Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2018, Article ID 7450607, 9 pages
https://doi.org/10.1155/2018/7450607
Research Article

Black Hole Thermodynamics and Generalized Uncertainty Principle with Higher Order Terms in Momentum Uncertainty

1Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Saltlake, Kolkata 700106, West Bengal, India
2Inter-University Centre for Astronomy & Astrophysics (IUCAA), Pune 411007, India
3Department of Physics, West Bengal State University, Barasat, Kolkata 700126, India
4Department of Physics, Kandi Raj College, Kandi, Murshidabad 742137, India

Correspondence should be addressed to Sunandan Gangopadhyay; moc.liamg@yayhdapognag.nadnanus

Received 31 January 2018; Revised 16 April 2018; Accepted 23 April 2018; Published 7 June 2018

Academic Editor: George Siopsis

Copyright © 2018 Sunandan Gangopadhyay and Abhijit Dutta. 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 study the modification of thermodynamic properties of Schwarzschild and Reissner-Nordström black hole in the framework of generalized uncertainty principle with correction terms up to fourth order in momentum uncertainty. The mass-temperature relation and the heat capacity for these black holes have been investigated. These have been used to obtain the critical and remnant masses. The entropy expression using this generalized uncertainty principle reveals the area law up to leading order logarithmic corrections and subleading corrections of the form . The mass output and radiation rate using Stefan-Boltzmann law have been computed which show deviations from the standard case and the case with the simplest form for the generalized uncertainty principle.

1. Introduction

The consistent unification of quantum mechanics (QM) with general relativity (GR) is one of the major tasks in theoretical physics. GR deals with the definition of world-lines of particles, which is in contradiction with QM since it does not allow the notion of trajectory due to the presence of an uncertainty in the determination of the momentum and position of a quantum particle. It has been the aim to unify these two theories into one theory known as quantum gravity. It is quite interesting that all approaches towards quantum gravity such as black hole physics [13], string theory [4, 5], or even Gedanken experiment [6] predict the existence of a minimum measurable length. The occurrence of such a minimal length also arises in various theories of quantum gravity phenomenology, namely, the generalized uncertainty principle (GUP) [5, 7, 8], modified dispersion relation (MDR) [912], and deformed special relativity (DSR) [13], to name a few. It is now widely accepted that Heisenberg uncertainty principle would involve corrections from gravity at energies close to the Planck scale. Thus, emergence of a minimal length seems to be inevitable when gravitational effects are taken into account. There has been a lot of work incorporating the existence of a minimal length scale in condensed matter and atomic physics experiments such as Lamb Shift, Landau levels and the scanning Tunneling Microscope [1419], loop quantum gravity [2022], noncommutative geometry [23], computing Planck scale corrections to the phenomena of superconductivity and quantum Hall effect [24], and understanding its consequences in cosmology [25, 26].

The incorporation of the GUP to study black hole thermodynamics has been another interesting area of active research [2738]. It has been observed that the GUP reveals a self-complete characteristic of gravity which basically amounts to hiding any curvature singularity behind an event horizon as a consequence of matter compression at the Planck scale [3941]. Further, the effects of the GUP have also been considered in the tunneling formalism for Hawking radiation to evaluate the quantum-corrected Hawking temperature and entropy of a Schwarzschild black hole [4246]. In our earlier findings [3236], we have studied the modification of thermodynamic properties, namely, the temperature, heat capacity, and entropy of black holes due to the simplest form of the GUP. Interestingly the correction to the Schwarzschild black hole temperature due to quadratic and linear-quadratic GUP has also been compared with the corrections from the quantum Raychaudhuri equation [47]. Very recently the Lorentz-invariance-violating class of dispersion relations has been applied to study the thermodynamics of black holes [48]. It would therefore be interesting to compare these results with those coming from the GUP.

