# GUP-Corrected van der Waals Black Holes

###### Abstract

In this paper, we study the generalized uncertainty principle (GUP) effects for the van der Waals (vdW) black holes. We use the GUP-corrected black hole temperature to obtain the modified vdW black hole solution. We also study the thermodynamics and phase transition of GUP-corrected vdW black holes. We show that small black holes are unstable when GUP is taken into account.

###### pacs:

04.60.Bc, 04.70.s, 05.70.a^{†}

^{†}preprint:

## I Introduction

Since the revolutionary papers of Bekenstein and Hawking, black hole thermodynamics has been one of the most important subject among the researchers in the scientific community Bekenstein1972 (); Bekenstein1973 (); Bardeen1973 (); Hawking1974 (); Bekenstein1974 (); Hawking1975 (). Considering black holes with temperature and entropy gives us new opportunities to explore many interesting thermodynamic phenomena. Furthermore, black hole thermodynamics gives the first hints of quantum gravity. It also gives the fundamental links between general relativity, thermodynamics and quantum mechanics. One may naturally ask whether black hole as a thermodynamic system shares any similarities with general thermodynamic system or not. These similarities become more clear and certain for black holes in anti-de Sitter (AdS) spacetime. Black holes in AdS spacetime have been studied widely in the literature since the pioneer paper of Hawking and Page Hawking1983 (). They found a first order phase transition between Schwarzschild AdS black hole and thermal AdS space. Interestingly when Schwarzschild AdS black hole is generalized to charged or rotating case, it shows the vdW fluids like behaviors. The authors of Chamblin1999a (); Chamblin1999b () studied the thermodynamics of charged AdS black holes and they found vdW like first order smalllarge black hole phase transition. This type of phase transition becomes more clear in the extended phase space where the cosmological constant is considered as a thermodynamic pressure. Once treating the cosmological constant as a thermodynamic pressure,

(1) |

naturally gives its conjugate quantity as a thermodynamic volume

(2) |

The charged AdS black hole thermodynamics and phase transition were studied by Kubiznak and MannKubiznak2012 (). They showed the charged AdS first order smalllarge black hole phase transition has the same characteristic behaviour with vdW fluids. They were also obtained critical exponents which coincide the exponents of vdW fluids. Up to now, their study has been extended for various solutions of black holes in the AdS spacetime Gunasekaran2012 (); Spallucci2013 (); Zhao2013 (); Behlaj2013 (); Cai2013 (); Mo2014 (); Xu2014 (); Li2014 (); Ma2014 (); Belhaj2015 (); Kubiznak2015 (); Hennigar2015 (); Caceres2015 (); Wei2016 (); Hendi2016 (); Momeni2017 (); Ovgun2018 (); Sun2018 (); Jamil2018 (); Nam2018a (); Nam2018b (); Kuang2018 (); Zhang2018 (); Okcu2017 (); Okcu2018 (); Mo2018 (); Zhao2018 (); Yekta2019 ().(One can refer to the recent comprehensive reviews Altamirano2014 (); Kubiznak2017 () and references therein.)

Based on the above fact, one may consider four dimensional metric form of the van der waals fluids in the context of Einstein gravity. Therefore, Rajagopal et al. Rajagopal2014 () obtained vdW black hole solution that has the same thermodynamics with vdW fluids.(See Pradhan2016 (); Hu2017 () for the thermodynamics of vdW black holes.) In their interesting paper, they also argued the corresponding stress energy tensor for their solution. They found stress energy tensor obeys energy conditions for a certain range of metric parameter. They also showed their solution is interpreted as near horizon metric. Following the methods in Rajagopal2014 () , a few AdS black hole solutions, which match the thermodynamics of a certain equation of states, were proposed in the literature. In Delsate2015 () , Delsate and Mann generalized the vdW solution in d dimensions. In Setare2015 () , Setare and Adami obtained Polytropic black hole solution which has the identical thermodynamics with that of the polytropic gas. Interestingly, under the small effective pressure limit, Abchouyeh et al. Abchouyeh2017 () obtained Anyon black hole solution which corresponds to thermodynamics of Anyon vdW fluids. Anyons are the particles that have the intermediate statistics between Fermi-Dirac and Bose-Einstein statistics. Then exact solution of Anyon black hole Xu2018 () was obtained by Xu. Finally, Debnath constructed a black hole solution whose thermodynamics matches the thermodynamics of modified Chaplygin gas Debnath2018 ().

