Advances in High Energy Physics
Volume 2018 (2018), Article ID 8728564, 7 pages
https://doi.org/10.1155/2018/8728564

Research Article

Quantum Gravity Effect on the Tunneling Particles from 2 + 1-Dimensional New-Type Black Hole

Ganim Gecim and Yusuf Sucu

Department of Physics, Faculty of Science, Akdeniz University, 07058 Antalya, Turkey

Correspondence should be addressed to Yusuf Sucu; ysucu@akdeniz.edu.tr

Received 13 October 2017; Revised 10 January 2018; Accepted 24 January 2018; Published 6 March 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Ganim Gecim and Yusuf Sucu. 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.

We investigate the generalized uncertainty principle (GUP) effect on the Hawking temperature for the 2 + 1-dimensional new-type black hole by using the quantum tunneling method for both the spin-1/2 Dirac and the spin-0 scalar particles. In computation of the GUP correction for the Hawking temperature of the black hole, we modified Dirac and Klein-Gordon equations. We observed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the graviton mass and the intrinsic properties of the tunneling particle, such as total angular momentum, energy, and mass. Also, we see that the Hawking temperature was found to be probed by these particles in different manners. The modified Hawking temperature for the scalar particle seems low compared with its standard Hawking temperature. Also, we find that the modified Hawking temperature of the black hole caused by Dirac particle’s tunneling is raised by the total angular momentum of the particle. It is diminishable by the energy and mass of the particle and graviton mass as well. These intrinsic properties of the particle, except total angular momentum for the Dirac particle, and graviton mass may cause screening for the black hole radiation.

1. Introduction

Black hole radiation is theoretically very important phenomenon for researchers who attempt to merge the gravitation with the thermodynamics and the quantum mechanics [1–8]. With the formulation of the quantum field theory in curved space-time based on the framework of the standard Heisenberg uncertainty principle, it was proved that a black hole can emit particles that are created by the quantum vacuum fluctuation near its outer horizon [6–8]. Since then, many alternative methods have been proposed to derive the black hole radiation known as Hawking radiation in the literature. For instance, the semiclassical method, based on quantum tunneling process of a particle across the outer horizon of a black hole from inside to outside, can be used to derive the Hawking radiation. The method implies two different approaches to compute the imaginary part of the action (), which is the classically forbidden trajectory of a particle across the outer horizon: the null geodesic [9–12] and the Hamilton-Jacobi [13–15]. In both approaches, the tunneling probability of a particle from a black hole, , is defined in terms of the classical action, as [9–15]. By using the semiclassical method, lots of studies about the Hawking radiation of a black hole as quantum tunneling process of a point-like particle have been carried out in the literature [16–36]. On the other hand, the above-mentioned studies on this method provide no specific information on type of the particle that is tunneled from a black hole. That is because the Hawking radiation does not depend on the intrinsic properties, such as mass, total (orbital + spin) angular momentum, energy, and charge of the tunneling point-like particle.

The existence of a minimal observable length which can be identified by the order of the Planck scale is a characteristic of the candidate theories of quantum gravity, such as string theory, loop quantum gravity, and noncommutative geometry [37–41]. This length leads us to a generalized uncertainty principle (GUP) instead of the standard Heisenberg uncertainty principle, because a particle is not a point-like particle in the context of these candidate theories anymore. Therefore, the uncertainty on the momentum of a particle increases and thus the standard Heisenberg uncertainty principle can be generalized as follows:

where , is the Planck mass, and is the dimensionless parameter [42–45]. The commutation relations between a particle position, , and momentum, , are modified as follows:

where and represent the modified position and momentum operators, respectively, and their definitions are as follows:

and in (3) are the standard position and momentum operators, respectively, and [46]. Then, the modified energy expression becomes

for which the energy mass shell condition, , is used. From these relations, the square of the momentum operator can be derived as follows:

where the higher order terms of the parameter are neglected.

The GUP relations are of great help to understand the nature of a black hole since quantum effects are the essential effects near the event horizon of a black hole. Recently, to investigate the quantum effects under the GUP relations, the thermodynamics properties of various black holes have been studied by using the quantum tunneling process of particles with various spins [46–67]. These studies indicate that the modified Hawking radiation depends not only on the black hole’s properties but also on the intrinsic properties of the tunneling particle.

