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In this article, close to the Planck scale, we discuss the remnant and residue entropy from a Rutz-Schwarzschild black hole in the frame of Finsler geometry. Employing the corrected Hamilton-Jacobi equation, the tunneling radiation of a scalar particle is presented, and the revised tunneling temperature and revised entropy are also found. Taking into account generalized uncertainty principle (GUP), we analyze the remnant stability and residue entropy based on thermodynamic phase transition. In addition, the effects of the Finsler perturbation parameter, GUP parameter, and angular momentum parameter on remnant and residual entropy are also discussed.

Although a complete self-consistent theory of quantum gravity has not been established, it is an effective way to understand the behavior of gravity by combining various models of quantum gravitational effects. In various quantum gravitation models, such as string theory, loop quantum gravity, and non-commutative geometry, it is believed that there exists a minimum observable length, and this minimum observable length should have the order of the Planck scale. Considering a minimum observable length, a Hilbert space representation of quantum mechanics has been formulated by Kempf et al. in [

Recently, based on GUP, people began to study the quantum tunneling and remnant of black hole. Combining the GUP with the minimum observed length into the tunneling method of Parikh and Wilczek, Nozari and Mehdipour [

Although Einstein’s general relativity described by Riemannian geometry is one of the most successful gravitational theories, it still has some problems in explaining the accelerating expansion of the universe and establishing a complete theory of quantum gravity. One has considered that the difficulties caused by general relativity may have something to do with the mathematical tools it uses. So, people try to establish a modified gravitational theory described by Finsler geometry. Finsler geometry is the most general differential geometry, which regards Riemannian geometry as its special case, and it is just Riemannian geometry without quadratic restriction [

In this section, considering the effect of GUP, applying with the corrected Hamilton-Jacobi equation, we will focus on investigating the scalar particle’s tunneling radiation from a Rutz-Schwarzschild black hole. Under the frame of Finsler geometry, Rutz constructed the generalized Einstein field equation and derived a Finsler black hole solution. The metric (Rutz-Schwarzschild black hole) is given by [

By taking into account the effect of GUP, in a curved spacetime, the revised Hamilton-Jacobi equation for the motion of scalar particles can be expressed as [

On the basis of the above scalar particle’s tunneling radiation, considering the GUP, now we focus on discussing on the remnant and entropy at the end of evaporation. Since all the tunneling particles at the event horizon can be regarded as massless, the mass of scalar particles is no longer considered in the following process. According to the uncertainty relation

As Figures

The figure shows the temperature

The figure shows the temperature

The figure shows the heat capacity

In Figure

In Figures

The figure shows the entropy

The figure shows the entropy

To see clearly the Finsler perturbation parameter and angular momentum parameter effect on remnant and residual entropy, we present Tables

Remnant and residual entropy with

| | |
---|---|---|

0.1 | 0.749748 | 3.29388 |

0.01 | 0.714856 | 1.90766 |

0.0001 | 0.711309 | 1.76666 |

Remnant and residual entropy with

| | |
---|---|---|

0.5 | 1.054090 | 22.8521 |

0.1 | 0.814124 | 7.17324 |

0.01 | 0.760867 | 3.95553 |

In addition, when

Based on GUP, using Parikh-Wilczek tunneling method, we continue to explore the quantum tunneling and remnant of a Rutz-Schwarzschild black hole in the frame of Finsler geometry. The Rutz-Schwarzschild metric in the Painleve coordinate system can be rewritten as

In conclusion, employing the tunneling radiation of a scalar particle and generalized uncertainty principle, we study the remnant and residue entropy from a Rutz-Schwarzschild black hole based on black hole thermodynamic. Firstly, based on the Hamilton-Jacobi equation revised by GUP, we present the modified tunneling temperature-uncertainty relation and modified entropy-uncertainty relation by using quantum tunneling method. Then, using the generalized uncertainty principle, we calculate the remnant and residual entropy after evaporation when the black hole reaches the Planck scale. Finally, based on black hole thermodynamic phase transition, a detailed analysis of whether there is a stable remnant and residual entropy in the final stage of evaporation is given; that is, the thermodynamic stability of a black hole is related to its thermal capacity and temperature. In order to ensure the thermal stability of a black hole, when the black hole evaporates to the Planck scale, the black hole with negative heat capacity must be transformed into a black hole with positive heat capacity. As a result, at remnant, the modified tunneling temperature and heat capacity tend to zero and modified entropy reaches the minimum value, which imply the black holes are in thermal equilibrium with the outside environment. In addition, the effects of the Finsler perturbation parameter and angular momentum parameter on remnant and residual entropy are also discussed.

The emergence of Finsler black hole solution infuses new vitality into general relativity and puts forward a new research and development idea for black hole physics theory. In this paper, we consider a simple Finsler black hole. For the Rutz-Schwarzschild black hole, we investigate the remnant and residue entropy based on the scalar particles’ tunneling radiation via semiclassical Hamilton-Jacobi method. In view of the size of the Finsler perturbation parameter

We also note that, the expression of GUP is not unique, which gives rise to different correction on tunneling radiation. Therefore, the remnant and residue entropy should also be distinct from that in the frameworks of other GUPS. We are going to discuss the related issues in further.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work is supported in part by the National Natural Science Foundation of China (Grant no. 11703018) and Natural Science Foundation of Liaoning Province, China (Grant no. 20180550275).