One can also consider above-mentioned black hole solution in the context of quantum gravity. Effects of quantum gravity are no longer negligible near the Planck scale and therefore various quantum gravity approaches suggest that thermodynamics of black hole should be modified Govindarajan2001 (); Mann1998 (); Sen2013 (); Das2002 (). Since a black hole is a gravitational system, quantum effects of such a system provide information about quantum gravity. Motivated by this fact, Upadhyay and Pourhassan Upadhyay2019 () studied the modification of dimensional vdW black hole for the thermal fluctuations interpreted as an quantum effects. They investigated thermal fluctuations effects on the thermodynamics of dimensional vdW black hole. In this paper, we want to explore the generalized uncertainty principle effects for the four dimensional vdW black holes. GUP is one of the phenomenological quantum gravity models and is considered as an modification of standard uncertainty principle Maggiore1993 (); Kempf1995 (). Therefore, it is possible to modified the thermodynamics of black hole by taking into account the quantum gravity effects near the Planck scaleAdler2001 (); Ali2012 (); Gangopadhyay2014 (); Abbasvandi2016 ().

The paper is arranged as follows: We first review the heuristic derivation of modified Hawking temperature, which was proposed by Xiang and Wen Xiang2009 (), with the generalized uncertanity principle. Next, we modify the vdW solution by using the modified Hawking temperature and then we check the energy conditions for stress energy tensor in Sect. III. In Sect. IV, we investigate the GUP corrected thermodynamics quantities and phase transition. Finally, we discuss our results in Sect. V. (We use the units ).

## Ii GUP-Corrected Black Hole Temperature

In this section, we will briefly review a generic GUP correction approach to semi-classical Hawking Temperature Xiang2009 (). The simplest form of GUP is given by

(3) |

where is a positive constant. In order to get correction to black hole thermodynamics, we need to solve the inequality in Eq.(3) for the momentum uncertanity . The solution is given by

(4) |

where we choose the lower bound of the inequality since we can recover the standard uncertainty principle in the limit . Series expanison of Eq.(4) yields

(5) |

Therefore, we can obtain by using the above statement

(6) |

where the rhs of the inequality can be considered as an effective Planck constant . On the other hand, when the black hole absorbs a particle, the smallest increase of black hole area is given by

(7) |

Taking the uncertanity of position and using the Eq.(6) with the Eq.(7), one can obtain the increase of the area as

(8) |

where stands for event horizon of black hole and is a calibration factor which can be obtained in the limit . Moreover, when the particle is absorbed, the minimum increase in black hole entropy is given . So we can obtain

(9) |

Using the temperature of black hole with Eq.(9), one can find

(10) |

where is the surface gravity of the black hole. In order to find , we should check the GUP-modified temperature in the limit . The Eq.(10) should give the standard result when goes to zero. Therefore we find the calibration factor equals . Finally we find

(11) |

which is the GUP modified temperature for static and spherically symmetric black holes.

In this section, we briefly review the generic GUP correction to black hole temperature. In the next section, we will use Eq.(11) to modify the vdW black hole solution. For the clarity of the following discussions, we choose in the rest of the paper.