The new-type black hole is one of the important results of the New Massive Gravity, which is 2 + 1-dimensional gravity and graviton in this theory has a mass [68]. In the framework of the standard Heisenberg uncertainty principle, the Hawking radiation of the new-type black holes had been studied by using the quantum tunneling process of the scalar, Dirac, and vector boson particles [23, 36]. In these studies, it was shown that the Hawking radiation only depends on the properties of the black hole and is independent of the properties of the tunneling point particles, that is, all these particles tunnel from the black holes in the same way. This indicates that even if an observer in enough or safe (i.e., infinite) distant from a black hole may detect Hawking radiation of the black hole, the observer cannot determine what kind of particles compose the radiation. Therefore, in this study, we will investigate whether the properties of the tunneling particles will affect the Hawking radiation of the black hole by using the quantum tunneling process of both the scalar and Dirac particles in the framework of the GUP.

The organization of this work is as follows: in Section 2, we modify the Dirac equation with respect to the GUP relations. Subsequently, using the modified Dirac equation, we calculate the tunneling probability of the Dirac particle by using the Hamilton-Jacobi method, and, then, we find the modified Hawking temperature of the black hole. In Section 3, the standard Klein-Gordon equation is rewritten under the GUP for the 2 + 1-dimensional new-type black hole. Subsequently, the tunneling probability of the scalar particle from the black hole and the modified Hawking temperature of the black hole are calculated, respectively. In conclusion, we evaluate and summarize the results.

2. Dirac Particle’s Tunneling in the New-Type Black Hole

Using the GUP relations, the standard Dirac equation given in [69] can be modified as follows:

and its explicit form is

where the is the modified Dirac spinor, is mass of the Dirac particle, are the space-time-dependent Dirac matrices, and are the spin affine connection for spin-1/2 particle [69]. The space-time background of the new-type black hole is given by

where is the AdS₃ radius defined as where is cosmological constant and is graviton mass, and is defined in terms of the outer, , and inner, , horizons radius, respectively. The black hole’s horizons are located at , where and are two constant parameters [23, 68]. Using (8), the spinorial affine connections are derived as follows [23]:

To calculate the tunneling probability of a Dirac particle from the black hole, we use the following ansatz for the modified wave function:

where and are the functions of space-time. is the classical action term for particle trajectory. Inserting (9) and (10) in (7), we obtain the following equations for the leading order in and as follows:

These two equations have nontrivial solutions for and when the determinant of the coefficient matrix is vanished. Accordingly, neglecting of the terms containing higher order of the parameter provides

Due to the commuting Killing vectors and , we can separate , in terms of the variables , , and , as , where and are the energy and angular momentum of the particle, respectively, and [57]. Using these definitions in (12), the integral of the radial equation, , becomes

where is an abbreviation and it is

Then, by integrating the radial equation, are obtained as

where the abbreviation is

On the other hand, the tunneling probabilities of particles crossing the outer horizon are given by

Hence, the tunneling probability of the Dirac particle is given by

where [70, 71] and . Then, the modified Hawking temperature of the Dirac particle, , is obtained as

where to find the temperature it used the following relation [72–75]:

If is at first expanded in terms of the powers and second neglected the higher order of the terms, then the modified Hawking temperature of the black hole is obtained as

From this result, we see that the modified Hawking temperature includes not only the mass parameter of the black hole, but also the AdS₃ radius, , (and, hence, the graviton mass) and the angular momentum, energy, and mass of the tunneled Dirac particle. On the other hand, in the case of , the modified Hawking temperature is reduced to the standard temperature obtained by quantum tunneling process of the point particles with spin-0, spin-1/2, and spin-1, respectively [23, 36].

3. Scalar Particle’s Tunneling in the New-Type Black Hole

To investigate the quantum gravity effects on the tunneling process of the scalar particles from the black hole, by using the GUP relations, the standard Klein-Gordon equations are modified as follows:

and its explicit form of the modified Klein-Gordon equation is written as follows:

where and are the modified wave function and mass of the scalar particle, respectively. Then, the modified Klein-Gordon equation in the new-type black hole background becomes as follows:

To consider the tunneling radiation of the black hole with (24), we employ the modified wave function of the scalar particle as

where is a constant. Substituting (25) into (24) and neglecting the higher order terms of , we get the equation of motion of the scalar particle as

