Advances in High Energy Physics
Volume 2016 (2016), Article ID 7058764, 7 pages
http://dx.doi.org/10.1155/2016/7058764

Research Article

Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation

Shu-Zheng Yang,¹ Kai Lin,2,3 Jin Li,⁴ and Qing-Quan Jiang¹

¹Department of Astronomy, China West Normal University, Nanchong, Sichuan 637002, China
²Instituto de Física e Química, Universidade Federal de Itajubá, 37500-903 Itajubá, MG, Brazil
³Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970 São Paulo, SP, Brazil
⁴Department of Physics, Chongqing University, Chongqing 400030, China

Received 20 April 2016; Accepted 8 June 2016

Academic Editor: Giulia Gubitosi

Copyright © 2016 Shu-Zheng Yang 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.

Recently the modified Dirac equation with Lorentz invariance violation has been proposed, which would be helpful to resolve some issues in quantum gravity theory and high energy physics. In this paper, the modified Dirac equation has been generalized in curved spacetime, and then fermion tunneling of black holes is researched under this correctional Dirac field theory. We also use semiclassical approximation method to get correctional Hamilton-Jacobi equation, so that the correctional Hawking temperature and correctional black hole’s entropy are derived.

1. Introduction

In 1974, Hawking proved black holes could radiate Hawking radiation once considering the quantum effect near the horizons of black holes [1, 2]. This theory indicates that black hole would be viewed as thermodynamic system, so that black hole physics can be connected closely with gravity, quantum theory, and thermodynamics physics. According to the view point of quantum tunneling theory, the virtual particles inside of black hole could cross the horizon due to the quantum tunneling effect and become real particles and then could be observed by observers as Hawking radiation. Wilczek et al. proposed a semiclassical method to study the quantum tunneling from the horizon of black hole [3–21]. Along with this method, the Hamilton-Jacobi method was applied to calculate the Hawking tunneling radiation. According to the Hamilton-Jacobi method, the wave function of Klein-Gordon equation can be rewritten as (where is semiclassical action) and Hamilton-Jacobi equation is obtained via semiclassical approximation. Using the Hamilton-Jacobi equation, the tunneling rate could be calculated by the relationship (where is the tunneling rate at the horizon of black hole), and then the Hawking temperature can be determined. People have applied this method to research Hawking tunneling radiation of several static, stationary, and dynamical black holes.

However, since the Hamilton-Jacobi equation is derived from Klein-Gordon equation, original Hamilton-Jacobi method just can be valid for scalar particles in principle. Therefore, Kerner and Mann studied fermions tunneling of black hole by a new method [22–32], which assumes the wave function of Dirac equation as spin-up and spin-down and then calculates the fermions tunneling, respectively. Nevertheless, this method is still impossible to apply in arbitrary dimensional spacetime. Our work in 2009 showed that the Hamilton-Jacobi equation can also be derived from Dirac equation via semiclassical approximation, so we proved that the Hamilton-Jacobi equation can be used to study the fermions tunneling directly [33–35].

On the other hand, as the basis of general relativity and quantum field theory, Lorentz invariance is proposed to be spontaneously violated at higher energy scales. A possible deformed dispersion relation is given by [36–44]

where and are the energy and momentum of particle and is “minimal length” with the order of the Plank length. The work of spacetime foam Liouville string models has introduced this relation with , and people also proposed quantum equation of spineless particles by using this relation. Recently, Kruglov considers the deformed dispersion relation with and proposes modified Dirac equation [45]:

where is ordinary gamma matrix and is space coordinate, while is spacetime coordinate. The effect of the correctional term would be observed in higher energy experiment.

In this paper, we try to generalize the modified Dirac equation in curved spacetime and then study the correction of Hawking tunneling radiation. In Section 2, the modified Dirac equation in curved spacetime is constructed and then the modified Hamilton-Jacobi equation is derived via semiclassical approximation. We apply the modified Hamilton-Jacobi equation to the fermions tunneling radiation of -dimensional black string and higher dimensional BTZ-like black strings in Sections 3 and 4, respectively, and Section 5 includes some conclusion and the discussion about the correction of black hole’s entropy.

2. Modified Dirac Equation and Hamilton-Jacobi Equation in Curved Spacetime

As we all know, the gamma matrix and partial derivative should become gamma matrix in curved spacetime and covariant derivative, respectively, namely,

where satisfy the relationship , is charged term of Dirac equation, and is spin connection. According to this transformation, we can construct the modified Dirac equation in curved spacetime as

where we choose but , while in (1) and (2). It is assumed that , so that the correctional term is very small.