The above studies motivate us to investigate the modification of thermodynamic properties for Schwarzschild and Reissner-Nordström (RN) black holes using the form of the GUP proposed in [30]. This GUP involves higher order terms in the momentum uncertainty. We compute the remnant and critical masses analytically for these black holes below where the temperature becomes ill-defined. We then use the Stefan-Boltzmann law to estimate the mass and the energy output as a function of time. We finally compute the entropy and obtain the well-known area theorem containing corrections from the GUP with higher order terms in momentum uncertainty.

The paper is organized as follows. In Section 2, we study the thermodynamics of Schwarzschild black hole taking into account the effect of the GUP, with higher order terms in momentum uncertainty. In Section 2.1, we also obtain the mass and radiation rate characteristics for the Schwarzschild black hole as a function of time by using the Stefan-Boltzmann law. In Section 3, we study the thermodynamics of Reissner-Nordström black holes taking into account the effect of the GUP. Finally, we conclude in Section 4.

2. Thermodynamics of Schwarzschild Black Hole

In this paper, we work with the following form of the GUP [49]:where is the Planck length (). Keeping terms up to fourth order in momentum uncertainty, we haveWe now consider a Schwarzschild black hole of mass . In the vicinity of the event horizon of the black hole, let a pair (particle-antiparticle) production occurs. For simplicity we consider the particle to be massless. The particle with negative energy falls inside the horizon and that with positive energy escapes outside the horizon and gets observed by some observer at infinity. The momentum of the emitted particle (p), which also characterizes the temperature (T), is of the order of its uncertainty in momentum . Consequentlywhere is the speed of light and is the Boltzmann constant.

The Hawking temperature of the black hole will be equal to the temperature of the particle when thermodynamic equilibrium is reached. The uncertainty in the position of a particle near the event horizon of the Schwarzschild black hole will be of the order of the Schwarzschild radius of the black holewhere is a calibration factor, is the Schwarzschild radius, and is Newton’s universal gravitational constant.

To relate the Hawking temperature of the black hole with the mass of the black hole, we consider the saturated form of the GUP (2)Substituting (3) and (4) in (5) giveswhere the relations and ( being the Planck mass) have been used. In the absence of corrections due to GUP, (6) reduces toThe value of now gets fixed to by comparing this expression with the semiclassical Hawking temperature [50, 51]. The mass-temperature relation (6) finally takes the formThe heat capacity of the black hole therefore readsThe above expression takes the formafter the following notations are introducedThe mass of the black hole decreases due to radiation from the black hole. This leads to an increase in the temperature of the black hole. It can be observed from (7) and (10) that there exists a finite temperature at which the heat capacity vanishes. To find out this temperature, we set C = 0. This givesSolving this, we getwhere the positive sign before the square root has been taken so that the above result reduces to corresponding result when [33].

Finally we get the expression for to beNow in terms of , the mass-temperature relation (8) can be represented asThe remnant mass can now be obtained by substituting (15) in (16). This yieldsReassuringly the above result reduces to the result in limit [33]Now for , , the remnant mass is given byAlso for , the mass-temperature relation (16) readsThe solution of this biquadratic equation in yieldsThe above relation readily implies the existence of a critical mass below which the temperature will be a complex quantityThis demonstrates that the remnant and critical masses are equal.

At this point, we would like to make a comment. It can be observed from the above analysis that analytical expressions for the remnant and critical masses can be obtained even if one retains terms of order of in the momentum uncertainty. This is because it leads to an equation of the form when the condition is imposed. This equation can be solved analytically to obtain the remnant mass. If we keep terms beyond this order in momentum uncertainty, analytical expressions for the remnant and critical masses can not be obtained.

The black hole entropy from the first law of black hole thermodynamics is given byTo obtain the entropy S in terms of the mass M of the black hole, we need to consider (16) to obtain an expression for the temperature T in terms of mass M.

Equation (16) yields up to Now the entropy expression in terms of the mass can be written aswhere is the semiclassical Bekenstein-Hawking entropy for the Schwarzschild black hole. In terms of the black hole horizon area , the above entropy expression can be written asThis completes our discussion of the effect of the GUP on the thermodynamic properties of the Schwarzschild black hole. In Figure 1, we present the plot of the entropy of the black hole versus the horizon area for the GUP case and compare it with the standard case.