Using , where and are the energy and angular momentum of the particle, respectively, and [57], then, the radial integral, , becomes as follows:

where the abbreviation is

And, are computed as

with the abbreviation being

where is outgoing and is incoming solutions of the radial part. Then, using (17), the tunneling probability of the scalar particle is calculated as

and, subsequently, using (20), the modified Hawking temperature of the scalar particle, , becomes as

Furthermore, neglecting the higher order of the terms in the expanding form of in terms of , we find the modified Hawking temperature of the black hole as follows:

This result indicates that the modified Hawking temperature is related not only to the mass parameter of the black hole, but also to the AdS₃ radius, , (and, hence, to the graviton mass) and angular momentum, energy, and mass of the tunneling scalar particle. Furthermore, as can be seen from (21) and (33), the Hawking temperature probed by a Dirac particle is higher than that of a scalar particle: for and . On the other hand, in the case of , the modified Hawking temperature reduced to the standard temperature obtained by quantum tunneling process of the point particles with spin-0, spin-1/2, and spin-1, respectively [23, 36].

4. Concluding Remarks

In this study, we investigated the quantum gravity effect on the tunneled both spin-0 scalar and spin-1/2 Dirac particles from new-type black hole in the context of 2 + 1-dimensional New Massive Gravity. For this, at first, using the GUP relations, we modified the Klein-Gordon and Dirac equations that describe the spin-0 scalar and spin-1/2 Dirac particles, respectively. Then, using the Hamilton-Jacobi method, the tunneling probabilities of the these particles are derived, and, subsequently, the corrected Hawking temperature of the black hole is calculated. We find that the modified Hawking temperature not only depends on the black hole’s properties but also depends on the emitted particle’s mass, energy, and total angular momentum. Also, it is worth mentioning that the modified Hawking temperature depends on mass of the graviton, that is, quantum particle which mediates gravitational radiation in the context of New Massive Gravity. As can be seen from (21), the Hawking temperature of the Dirac particle increases by the total angular momentum of the particle while it decreases by the energy and mass of the particle and the graviton mass.

In addition, we can summarize some important results as follows:

(i)In (33), the modified Hawking temperature of the scalar particle is lower than the standard Hawking temperature.(ii)However, in (21), as , the modified Hawking temperature of the Dirac particle is higher than the standard Hawking temperature. Furthermore, when , the modified Hawking temperature is lower than the standard Hawking temperature. If , then the GUP effect is canceled, and the Hawking temperature of the Dirac particle reduces to the standard Hawking temperature.(iii)According to (21) and (33), the modified Hawking temperature of the new-type black hole probed by tunneling Dirac particle is higher than that of scalar particle:

where we adopt that the mass of the Dirac particle is equivalent to the mass of the scalar particle, that is, .(iv)The new-type black hole is classified as six classes according to the signatures of the parameters and , and, hence, it exhibits different physical and mathematical properties. For example, it reduced to the static BTZ black hole in the case of and . In this context, according to tunneling of the scalar and Dirac particles, the modified Hawking temperature of the static BTZ black hole is

respectively. Here, is used and is the standard Hawking temperature of the static BTZ black hole in the context of the 2 + 1-dimensional New Massive Gravity theory [23].(v)In the absence of the quantum gravity effect, that is, , the modified Hawking temperature is reduced to the standard temperature obtained by quantum tunneling of the massive spin-0, spin-1/2, and spin-1 point particles [23, 36].

Finally, in the context of GUP, we have seen that the graviton and the tunneling particle masses have an effect decreasing the Hawking temperature in both scalar and Dirac particle tunneling process. On the other hand, the total angular momentum has different effect on the Hawking temperature for a type of tunneling particle. For a scalar particle, it results in decrease in the temperature whereas it provides an increase in the temperature for a Dirac particle. These results show that the intrinsic properties of the particle, except total angular momentum for the Dirac particle, and graviton mass may cause screening for the black hole radiation.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Akdeniz University, Scientific Research Projects Unit.

References

J. M. Greif, Junior Thesis, Princeton University, 1969.

B. Carter, “Rigidity of a black hole,” Nature Physical Science, vol. 238, no. 83, pp. 71-72, 1972.

J. D. Bekenstein, “Black holes and entropy,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 7, no. 8, pp. 2333–2346, 1973.

J. D. Bekenstein, “Generalized second law of thermodynamics in black-hole physics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 9, no. 12, pp. 3292–3300, 1974.