## Iii GUP-Corrected vdW Black Holes

vdW equation of state is a generalized version of ideal gas equation. It is given by

(12) |

Here denotes the specific volume, constant is a measure of the attraction between the particles and is a measure of the particle volume. One can use vdW equation for describing the behavior of liquidgas phase transition. In order to construct a solution whose thermodynamic matches with that of Eq.(12), we start with the following spherically symmetric ansatz for the metric

(13) |

(14) |

where the function can be obtained by using GUP-corrected black hole temperature. Now, we assume that the given metric is a solution of the Einstein field equation, . Here, we choose the stress energy tensor as an anisotropic fluid source in the following form

(15) |

where , and denote the components of the vielbein (), energy density and principle pressure, respectively. As a solution of Einstein field equation, we need a physically meaningful stress energy source for our metric ansatz. Therefore, we require our corresponding stress energy tensor should satisfy certain energy conditions such as weak, strong and dominance energy conditions. One can easily obtain the energy density and principal pressures as follows

(16) |

(17) |

where the prime denotes the derivative with respect to . The mass of black hole can be obtained by using the metric ansatz eq.(14)

(18) |

The thermodynamic quantities are

(19) |

(20) |

(21) |

(22) |

where we choose integration constant to make a dimensionless logarithmic term in Eq.(20). One can also define the specific volume of black hole by Altamirano2014 ()

(23) |

where and in dimensions . is proportional to the horizon area of black hole with . Therefore, one can obtain

(24) |

In order to construct a solution whose thermodynamics is identical with the the vdW fluids, we assume that and use the equality between Eq.(12) and Eq.(19). Therefore we can write

(25) |

where . The above equation can be written in the form , where the functions and depend on the functions and and their derivatives. To solve the Eq.(25), we should independently take the functions and equal zero,

(26) |

(27) |

One can obtain

(28) |

For simplicity, we choose integration constant . So we obtain

(29) |

Now by substituting Eq.(29) into Eq.(26) and then expanding up to second order of , one can find the following equation:

(30) |

and this equation yields the solution

(31) | |||||

where we choose the suitable integration constant to make the dimensionless logarithmic terms. Using the solutions in Eq.(28) and Eq.(III), we can obtain modified functions and therefore we get the GUP-corrected vdW black hole solution. As it can be seen from Eq.(28) and Eq.(III), the result in Rajagopal2014 () are obtained in the limit .

For a valid physical solution, stress energy tensor should satisfy the certain energy conditions. Therefore, one should also check the energy conditions for finding a physically sensible solution Poisson2004 ():

(32) | |||||

(33) | |||||

(34) |

In Fig.(1), we show that it is possible to satisfy the all energy conditions near the outer of the horizon for the sufficiently small pressure. Since is positive for a sufficiently large , it is not displayed in the figure. It seems our solution is near horizon solution.

In the next section, we will investigate the thermodynamics of GUP-corrected vdW black holes.

## Iv Thermodynamics and Phase Transition of GUP-Corrected vdW Black Holes

Since we obtain the GUP-corrected function, we explore the GUP effects for the thermodynamic quantities of vdW black hole. In Fig.(2), we show both semiclassical and GUP-corrected entropies of vdW black hole. It is clearly obvious that GUP-corrected entropy is always smaller than semiclassical entropy. Moreover, modified entropy is a monotonically increasing function of for the region () since

(35) |

In Fig.(3), semiclassical and modified temperatures are plotted in term of . GUP-corrected temperature has an ustable branch for small black holes, while it has the same characteristic behaviour with semiclassical temperature due to negligible quantum gravity effects for the larger event horizons. The unstable branch corresponds to negative specific heat region. Therefore vdW black hole is thermodynamically unstable for the values of smaller event horizon in the presence of GUP effects. It is also clear that the temperature increases for the GUP correction.

Now, we investigate the behaviour of heat capacity for thermodynamic stability and possible phase transition of vdW black hole in Fig.(4). We observe that corrected heat capacity diverges. So it shows the stable-unstable black hole phase transition in the presence of GUP modification. Pradhan also reported a similar phase transition for vdW black hole in Pradhan2016 (), but the phase transition occurs at the negative values of event horizon. Therefore, it is not physically acceptable.

It is also interesting to study PV criticality of GUP-corrected vdW black hole. From Eq.(19) and corrected function, one can obtain the equation of state

(36) |

In order to obtain the critical points, we should solve the following two equations:

(37) |

Since the Eq.(36) is very complicated, we have to find critical points numerically. We set and . So we find , and . In Fig(5), we show the small-large black hole phase transition. Expect the unstable left branch, the phase diagram mostly shows the well-known behavior of liquid-gas system. Similar behaviors for the GUP corrected phase transition of charged AdS black hole were also showed in the paperSun2018 () by Sun and Ma.