Now let us use the modified Dirac equation to derive the modified Hamilton-Jacobi equation. Firstly, we rewrite the wave function of Dirac equation as [33–35]

where and are matrices and . In semiclassical approximation, we can consider that is very small, so that we can neglect the terms with after dividing by the exponential terms and multiplying by . Therefore, (4) is rewritten as

Considering the relationship

we can get

where

Now, multiplying both sides of (9) by the matrix , we can obtain

The second term of the above equation could be simplified again by (8), so the above equation can be rewritten as

where we can prove the relation

We always ignore terms because is very small. Now, let us exchange the position of and in (11) and consider the relation of gamma matrices ; then we can obtain

Namely,

where

Using the idea of (10)-(11) again, we can multiply both sides of (15) by the matrix , so that the equation becomes

The second term of the above equation could be simplified again by (15). Then, exchange and and use the relationship , so the above equation can be rewritten as

The condition that (17) has nontrivial solution required the determinant of coefficient in (17) should vanish, so we can directly get the equation

Consider the square root for left side of (18) and ignore all terms, so we can directly get the modified Hamilton-Jacobi equation:

Therefore, we find that the modified Dirac equation from Lorentz invariance violation could lead to the modified Hamilton-Jacobi equation, and the correction of Hamilton-Jacobi equation depends on the energy and mass of radiation fermions. Using the modified Hamilton-Jacobi equation, we then investigate the fermions Hawking tunneling radiation of -dimensional black string and -dimensional BTZ-like string in the following two sections.

3. Fermions Tunneling of -Dimensional Black String

The research of gravity in dimension can help people further understand the properties of gravity, and it is also important to construct the quantum gravity. Recently, Murata et al. have researched the -dimensional gravity with dilaton field, whose action is given by [46]

where , , and are, respectively, the dilaton field, 3-dimensional Planck mass, and the parameter with mass dimension. The static black string solution is given by

It is evident that the horizon of this black hole is , but the black string is unstable as (where is scale of compactification), so it is assumed that .

Now we research the fermions tunneling of this black hole, so the modified Hamilton-Jacobi equation in this spacetime is given by

where we have set , yielding the radial Hamilton-Jacobi equation as

where the constant is from separation of variables, and (23) finally can be written as

At the horizon of the black string, the above equation is integrated via residue theorem, and we can get

and the fermions tunneling rate

The relationship between tunneling rate and Hawking temperature required

where and are the modified and nonmodified Hawking temperature of the -dimensional black string, respectively, and term is the correction.

4. Fermions Tunneling of Higher Dimensional BTZ-Like Black Strings

As we all know that the linear Maxwell action fails to satisfy the conformal symmetry in higher dimensional spacetime [47, 48], so Hassaïne and Martínez proposed gravity theory with nonlinear Maxwell field in arbitrary dimensional spacetime:

where is cosmological constant. Hendi researched -dimensional static black strings solution with , , and , and it is charged BTZ-like solutions [49], whose metric is

where

and the electromagnetic potential is

As , this solution is no other than the static charged BTZ solution. We will study the Hawking radiation and black hole temperature at the event horizon of this black string. In (19), we can set , where are the space coordinates excluding the radial coordinate, so that the modified Hamilton-Jacobi equation is given by

and the radial equation with constant is

Therefore, at the horizon of the black string, and we finally get

where . This means that the fermions tunneling rate is

and the Hawking temperature is

where and are the modified and nonmodified Hawking temperature of the -dimensional BTZ-like black string, respectively, and term is the correction.

5. Conclusions

In this paper, we consider the deformed dispersion relation with Lorentz invariance violation and generalize the modified Dirac equation in curved spacetime. The fermions tunneling radiation of black strings is researched, and we find that the modified Dirac equation could lead to Hawking temperature’s correction, which depends on the correction parameter and particle mass in the modified Dirac equation. Next we will discuss the correction of black hole entropy in this theory.

The first law of black hole thermodynamics requires

where and are electromagnetic potential and rotating potential, so the nonmodified entropy of black hole is [1, 2, 50, 51]

From the above results, we know that the relationship between modified and nonmodified Hawking temperature is , since the nonmodified black hole entropy is given by

where we can ignore because . Equation (39) shows that the correction of black hole entropy depends on , which is independent from time and space coordinates. However, it is possible that depends on other parameters in curved spacetime, and it is very interesting that depends on . In particular, as , we can get the logarithmic correction of black hole entropy

In quantum gravity theory, the logarithmic correction has been researched in detail [52–69], and according to [70], it is required that the coefficient of logarithmic correction should be in -dimensional spacetime, so it indicates that could be .

On the other hand, from the deformed dispersion relation (1) with , it is implied that the Klein-Gordon equation could be given by

so the generalized uncharged Klein-Gordon equation in static curved spacetime is

and using the semiclassical approximation with , the modified Hamilton-Jacobi equation in scalar field is given by

Namely,

Contrasting (19) and (44) as uncharged case, we find that the correctional terms of Dirac field and scalar field are very different. The fact implies that the corrections of Hawking temperature and black hole entropy from Hawking tunneling radiation with different spin particles could be different, and this conclusion could be helpful to suggest a new idea to research the black hole information paradox. Work in these fields is currently in progress.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by FAPESP no. 2012/08934-0, CNPq, CAPES, and National Natural Science Foundation of China no. 11573022, no. 11205254, no. 11178018, and no. 11375279.

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