J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics, vol. 31, pp. 161–170, 1973.

S. W. Hawking, “Black hole explosions?” Nature, vol. 248, no. 5443, pp. 30-31, 1974.

S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975.

S. W. Hawking, “Black holes and thermodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 13, no. 2, pp. 191–197, 1976.

P. Kraus and F. Wilczek, “Self-interaction correction to black hole radiance,” Nuclear Physics B, vol. 433, no. 2, pp. 403–420, 1995.

P. Kraus and F. Wilczek, “Effect of self-interaction on charged black hole radiance,” Nuclear Physics B, vol. 437, no. 1, pp. 231–242, 1995.

M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” Physical Review Letters, vol. 85, no. 24, pp. 5042–5045, 2000.

M. K. Parikh, “New coordinates for de Sitter space and de Sitter radiation,” Physics Letters B, vol. 546, no. 3-4, pp. 189–195, 2002.

M. Angheben, M. Nadalini, L. Vanzo, and S. Zerbini, “Hawking radiation as tunneling for extremal and rotating black holes,” Journal of High Energy Physics, vol. 2005, no. 5, article 014, 2005.

K. Srinivasan and T. Padmanabhan, “Particle production and complex path analysis,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 60, no. 2, Article ID 024007, 20 pages, 1999.

S. Shankaranarayanan, K. Srinivasan, and T. Padmanabhan, “Method of complex paths and general covariance of Hawking radiation,” Modern Physics Letters A, vol. 16, no. 9, pp. 571–578, 2001.

I. Sakalli and A. Övgün, “Black hole radiation of massive spin-2 particles in (3+1) dimensions,” The European Physical Journal Plus, vol. 131, no. 6, article 184, 13 pages, 2016.

A. Yale and R. B. Mann, “Gravitinos tunneling from black holes,” Physics Letters B, vol. 673, no. 2, pp. 168–172, 2009.

R. Li and J.-R. Ren, “Dirac particles tunneling from BTZ black hole,” Physics Letters B, vol. 661, no. 5, pp. 370–372, 2008.

R. Kerner and R. B. Mann, “Tunnelling, temperature, and Taub-NUT black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 73, no. 10, Article ID 104010, 11 pages, 2006.

R. Kerner and R. B. Mann, “Fermions tunnelling from black holes,” Classical and Quantum Gravity, vol. 25, no. 9, Article ID 095014, 17 pages, 2008.

R. Kerner and R. B. Mann, “Tunnelling from Gödel black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 75, no. 8, Article ID 084022, 9 pages, 2007.

G. Gecim and Y. Sucu, “Hawking radiation of topological massive warped-AdS3 black hole families,” https://arxiv.org/abs/1406.0290v2.

G. Gecim and Y. Sucu, “Tunnelling of relativistic particles from new type black hole in new massive gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 2, article 023, 2013.

G. Gecim and Y. Sucu, “Dirac and scalar particles tunnelling from topological massive warped-AdS3 black hole,” Astrophysics and Space Science, vol. 357, article 105, 2015.

S. I. Kruglov, “Black hole radiation of spin-1 particles in (1+2) dimensions,” Modern Physics Letters A, vol. 29, no. 39, Article ID 1450203, 5 pages, 2014.

S. I. Kruglov, “Black hole emission of vector particles in (1+1) dimensions,” International Journal of Modern Physics A, vol. 29, no. 22, Article ID 1450118, 10 pages, 2014.

H. Gursel and I. Sakalli, “Hawking radiation of massive vector particles from warped AdS3 black hole,” Canadian Journal of Physics, vol. 92, no. 2, pp. 147–149, 2016.

I. Sakalli and A. Ovgun, “Hawking radiation of spin-1 particles from a three-dimensional rotating hairy black hole,” Journal of Experimental and Theoretical Physics, vol. 121, no. 3, pp. 404–407, 2015.

G.-R. Chen, S. Zhou, and Y.-C. Huang, “Vector particles tunneling from BTZ black holes,” International Journal of Modern Physics D, vol. 24, no. 1, Article ID 1550005, 7 pages, 2015.

G.-R. Chen and Y.-C. Huang, “Hawking radiation of vector particles as tunneling from the apparent horizon of Vaidya black holes,” International Journal of Modern Physics A, vol. 30, no. 15, Article ID 1550083, 9 pages, 2015.

