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Based on the generalized uncertainty principle (GUP) , the researchers find that the quantum gravity affects the Klein-Gordon equation exactly. Hence, the Klein-Gordon equation which is corrected by GUP will be more suitable on the expression of the tunneling behavior. Then, the corrected Hawking temperature of the GHS black hole is obtained. After analyzing this result, we find out that the Hawking temperature is not only related to the mass of black hole, but also related to the mass and energy of outgoing fermions. Finally, we infer that the Hawking radiation will be stopped, and the remnants of black holes exist naturally.

In 1974, Hawking pointed out that Hawking radiation could be regarded as a process of tunneling where the vacuum fluctuation is caused at the event horizon of black holes [

In 2000, the development of quantum gravitational theory came into a period of rapid development. And the significant representative was the supergravity theory and loop quantum gravity theory. More and more evidences implied that the generalized uncertainty principle (GUP) could describe the minimum measurable length [

The expression of GUP is derived as [

where,

Also, many other various modifications are referred to [

The aim of this paper is to study the tunneling radiation of scalar particles in the GHS black hole with the Klein-Gordon equation near the horizon. The rest of this paper proceeds as follow: Section 2 modifies the Klein- Gordon equation; Section 3 studies the Hawking radiation of the GHS black hole with the Klein-Gordon equation; Section 4 is only a conclusion.

In this section, we will discuss the modified Klein-Gordon equation by the quantum gravity in carved spacetime. In 1995, Kempf put forward the GUP, and the expression can be found in Equation (2). In Equation (3), the canonical commutation relations which express as

To studied the effect which the quantum gravity have on the Klein-Gordon equation, we expand the Klein- Gordon equation as two parts, the left hand is relate to the square of energy, and the right hand is relate to the square of coordinate. So we rewrite this equation as,

In reference [

In above equation, the mass-energy shell condition

Carefully analysis on the modified Klein-Gordon equation, it is obvious for us that the quantum gravity has an important influence on the dynamic behavior of scalar particles. In the following section, we will aim at the tunneling behavior of scalar particles of the GHS black hole with the Klein-Gordon equation.

In this section, we are devoted to study tunneling radiation of scalar particles across the horizon of the GHS black hole in string theory by using modified Klein-Gordon equation. The GHS black hole is a member of a family of solutions to low energy string theory, the action can be written as,

where

Here,

And,

The symbol

where

The equation of motion of scalar particles is obtained,

For the convenience of later calculation, the higher order terms of

Here,

And after we considered the tunneling behavior is the radial, so the following form is obtained,

Here,

where,

Taking conditions

It is easy for us to calculate the roots of the above equation, after we substituted Equations (11), (12) into Equation (24), so the solution of this quartic equation at the horizon is,

With the path integral, substituting the metric of the GHS black hole into the above equation, we can finally find that the value of the GHS black hole,

And, ± can be related to outgoing/ingoing particles of the GHS black hole, the symbol

The relation between the tunneling rate and the action is,

So the corrected tunneling rate of the GHS black hole near the event horizon can be express as,

Therefore, it is easy for us to obtain the corrected Hawking temperature of the GHS black hole near the event horizon,

Neglecting the higher order terms of

And,

The corrected Hawking temperature and corrected tunneling rate of GHS black hole are obtained in Equations (29)-(32). It is obvious for us to see that the corrected Hawking temperature is not related to the metric of the black hole, but related to the mass and energy of the outgoing particles. So, the quantum gravity has an important influence on the Hawking radiation of the GHS black hole. Carefully analysis on results in Equation (31), it gives us the information that the quantum correction slows down the increase of temperature during the tunneling radiation, so the Hawking radiation will stop at some particular temperature. In this way, it imply us that residuum of GHS black hole is left certainly. Now, letâ€™s focus on the residuum of black holes. Considering the massless particle and the condition

The residue mass and the upper limit value of temperature in black hole can be express as,

Here,

In this paper, we aim at the quantum tunneling radiation of scalar particles of the GHS black hole with the consideration of quantum gravity. The results in Equation (31) imply that the tunneling radiation is not only related to the mass of the GHS black hole, but also related to the mass and energy of the outgoing particle. So we can realize that this correction on the Hawking radiation is very important, and can not be neglected. According to the Equation (31), it indicates that parameters

Now, the cosmic scale and microscale are all important in the research of the physics fields. The conclusion in this paper supports this viewpoint, and the present studies indicate that the quantum gravity is worth studying. Until now, it has attracted much more attention of theory physicists and become one of the most important issues in the astronomy and theoretical physics. In the future work, we will try more and more efforts on this issue.

Major Project of Education Department in Sichuan Province, No: 15ZA0325.