As a result, instabilities occur for the small black holes when GUP effects are taken into account.

## V Conclusions and Discussions

In this paper, we have considered the GUP-correction and obtained the GUP-corrected vdW black hole solution. We have also investigated the thermodynamics quantities and the phase transition of that black hole. Black hole thermodynamics should be modified since the quantum gravity effects are taken into account near the Planck scale. Therefore we have considered GUP modification for the vdW black holes. In order to obtain the solution, we have used the GUP-corrected black hole equation of state with that of the vdW fluids. Then, we have found the modified function.

Moreover, we have found the modified thermodynamic quantities. We have showed that GUP-corrected entropy is always smaller than the semiclassical entropy. Modified temperature also increases in the presence of GUP-correction. In order to investigate any phase transitions and thermodynamical stabilities, we have analyzed the specific heat. Interestingly we have found stable-unstable black hole phase transition in the GUP-corrected vdW black holes. Finally, we have presented the P-V criticality of GUP-corrected vdW black holes.

In a summary, we have revealed some properties of thermodynamics and phase transition of vdW black holes due to the GUP-corrections. It is also interesting to study higher-dimensional vdW black holes for the GUP modifications since the importance of the particular dimensionsDelsate2015 () and our study can be generalized for the black hole solutions Abchouyeh2017 (); Setare2015 (); Xu2018 (); Debnath2018 () in this direction.