X.-Q. Li and G.-R. Chen, “Massive vector particles tunneling from Kerr and Kerr-Newman black holes,” Physics Letters B, vol. 751, pp. 34–38, 2015.

I. Sakalli and A. Övgün, “Quantum tunneling of massive spin-1 particles from non-stationary metrics,” General Relativity and Gravitation, vol. 48, no. 1, article 1, 10 pages, 2016.

G.-R. Chen, S. Zhou, and Y.-C. Huang, “Vector particles tunneling from four-dimensional Schwarzschild black holes,” Astrophysics and Space Science, vol. 357, no. 1, article 51, 2015.

R. Li and J. Zhao, “Hawking radiation of massive vector particles from the linear dilaton black holes,” The European Physical Journal Plus, vol. 131, article 249, 2016.

I. Sakalli and H. Gursel, “Quantum tunneling from rotating black holes with scalar hair in three dimensions,” The European Physical Journal C, vol. 76, no. 6, article 318, 2016.

G. Gecim and Y. Sucu, “Massive vector bosons tunnelled from the (2+1)-dimensional black holes,” The European Physical Journal Plus, vol. 132, no. 3, article 105, 2017.

H. Hinrichsen and A. Kempf, “Maximal localization in the presence of minimal uncertainties in positions and in momenta,” Journal of Mathematical Physics, vol. 37, no. 5, pp. 2121–2137, 1996.

A. Kempf, “On quantum field theory with nonzero minimal uncertainties in positions and momenta,” Journal of Mathematical Physics, vol. 38, no. 3, pp. 1347–1372, 1997.

A. F. Ali, S. Das, and E. C. Vagenas, “Discreteness of space from the generalized uncertainty principle,” Physics Letters B, vol. 678, no. 5, pp. 497–499, 2009.

S. Das, E. C. Vagenas, and A. F. Ali, “Discreteness of space from GUP II: relativistic wave equations,” Physics Letters B, vol. 690, pp. 407–412, 2010, Erratum to: Physics Letters B, vol. 692, p. 342, 2010.

B. Carr, J. Mureika, and P. Nicolini, “Sub-Planckian black holes and the Generalized Uncertainty Principle,” Journal of High Energy Physics, vol. 2015, no. 7, article 52, 2015.

A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 52, no. 2, pp. 1108–1118, 1995.

S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer, and H. Stöcker, “Signatures in the Planck regime,” Physics Letters B, vol. 575, no. 1-2, pp. 85–99, 2003.

S. Hossenfelder, “Can we measure structures to a precision better than the Planck length?” Classical and Quantum Gravity, vol. 29, no. 11, Article ID 115011, 2012.

M. Faizal and S. I. Kruglov, “Deformation of the Dirac equation,” International Journal of Modern Physics D, vol. 25, no. 1, Article ID 1650013, 2016.

D. Chen, H. Wu, H. Yang, and S. Yang, “Effects of quantum gravity on black holes,” International Journal of Modern Physics A, vol. 29, no. 26, Article ID 1430054, 2014.

K. Nozari and S. Saghafi, “Natural cutoffs and quantum tunneling from black hole horizon,” Journal of High Energy Physics, vol. 2012, no. 11, article 005, 2012.

T. K. Jana and P. Roy, “Exact solution of the Klein-Gordon equation in the presence of a minimal length,” Physics Letters A, vol. 373, no. 14, pp. 1239–1241, 2009.

K. Nozari and M. Karami, “Minimal length and generalized Dirac equation,” Modern Physics Letters A, vol. 20, no. 40, pp. 3095–3103, 2005.

X.-Q. Li, “Massive vector particles tunneling from black holes influenced by the generalized uncertainty principle,” Physics Letters B, vol. 763, pp. 80–86, 2016.

D. Chen, H. Wu, and H. Yang, “Fermion's tunnelling with effects of quantum gravity,” Advances in High Energy Physics, vol. 2013, Article ID 432412, 6 pages, 2013.

D. Y. Chen, Q. Q. Jiang, P. Wang, and H. Yang, “Remnants, fermions' tunnelling and effects of quantum gravity,” Journal of High Energy Physics, vol. 2013, no. 11, article 176, 2013.

D. Y. Chen, H. W. Wu, and H. Yang, “Observing remnants by fermions' tunneling,” Journal of Cosmology and Astroparticle Physics, vol. 2014, no. 3, article 036, 2014.