Figure 1: The entropy versus area plot. Here the solid line (lower curve) represents the GUP case (considering only the first two terms in the right hand side of (26)), and the dashed line (upper curve) represents the standard case.
2.1. Energy Output as a Function of Time

Due to radiation of the black hole, the mass of the black hole reduces, while its temperature keeps on increasing. If one assumes that the energy loss is dominated by photons, then one can apply the Stefan-Boltzmann law to estimate the energy radiated as a function of timewhere is the Stefan-Boltzmann constant. In terms of Schwarzschild black hole mass M with the horizon area , the above equation implieswhere we have used .

We now write the above equation taking into account the effect of the GUP. Thus, considering the mass-temperature relation (24), the radiation rate takes the following form:where we have set and the characteristic time is being defined as . If refers to the initial mass at time , the solution of the above equation yields the mass-time relation. Up to , we havewhereIn Figures 2 and 3, we have plotted the mass of the black hole as a function of time and the radiation rate as a function of time.

Figure 2: The mass of the black hole versus time. The mass is in units of Planck mass and the time is in units of characteristic time. Here thin line (lower curve) represents GUP case (considering both and ), the dashed line (middle curve) represents GUP case (considering only ), and thick line (upper curve) represents the standard case.
Figure 3: The radiation rate of the black hole versus time. The rate is in units of Planck mass per characteristic time and the time is in units of characteristic time. Here thin line (upper curve) represents GUP case (considering both and ), the dashed line (middle curve) represents GUP case (considering only ), and thick line (lower curve) represents the standard case.

3. Thermodynamics of Reissner-Nordström Black Hole

In this section, we consider the Reissner-Nordström (RN) black hole of mass and charge . In this case, near the horizon of the black hole, the position uncertainty of a particle will be of the order of the RN radius of the black holewhere is the radius of the horizon of the RN black hole. Substituting the value of and from (3) and (32), the GUP (5) can be rewritten asOnce again, in the absence of correction due to GUP, (33) reduces toComparing the above relation with the semiclassical Hawking temperature yields the value of to beThis finally fixes the form of the mass-charge-temperature relation (33) to bewhere the identityhas been used.

The heat capacity of the black hole can now be calculated using relation (9) and equation (36):To express the heat capacity in terms of the mass, once again we make use of relation (12) to recast (36) in the formwhereNow to find out the temperature where the radiation process stops, we set C = 0. Equation (38) therefore yieldsfrom which solution of readswhere the positive sign before the square root has been taken to reproduce the result corresponding to the limit [33].

The remnant mass can now be computed by substituting (42) in (39). This would then giveThus, we finally obtain the following cubic equation for the remnant mass:whereSolution of (44) gives us the exact expression of remnant mass for the RN black hole. This yieldswhereThe above expression for the remnant mass reduces to the remnant mass for the Schwarzschild black hole (17) in the limit.

Finally, we proceed to compute the entropy of the RN black hole. To do that, we first obtain an expression of from (39) in terms of the mass and the charge of the RN black hole. This gives up to From this one now can calculate the entropy for the RN black hole using (38) and (48):where is the semiclassical Bekenstein-Hawking entropy for the RN black hole. In terms of the area of the horizon , the above equation can be written as which is the area theorem for the RN black hole with corrections from the GUP containing higher order terms in the momentum uncertainty.

We would like to conclude this section by mentioning that, in [52], it has been pointed out that there is a part of the information (leaking out of the black hole due to Hawking radiation) related to nonthermal GUP correlations. This insight may be important to provide a solution for the well-known information loss paradox and is worth investigating in future.