## References

- (1) J. D. Bekenstein, J. Lett. Nuovo Cimento 4 (1972) 737.
- (2) J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333.
- (3) J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161.
- (4) S. W. Hawking, Nature 248 (1974) 30.
- (5) J. D. Bekenstein, Phys. Rev. D 9 (1974) 3292.
- (6) S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.
- (7) S. W. Hawking and D. N. Page, Commun.Math.Phys. 87 (1983) 577.
- (8) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D 60 (1999) 064018.
- (9) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. D 60 (1999) 104026.
- (10) D. Kubiznak R. B. Mann, J. High Energy Phys. 2012 (2012) 033.
- (11) S. Gunasekaran, R. B. Mann and D. Kubiznak, J. High Energy Phys. 2012 (2012) 110.
- (12) E. Spallucci and A. Smailagic, Phys. Lett. B 723 (2013) 436.
- (13) R. Zhao, H.-H. Zhao, M.-S. Ma and L.-C. Zhang, Eur. Phys. J. C 73 (2013) 2645.
- (14) A. Behlaj, M. Chabab, H. E. Moumni, L. Medari and M. B. Sedra, Chinese Phys. Lett. 30 (2013) 090402.
- (15) R.-G. Cai, L.-M. Cao, L. Li. and R.-Q. Yang, J. High Energy Phys. 2013 (2013) 005.
- (16) J.-X. Mo, G.-Q. Li and W.-B. Liu, Phys. Lett. B 730 (2014) 111.
- (17) W. Xu and L. Zhao, Phys. Lett. B 736 (2014) 214.
- (18) G.-Q Li, Phys. Lett. B 735 (2014) 256.
- (19) M.-S. Ma, F. Liu and R. Zhao, Class. Quantum Grav. 73 (2014) 095001.
- (20) A. Belhaj, M. Chabab, H. E. Moumni, K. Masmar and M. B. Sedra, Int. J. Geom. Methods Mod. Phys 12 (2015) 1550017.
- (21) D. Kubiznak and R. Mann, Can. J. Phys. 93 (2015) 999.
- (22) R. A. Hennigar and R. B. Mann, Entropy 17 (2015) 8056.
- (23) E. Caceres, P. H. Nguyen and J.F. Pedraza, J. High Energy Phys. 2015 (2015) 184.
- (24) S.-W. Wei, P. Cheng and Y.-X. Liu, Phy. Rev. D 93 (2016) 084015.
- (25) S. H. Hendi, S. Panahiyan and B. E. Panah, J. High Energy Phys. 2016 (2016) 129.
- (26) D. Momeni, M. Faizal, K. Myrzakulov and R. Myrzakulov, Phys. Lett. B 765 (2017) 154.
- (27) A. Övgün, Adv. High Energy Phys. 2018 (2018) 8153721.
- (28) Z. Sun, M.-S. Ma, Europhys. Lett. 122 (2018) 60002.
- (29) M. Jamil, B. Pourhassan, A. Övgün and I. Sakalli, arXiv:1811.02193
- (30) C. H. Nam, Eur. Phys. J. C 78 (2018) 581.
- (31) C. H. Nam, Eur. Phys. J. C 78 (2018) 1016.
- (32) X.-M. Kuang, B. Liu and A. Övgün, Eur. Phys. J. C 78 (2018) 840.
- (33) M. Zhang, Nucl. Phys. B 935 (2018) 170.
- (34) Ö. Ökcü and E. Aydiner, Eur. Phys. J. C 77 (2017) 24.
- (35) Ö. Ökcü and E. Aydiner, Eur. Phys. J. C 78 (2018) 123.
- (36) J.-X. Mo, G.-Q. Li, S.-Q. Lan and X.-B. Xu, Phys. Rev. D 98 (2018) 124032.
- (37) Z.-W Zhao., Y.-H. Xiu and N. Li, Phys. Rev. D 98 (2018) 124003.
- (38) D. M. Yekta, A. Hadikhani and Ö. Ökcü, arXiv:1905.03057 (2019).
- (39) N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, Galaxies 2 (2014) 89.
- (40) D. Kubiznak, R. B. Mann and M. Teo, Class. Quantum Grav. 34 (2017) 063001.
- (41) A. Rajagopal, D. Kubiznak and R. B. Mann, Phys. Lett. B 737 (2014) 277.
- (42) A. Rajagopal, D. Kubiznak and R. B. Mann, Phys. Lett. B 737 (2014) 277.
- (43) P. Pradhan, Europhys. Lett. 116 (2016) 10001.
- (44) Y. Hu, J. Chen and Y. Wang, Gen. Relativ. Gravit. 49 (2017) 148.
- (45) T. Delsate and R. B. Mann, J. High Energy Phys. 2015 (2015) 70.
- (46) M. R. Setare and H. Adami, Phys. Rev. D 91 (2015) 084014.
- (47) M. A. Abchouyeh, B. Mirza and M. K. Takrami, Phys. Lett. B 780 (2018) 240.
- (48) W. Xu, Eur. Phys. J. C 78 (2018) 871.
- (49) U. Debnath, arXiv:1903.04379 (2019).
- (50) T. R. Govindarajan, R. K. Kaul and V. Suneeta , Class. Quantum Grav. 18 (2001) 2877.
- (51) R. B. Mann and S. N. Solodukhin, Nucl. Phys. B 523 (1998) 293.
- (52) A. Sen, J. High Energy Phys. 2013 (2013) 156.
- (53) S. Das, P. Majumdar and R. K. Bhaduri, Class. Quantum Grav. 19 (2002) 2355.
- (54) S. Upadhyay and B. Pourhassan, Prog. Theor. Exp. Phys. 2019 (2019) 1.
- (55) M. Maggiore, Phys. Lett. B 304 (1993) 65.
- (56) A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52 (1995) 1108.
- (57) R. J. Adler, P. Chen and D. I. Santiago, Gen. Relat. Gravit. 33 (2001) 2101.
- (58) A. F. Ali, J. High Energy Phys. 2012 (2012) 067.
- (59) S. Gangopadhyay, A. Dutta and A. Saha Gen. Relat. Gravit. 46 (1995) 1661.
- (60) N. Abbasvandi, M. J. Soleimani, S. Radiman and W. A. T. W. Abdullah, Int. J. Mod. Phys. A 31 (2016) 1650129.
- (61) L. Xiang and X. Q. Wen, J. High Energy Phys. 2009 (2009) 046.
- (62) E. Poisson, A. Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, 1st edn., (Cambridge University Press, Cambridge, 2004).