X.-X. Zeng and Y. Chen, “Quantum gravity corrections to fermions’ tunnelling radiation in the Taub-NUT spacetime,” General Relativity and Gravitation, vol. 47, no. 4, article 47, 2015.

H.-L. Li, Z.-W. Feng, and X.-T. Zu, “Quantum tunneling from a high dimensional Gödel black hole,” General Relativity and Gravitation, vol. 48, no. 2, article 18, 2016.

P. Wang, H. Yang, and S. Ying, “Quantum gravity corrections to the tunneling radiation of scalar particles,” International Journal of Theoretical Physics, vol. 55, no. 5, pp. 2633–2642, 2016.

Z.-Y. Liu and J.-R. Ren, “Fermions tunnelling with quantum gravity correction,” Communications in Theoretical Physics, vol. 62, no. 6, pp. 819–823, 2014.

B. Mu, P. Wang, and H. Yang, “Minimal length effects on tunnelling from spherically symmetric black holes,” Advances in High Energy Physics, vol. 2015, Article ID 898916, 2015.

G. Li and X. Zu, “Scalar particles’ tunneling and effect of quantum gravity,” Journal of Applied Mathematics and Physics, vol. 3, no. 2, pp. 134–139, 2015.

M. A. Anacleto, F. A. Brito, and E. Passos, “Quantum-corrected self-dual black hole entropy in tunneling formalism with GUP,” Physics Letters B, vol. 749, pp. 181–186, 2015.

R. Casadio, P. Nicolini, and R. da Rocha, “GUP Hawking fermions from MGD black holes,” https://arxiv.org/abs/1709.09704v1.

J. Sadeghi and V. Reza Shajiee, “Quantum tunneling from the charged non-rotating BTZ black hole with GUP,” The European Physical Journal Plus, vol. 132, no. 3, article 132, 2017.

I. Sakalli, A. Övgün, and K. Jusufi, “GUP assisted Hawking radiation of rotating acoustic black holes,” Astrophysics and Space Science, vol. 361, no. 10, article 330, 2016.

H.-L. Li and X.-T. Zu, “Black hole remnant and quantum tunnelling in three-dimensional Gödel spacetime,” Astrophysics and Space Science, vol. 357, no. 1, article 6, 2015.

Z. Feng, L. Zhang, and X. Zu, “The remnants in Reissner-Nordström-de Sitter quintessence black hole,” Modern Physics Letters A, vol. 29, no. 26, Article ID 1450123, 2014.

G. Gecim and Y. Sucu, “The GUP effect on tunnelling of massive vector bosons from the 2+1 dimensional black hole,” https://arxiv.org/abs/1704.03537v2.

G. Gecim and Y. Sucu, “The GUP effect on Hawking radiation of the 2+1 dimensional black hole,” Physics Letters B, vol. 773, pp. 391–394, 2017.

Y. Kwon, S. Nam, J.-D. Park, and S.-H. Yi, “Quasi-normal modes for new type black holes in new massive gravity,” Classical and Quantum Gravity, vol. 28, no. 14, Article ID 145006, 2011.

Y. Sucu and N. Ünal, “Exact solution of Dirac equation in 2+1 dimensional gravity,” Journal of Mathematical Physics, vol. 48, no. 5, Article ID 052503, 2007.

K. Lin and S. Z. Yang, “A simpler method for researching fermions tunneling from black holes,” Chinese Physics B, vol. 20, no. 11, Article ID 110403, 2011.

Y.-P. Hu, J.-Y. Zhang, and Z. Zhao, “Anote on the hawking radiation calculated by the quasiclassical tunneling method,” Modern Physics Letters A, vol. 25, no. 4, pp. 295–308, 2010.

R. Di Criscienzo and L. Vanzo, “Fermion tunneling from dynamical horizons,” Europhysics Letters, vol. 82, no. 6, Article ID 60001, 2008.

G. E. Volovik, Exotic Properties of Superfluid 3He, World Scientific, Singapore, 1992.

G. E. Volovik, “Simulation of a Panlevé-Gullstrand black hole in a thin 3He-A film,” Journal of Experimental and Theoretical Physics Letters, vol. 69, no. 9, pp. 705–713, 1999.

G. E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford, UK, 2003.