Advances in High Energy Physics
Volume 2018 (2018), Article ID 3131620, 11 pages
https://doi.org/10.1155/2018/3131620

Research Article

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

Ali Övgün,1,2 Wajiha Javed,³ and Riasat Ali³

¹Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso, Chile
²Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, Northern Cyprus, Mersin 10, Turkey
³Division of Science and Technology, University of Education, Township Campus, Lahore, Pakistan

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

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

Academic Editor: Saibal Ray

Copyright © 2018 Ali Övgün et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

Using the semiclassical WKB approximation and Hamilton-Jacobi method, we solve an equation of motion for the Glashow-Weinberg-Salam model, which is important for understanding the unified gauge-theory of weak and electromagnetic interactions. We calculate the tunneling rate of the massive charged W-bosons in a background of electromagnetic field to investigate the Hawking temperature of black holes surrounded by perfect fluid in Rastall theory. Then, we study the quantum gravity effects on the generalized Proca equation with generalized uncertainty principle (GUP) on this background. We show that quantum gravity effects leave the remnants on the Hawking temperature and the Hawking radiation becomes nonthermal.

1. Introduction

General relativity is analogously linked to the thermodynamics and quantum effects which strongly support it [1–3]. 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 radiate at temperature [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 [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 , , and 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]:

where is a mass of black hole, and are the Rastall geometric parameters, is the dust field structure parameter, and is a charge of black hole. Now, we can rewrite (1) in the following form:

where , , , and are given below:

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

where is a coefficients matrix, is particles mass, and is antisymmetric tensor, since

where is the vector potential of the charged black hole and and are the components of , is the charge of the particle, and is covariant derivative. The values of and are given by

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

to the Lagrangian (4) (where and correspond to particles action, for ) and neglecting the higher order terms, we get the following set of equations given below:

We can choose by using separation of variables technique, i.e.,

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

which provides “” as a matrix of order and its components are given by

where ,  , and . For the nontrivial solution and solving above equations one can yield

where + and − represent the outgoing and incoming particles, respectively, whereas, “” is the function which can be defined as

and is the angular velocity at event horizon.

By integrating (15) around the pole, we get

where the surface gravity of the charged black hole is given by

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

Now, we can calculate the by comparing the with the Boltzmann formula , and we get

The result shows that the is dependent on , the vector potential components ( and ), energy , angular momentum , and mass of black hole ; and are the Rastall geometric parameters; and and 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:

where is a mass of black hole, is the negative radiation structure parameter, and is a charge of black hole.

Following the procedure given in Section 2.1 for this line element, we can obtain the surface gravity of this charged black hole surrounded by the radiation field in the following form:

where . The tunneling rate of particles can be calculated as

and the corresponding Hawking temperature at horizon can be obtained as

This temperature depends on radiation structure parameter , mass , and black hole charge .

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:

where 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

The corresponding tunneling probability

and Hawking temperature

are derived and given in the above expressions. The Hawking temperature depends on , , and , 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]:

where 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.,

Moreover, the required tunneling probability of particles

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

This temperature depends on , , and , 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]

where is a phantom field structure parameter. For vector potential of this black hole, the surface gravity can be derived as

The tunneling probability of particles

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

The Hawking temperature depends on , , and ; 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 given by [63]

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

with

It is noted that we use the Latin indices for the modified tensor as follows: ; on the other hand, for , we use the for the time coordinate. Moreover, we note that , where is the Planck mass and 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

where , , , and are given below:

Using the WKB method, we define the as follows:

where is defined as

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

where the s are defined as

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

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

where is a matrix, the elements of which are

where , , and .

It is noted that, for the condition of , we find the nontrivial solution of (51). First, we consider only the lowest order terms of and then calculate the . 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:

where

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

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

where .

It is quite clear that . We note that represents the radial function for the outgoing particles and is for the ingoing particles. Thus, the tunneling rate of bosons near the event horizon is

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

where 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:

where original Hawking temperature is

This temperature depends on radiation structure parameter , mass , and black hole charge . 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:

with the original Hawking temperature

derived and given in above expressions. The Hawking temperature depends on , , and , 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:

with

This temperature depends on , , and , 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:

with

The Hawking temperature depends on , , and ; 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 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 of the black holes is associated with its mass and charged . 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 and tunneling probability quantities are not just sensitively dependent on the mass and charged of the black hole. The Hawking temperature , tunneling probability , and surface gravity are only dependent on the geometry (structure parameter) of black hole. Moreover, the corrected Hawking temperature 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.

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 Comisión Nacional de Ciencias y Tecnología of Chile (CONICYT) through FONDECYT Grant 3170035 (Ali Övgün).

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