4. Conclusions

In this paper, we have investigated the modifications of the various thermodynamic properties of Schwarzschild and Reissner-Nordström black holes using higher order momentum uncertainty terms in the GUP. We obtain the GUP modified mass-temperature relation. This then leads to the existence of a remnant mass thereby preventing the complete evaporation of the black hole. The expression for the remnant and critical masses have been obtained analytically. In this regard, we observe that analytical expressions for these masses can be obtained even if we keep terms of the order of in the momentum uncertainty. Beyond this, it is no longer possible to obtain analytical expression for the critical and remnant masses. We also compute the mass and energy outputs as functions of time using the Stefan-Boltzman law. We observe that these expressions get modified from the standard case as well as the case where the simplest form of the GUP is used. The expression for the entropy exhibits the well-known area theorem in terms of the horizon area in both cases up to leading order corrections from the GUP.

Data Availability

No data has been used.

Conflicts of Interest

There are no conflicts of interest with any funding agency regarding the publication of this manuscript.

Acknowledgments

Sunandan Gangopadhyay acknowledges the support by DST SERB, India, under Start Up Research Grant (Young Scientist), File no. YSS/2014/000180.

References

  1. M. Maggiore, “A generalized uncertainty principle in quantum gravity,” Physics Letters B, vol. 304, no. 1-2, pp. 65–69, 1993. View at Publisher · View at Google Scholar · View at Scopus
  2. D. J. Gross and P. F. Mende, “The high-energy behavior of string scattering amplitudes,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 197, no. 1-2, pp. 129–134, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. D. J. Gross and P. F. Mende, “String theory beyond the Planck scale,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 303, no. 3, pp. 407–454, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. D. Amati, M. Ciafaloni, and G. Veneziano, “Can spacetime be probed below the string size?” Physics Letters B, vol. 216, no. 1-2, pp. 41–47, 1989. View at Publisher · View at Google Scholar · View at Scopus
  5. K. Konishi, G. Paffuti, and P. Provero, “Minimum physical length and the generalized uncertainty principle in string theory,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 234, no. 3, pp. 276–284, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  6. F. Scardigli, “Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment,” Physics Letters B, vol. 452, no. 1-2, pp. 39–44, 1999. View at Publisher · View at Google Scholar
  7. M. Maggiore, “The algebraic structure of the generalized uncertainty principle,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 319, no. 1-3, pp. 83–86, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. Maggiore, “Quantum groups, gravity, and the generalized uncertainty principle,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 49, no. 10, pp. 5182–5187, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  9. G. Amelino-Camelia, M. Arzano, and A. Procaccini, “Severe constraints on the loop-quantum-gravity energy-momentum dispersion relation from the black-hole area-entropy law,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 70, no. 10, Article ID 107501, 2004. View at Publisher · View at Google Scholar
  10. G. Amelino-Camelia, A. Procaccini, and M. Arzano, “A glimpse at the flat-spacetime limit of quantum gravity using the Bekenstein argument in reverse,” International Journal of Modern Physics D, vol. 13, no. 10, pp. 2337–2343, 2004. View at Publisher · View at Google Scholar
  11. G. Amelino-Camelia, M. Arzano, Y. Ling, and G. Mandanici, “Black-hole thermodynamics with modified dispersion relations and generalized uncertainty principles,” Classical and Quantum Gravity, vol. 23, no. 7, pp. 2585–2606, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. D. Sadatian and H. Dareyni, “Correction of Cardy-Verlinde formula for fermions and bosons with modified dispersion relation,” Annals of Physics, vol. 380, pp. 71–77, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Magueijo and L. Smolin, “String theories with deformed energy-momentum relations, and a possible nontachyonic bosonic string,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 71, no. 2, Article ID 026010, 6 pages, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Das and E. C. Vagenas, “Universality of quantum gravity corrections,” Physical Review Letters, vol. 101, Article ID 221301, 4 pages, 2008. View at Publisher · View at Google Scholar
