub.H] and tunneling probability [GAMMA] quantities are not just sensitively dependent on the mass M and charged Q of the black hole. The Hawking temperature TH, tunneling probability r, and surface gravity [kappa] are only dependent on the geometry (structure parameter) of black hole. Moreover, the corrected Hawking temperature [T.sub.e-H] = [T.sub.0] (1 - [beta][XI]) has been calculated with the effect of quantum gravity. The Hawking temperature is increased if one uses the quantum gravity effects, but then these effects are canceled in some point and black hole remnants occur.

https://doi.org/10.1155/2018/3131620

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Comision Nacional de Ciencias y Tecnologla of Chile (CONICYT) through FONDECYT Grant No 3170035 (Ali Ovgun).

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Ali Ovgun (iD), (1,2) Wajiha Javed, (3) and Riasat Ali (3)

(1) Instituto de Fisica, Pontificia Universidad Catolica de Valparaiso, Casilla 4950, Valparaiso, Chile

(2) Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, Northern Cyprus, Mersin 10, Turkey

(3) Division of Science and Technology, University of Education, Township Campus, Lahore, Pakistan

Correspondence should be addressed to Ali Ovgun; ali.ovgun@pucv.cl

Received 17 April 2018; Revised 25 June 2018; Accepted 10 July 2018; Published 18 July 2018

Academic Editor: Saibal Ray
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Title Annotation:Research Article
Author:Ovgun, Ali; Javed, Wajiha; Ali, Riasat
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
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