Advances in High Energy Physics
Volume 2016 (2016), Article ID 4647069, 6 pages
http://dx.doi.org/10.1155/2016/4647069

Research Article

Fermions Tunneling from Higher-Dimensional Reissner-Nordström Black Hole: Semiclassical and Beyond Semiclassical Approximation

ShuZheng Yang,¹ Dan Wen,1,2,3 and Kai Lin1,4

¹Department of Astronomy, China West Normal University, Nanchong, Sichuan 637002, China
²College of Physics and Information Science, Hunan Normal University, Changsha, Hunan 410081, China
³Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista, 12516-410 Guaratinguetá, SP, Brazil
⁴Instituto de Física e Química, Universidade Federal de Itajubá, 37500-903 Itajubá, MG, Brazil

Received 20 April 2016; Accepted 25 July 2016

Academic Editor: Mir Faizal

Copyright © 2016 ShuZheng 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.

Based on semiclassical tunneling method, we focus on charged fermions tunneling from higher-dimensional Reissner-Nordström black hole. We first simplify the Dirac equation by semiclassical approximation, and then a semiclassical Hamilton-Jacobi equation is obtained. Using the Hamilton-Jacobi equation, we study the Hawking temperature and fermions tunneling rate at the event horizon of the higher-dimensional Reissner-Nordström black hole space-time. Finally, the correct entropy is calculation by the method beyond semiclassical approximation.

Hawking radiation is an important prediction in modern gravitation theory [1–5]. Recently, Kraus et al. proposed quantum tunneling theory to explain and study Hawking radiation [6–31], and then semiclassical Hamilton-Jacobi method is put forward to research the properties of scalar particles’ tunnels [32–36]. In 2007, Kerner and Mann investigated the 1/2 spin fermion tunneling from static black holes [37]. In their work, the spin up and spin down cases are researched, respectively, and the radial equations are obtained, so that they can finally determine the Hawking temperature and tunneling rate at the event horizon. Subsequently, Kerr and Kerr-Newman black holes cases, the charged dilatonic black hole case, the de Sitter horizon case, the BTZ black hole case, 5-dimensional space-time cases, and several nonstationary black hole cases were all researched, respectively [38–48], and we used Hamilton-Jacobi method to study the fermion tunneling from higher-dimensional uncharged black holes [49, 50]. However, up to now, no one has studied higher-dimensional charged black holes cases, so we set out to research that case. In our work, we developed the Kerner and Mann method and proved that the semiclassical Hamilton-Jacobi equation can be obtained not only with the Klein-Gordon equation of curved space-time, but also with the Dirac equation in curved space-time. Applying the Hamilton-Jacobi equation, we can then obtain semiclassical Hawking temperature and tunneling rate at the event horizon of higher-dimensional Reissner-Nordström black hole.

In modern physics theory, the concept of an extra dimension can help to solve some theoretical issues, so several higher-dimensional metrics of curved space-time were investigated. The metric of static charged -dimensional Reissner-Nordström black hole is given by [10, 51–55]

where is the metric of -dimensional sphere

where and are mass and electric charge of black hole, and the electromagnetic potential is

where is volume of unit -sphere (we can adopt the units ). The outer/inner horizon located at

Obviously, at the horizons, the equation should be satisfied. However, the physical property near the inner horizon cannot be researched, so we just study the fermion tunneling at the outer event horizon of this black hole. The charged Dirac equation in curved space-time is

where

where and are mass and electric charge of the particles, and is the covariant derivative of tetrad . The gamma matrices in curved space-time need to be satisfied

After the gamma matrices are defined, we choose the gamma matrices in ()-dimensional flat space-time as

where and are unit matrices and flat gamma matrices with order and is the order of the matrices in even (odd) dimensional space-time. Corresponding to the flat case, the gamma matrices can be chosen as

Now, let us simplify the Dirac equation via semiclassical approximation, and rewrite the spinor function as

where and are column matrices with order and is classical action. Via semiclassical approximation method, Substituting (10) into (5) and dividing the exponential term and multiplying by , we can get

Solving (11), we have

It is evident that the coefficient matrices of (16) must vanish, when and have nontrivial solutions. Due to the fact that , we can write the condition that determinant of coefficient vanish as

From the relation of flat gamma matrices , we can obtain the semiclassical Hamilton-Jacobi equation in -dimensional Reissner-Nordström space-time

Using the Hamilton-Jacobi equation, in charged static space-time, we can separate the variables for the action as

and the Hamilton-Jacobi equation is broken up as

where (20) and (21) are radial and nonradial equations, respectively, and is a constant. However, we only research on the radial equation, because the tunnel at the event horizon is radial. From (20), we can get

Near the event horizon, the radial action is given by

where is part of the outgoing solution, while is the part of incoming solution, and

So the tunneling rate is

where represents the imaginary part of the function, and the Hawking temperature is

However, above calculation is worked on semiclassical approximation, because we ignored all higher order terms of . Recently, Banerjee and Majhi proposed a new method beyond semiclassical approximation to research the quantum tunneling, and their works show that the conclusion should be corrected [56–65], and this correct entropy may be applied in quantum gravity theory. Now let us generalize this work in higher-dimensional Reissner-Nordström black hole space-time.

Because the tetrad in the space-time are given by

so that is

It means the Dirac equation becomes

and this equation can be simplified at event horizon

because at event horizon, and is tortoise coordinate. On the other hand, the space-time background is static, and can be rewritten as

where and are matrices with and is frequency or energy of Dirac particle. Finally, applying the definitions of and , we get

here and are th elements of matrices and , respectively. Above equation becomes

so

At event horizon and depends on the position , so above equation implies

and the solution is

Above relation means and can be rewritten as

and are constants.

Next, let us use the method beyond semiclassical approximation to expand and as

so we get

Therefore, the determinants of matrices vanish:

In order to calculate the tunneling rate and Hawking temperature, we rewrite (, is area of black hole, and are dimensionless constant parameters) since the forms of are the same, so the total radial action is

The tunneling rate of Dirac particle at event horizon is given by

From the relation between tunneling rate and Hawking radiation, we get the temperature of black holes

Finally, the laws of black hole thermodynamics request

and is entropy of semiclassical approximation, and this result shows the correction of entropy is logarithmic correction.

In this paper, we studied fermions tunneling from higher-dimensional Reissner-Nordström black holes and obtained the Hamilton-Jacobi equation from charged Dirac equation. This work shows that semiclassical Hamilton-Jacobi equation can describe the property of both 0 spin scalar particles and 1/2 spin fermions. In this work, we did not emphasize dimensions of space-time larger than () dimensions, so the method also can be used in the research of ()-dimensions and lower cases.

As we all know, the information loss is an open problem in black hole physics, and the information of particles maybe vanishes at the singularity. In order to solve this difficulty, Horowitz and Maldacena proposed a boundary condition, which is called the black hole final state, at singularity of black hole to perfectly entangle the incoming Hawking radiation particles and the collapsing matter [66, 67]. Due to the boundary condition, any particle which is falling into the black holes completely annihilates. It is a new and interesting idea to investigate the black hole physics and quantum gravity, so we also will work on this area in the future.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported in part by FAPESP no. 2012/08934-0, CNPq, CAPES, and National Natural Science Foundation of China (no. 11573022 and no. 11375279).

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