  15. S. Das and E. C. Vagenas, “Phenomenological implications of the generalized uncertainty principle,” Canadian Journal of Physic, vol. 87, no. 3, pp. 233–240, 2009. View at Publisher · View at Google Scholar
  16. S. Das and E. C. Vagenas, “Reply to ‘Comment on ‘Universality of Quantum Gravity Corrections,” Physical Review Letters, vol. 101, 2010. View at Google Scholar
  17. A. F. Ali, S. Das, and E. C. Vagenas, “Discreteness of space from the generalized uncertainty principle,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 678, no. 5, pp. 497–499, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. M. Wang, “Erratum: “Solitary wave solutions for variant Boussinesq equations'' [Phys.\ Lett.\ A {\bf 199} (1995), no.\ 3-4, 169--172],” Physics Letters A, vol. 212, no. 6, 353 pages, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. F. Ali, S. Das, and E. C. Vagenas, “Proposal for testing quantum gravity in the lab,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 84, no. 4, Article ID 044013, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. R. Gambini and J. Pullin, “Nonstandard optics from quantum space-time,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 59, no. 12, 124021, 4 pages, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Alfaro, H. A. Morales-Técotl, and L. F. Urrutia, “Quantum gravity corrections to neutrino propagation,” Physical Review Letters, vol. 84, no. 11, pp. 2318–2321, 2000. View at Publisher · View at Google Scholar
  22. G. Amelino-Camelia, L. Smolin, and A. Starodubtsev, “Quantum symmetry, the cosmological constant and Planck-scale phenomenology,” Classical and Quantum Gravity, vol. 21, no. 13, pp. 3095–3110, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. F. Girelli, E. R. Livine, and D. Oriti, “Deformed special relativity as an effective flat limit of quantum gravity,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 708, no. 1-3, pp. 411–433, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. S. Das and R. B. Mann, “Planck scale effects on some low energy quantum phenomena,” Physics Letters B, vol. 704, no. 5, pp. 596–599, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. S. Basilakos, S. Das, and E. C. Vagenas, “Quantum gravity corrections and entropy at the planck time,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 9, article no. 027, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. W. Chemissany, S. Das, A. F. Ali, and E. C. Vagenas, “Effect of the generalized uncertainty principle on post-inflation preheating,” Journal of Cosmology and Astroparticle Physics, vol. 2011, no. 12, p. 17, 2011. View at Publisher · View at Google Scholar · View at Scopus
  27. F. Scardigli, “Some heuristic semi-classical derivations of the Planck length, the Hawking effect and the Unruh effect,” Il Nuovo Cimento B, vol. 110, no. 9, pp. 1029–1034, 1995. View at Publisher · View at Google Scholar · View at Scopus
  28. K. Nozari and S. H. Mehdipour, “Gravitational uncertainty and black hole remnants,” Modern Physics Letters A, vol. 20, no. 38, pp. 2937–2948, 2005. View at Publisher · View at Google Scholar · View at Scopus
  29. K. Nozari and S. H. Mehdipour, “Black holes remnants in extra dimensions and dark matter,” International Journal of Modern Physics A, vol. 21, no. 23-24, pp. 4979–4992, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  30. R. Banerjee and S. Ghosh, “Generalised uncertainty principle, remnant mass and singularity problem in black hole thermodynamics,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 688, no. 2-3, pp. 224–229, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. 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. View at Publisher · View at Google Scholar · View at Scopus
  32. S. Gangopadhyay, A. Dutta, and A. Saha, “Generalized uncertainty principle and black hole thermodynamics,” General Relativity and Gravitation, vol. 46, no. 2, Art. 1661, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. A. Dutta and S. Gangopadhyay, “Remnant mass and entropy of black holes and modified uncertainty principle,” General Relativity and Gravitation, vol. 46, no. 6, Art. 1747, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. S. Gangopadhyay, A. Dutta, and M. Faizal, “Constraints on the generalized uncertainty principle from black-hole thermodynamics,” EPL (Europhysics Letters), vol. 112, no. 2, Article ID 20006, 2015. View at Publisher · View at Google Scholar
  35. M. A. Anacleto, F. A. Brito, G. C. Luna, E. Passos, and J. Spinelly, “Quantum-corrected finite entropy of noncommutative acoustic black holes,” Annals of Physics, vol. 362, pp. 436–448, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. A. Dutta and S. Gangopadhyay, “Thermodynamics of Black Holes and the Symmetric Generalized Uncertainty Principle,” International Journal of Theoretical Physics, vol. 55, no. 6, pp. 2746–2754, 2016. View at Publisher · View at Google Scholar · View at Scopus
  37. S. Zhou and G.-R. Chen, “Corrected black hole thermodynamics in Damour-Ruffini's method with generalized uncertainty principle,” International Journal of Modern Physics D: Gravitation, Astrophysics, Cosmology, vol. 26, no. 7, 1750062, 8 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  38. Z. W. Feng, H. L. Li, X. T. Zu, and S. Z. Yang, “Quantum corrections to the thermodynamics of Schwarzschild-Tangherlini black hole and the generalized uncertainty principle,” The European Physical Journal C, vol. 76, article 212, 2016. View at Publisher · View at Google Scholar
  39. G. Dvali, S. Folkerts, and C. Germani, “Physics of trans-Planckian gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 84, no. 2, Article ID 024039, 2011. View at Publisher · View at Google Scholar · View at Scopus
  40. G. Dvali and C. Gomez, “Ultra-high energy probes of classicalization,” Journal of Cosmology and Astroparticle Physics, vol. 2012, no. 7, article 015, 2012. View at Publisher · View at Google Scholar
  41. M. Isi, J. Mureika, and P. Nicolini, “Self-completeness and the generalized uncertainty principle,” Journal of High Energy Physics, no. 11, 139, front matter+17 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  42. K. Nozari and S. H. Mehdipour, “Hawking radiation as quantum tunneling from a noncommutative Schwarzschild black hole,” Classical and Quantum Gravity, vol. 25, no. 17, Article ID 175015, 2008. View at Publisher · View at Google Scholar · View at Scopus
  43. K. Nozari and S. H. Mehdipour, “Quantum gravity and recovery of information in black hole evaporation,” EPL (Europhysics Letters), vol. 84, no. 2, Article ID 20008, 2008. View at Publisher · View at Google Scholar · View at Scopus
  44. M. Dehghani, “Hawking tunneling radiation of the spherically symmetric black holes at the Planck scale,” Astrophysics and Space Science, vol. 357, no. 2, article no. 169, 2015. View at Publisher · View at Google Scholar · View at Scopus
  45. S. Gangopadhyay, “Minimal Length Effects in Black Hole Thermodynamics from Tunneling Formalism,” International Journal of Theoretical Physics, vol. 55, no. 1, pp. 617–624, 2016. View at Publisher · View at Google Scholar · View at Scopus
  46. S.-Z. Yang, Z.-W. Feng, and H.-L. Li, “The tunneling radiation from non-stationary spherical symmetry black holes and the Hamilton-Jacobi equation,” International Journal of Theoretical Physics, vol. 56, no. 2, pp. 546–553, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. E. C. Vagenas, L. Alasfar, S. M. Alsaleh, and A. F. Ali, “The GUP and quantum Raychaudhuri equation,” Nuclear Physics B, vol. 931, pp. 72–78, 2018. View at Publisher · View at Google Scholar
  48. G.-P. Li, J. Pu, Q.-Q. Jiang, and X.-T. Zu, “An application of Lorentz-invariance violation in black hole thermodynamics,” The European Physical Journal C, vol. 77, no. 10, article no. 666, 2017. View at Publisher · View at Google Scholar · View at Scopus
  49. K. Nozari and A. S. Sefiedgar, “Comparison of approaches to quantum correction of black hole thermodynamics,” Physics Letters B, vol. 635, no. 2-3, pp. 156–160, 2006. View at Publisher · View at Google Scholar
  50. S. W. Hawking, “Black hole explosions?” Nature, vol. 248, no. 5443, pp. 30-31, 1974. View at Publisher · View at Google Scholar · View at Scopus
  51. S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. K. Nozari and S. H. Mehdipour, “Parikh-Wilczek tunneling from noncommutative higher dimensional black holes,” Journal of High Energy Physics. A SISSA Journal, no. 3, 061, 24 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus