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Black hole thermodynamics establishes a deep and satisfying link to gravity, thermodynamics, and quantum theory. And, the thermodynamic property of black hole is essentially a quantum feature of gravity. In this paper, in order to study the influence of the quantum gravity effect on the quantum properties of black hole, we study the thermodynamics and its quantum correction to a non-commutative black hole. First of all, the temperature of the non-commutative Schwarichild black hole is calculated by using three different methods: surface gravity, tunneling effects and the first law of black hole thermodynamics. It is found that the same hole temperature is obtained by means of the surface gravity and tunneling effects. However, by using the first law of black hole thermodynamics, different results are derived from the first two methods. Therefore, we incline to the result obtained by surface gravity and tunneling effects, and the temperature obtained by the thermodynamic law needs modifying. That is, for the non-commutative black hole, there is a contradiction to the first law of thermodynamics. To calculate the temperature and other thermodynamic quantities for the non-commutative Schwarichild black hole, we use the corrected first law of black hole thermodynamics proposed in the literature. It is found that the black hole temperature derived by the corrected first law is the same as the temperature obtained by the surface gravity and the tunneling model, and the black hole entropy still follows Beckenstein-Hawking area law. Also, the heat capacity of the black hole is obtained and analyzed. It is seen that when the horizon radius and non-commutative parameter satisfy the particular conditions, the heat capacity is positive and the non-commutative black holes are thermodynamically stable. This is a different result from that of the usual Schwarichild black hole. Further, by studying the influence of generalized uncertainty principle on non-commutative black hole thermodynamics, the quantum corrections from generalized uncertainty principle for temperature, entropy and heat capacity of the non-commutative Schwarzschild black hole are given. It is found that with considering this quantum gravity effect, the obtained black hole entropy contains the item of are alogarithm. If the effect of the generalized uncertainty principle is neglected, the corrected black hole entropy can return to that in the usual case of Beckenstein-Hawing area law. Similarly, the corrected black hole temperature and heat capacity can also return to their counterparts in the case of usual Schwarzschild black hole when this quantum gravity effect is ignored.
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Keywords:
- noncommutative black holes /
- corrected first law of black hole thermodynamics /
- generalize uncertainty principle /
- quantum corrections
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[1] Hawking S W 1974 Nature 248 30
Google Scholar
[2] Hawking S W 1975 Commun. Math. Phys. 43 199
Google Scholar
[3] Bekenstein J D 1973 Phys. Rev. D 7 2333
[4] Wald R M 2001 Living Rev. Rel. 4 6
Google Scholar
[5] 蔡荣根, 曹利明 2016 科学通报 61 2083
Cai R G, Cao L M 2016 Chin. Sci. Bull. 61 2083
[6] Garay L J 1995 Int. J. Mod. Phys. A 10 145
Google Scholar
[7] Maggiore M 1993 Phys. Lett. B 304 65
Google Scholar
[8] Kempf A, Mangano G, Mann R B 1995 Phys. Rev. D 52 1108
Google Scholar
[9] Tawfik A, Diab A 2014 Int. J. Mod. Phys. D 23 1430025
Google Scholar
[10] Chang L N, Minic D, Okamura N, Takeuchi T 2002 Phys. Rev. D 65 125028
Google Scholar
[11] Li X 2002 Phys. Lett. B 540 9
Google Scholar
[12] Liu C Z 2008 Sci. China Ser. G: Phys. Mech. Astron. 51 113
Google Scholar
[13] Gangopadhyay S, Dutta A, Faizal M 2015 EPL 112 20006
Google Scholar
[14] Li H L, Song D W, Li W 2019 Gen. Relat. Gravit. 51 20
Google Scholar
[15] Chen D, Wu H, Yang H 2013 Adv. High. Energy Phys. 43 24126
[16] Zhao R, Zhang S L 2006 Phys. Lett. B 641 318
Google Scholar
[17] Majumder B 2011 Phys. Lett. B 703 402
Google Scholar
[18] Banerjee R, Ghosh S 2010 Phys. Lett. B 688 224
Google Scholar
[19] Medved A J M, Vagenas E C 2004 Phys. Rev. D 70 124021
Google Scholar
[20] Anacleto M A, Brito F A, Passos E 2015 Phys. Lett. B 749 181
Google Scholar
[21] Yoon M, Ha J, Kim W 2007 Phys. Rev. D 76 047501
Google Scholar
[22] Anacleto M A, Bazeia D, Brito F A, Mota-Silva J C 2016 Adv. High. Energy Phys. 11 8465759
[23] Maluf R V, Neves J C S 2018 Phys. Rev. D 97 104015
Google Scholar
[24] Casadio R, Nicolini P, da Rocha R 2018 Class. Quant. Grav. 35 185001
Google Scholar
[25] Myung Y S, Kim Y W, Park Y J 2007 Phys. Lett. B 645 393
Google Scholar
[26] Witten E 1996 Nucl. Phys. B 460 335
Google Scholar
[27] Szabo R J 2003 Phys. Rept. 378 207
Google Scholar
[28] Nasseri F 2005 Gen. Relat. Gravit. 37 2223
Google Scholar
[29] Nicolini P, Smailagic A, Spallucci E 2006 Phys. Lett. B 632 547
Google Scholar
[30] Lopez-Dominguez J C, Obregon O, Sabido M, Ramirez C 2006 Phys. Rev. D 74 084024
Google Scholar
[31] Chaichian M, Tureanu A, Zet G 2008 Phys. Lett. B 660 573
Google Scholar
[32] Kobakhidze A 2009 Phys. Rev. D 79 047701
Google Scholar
[33] Nicolini P 2005 J. Phys. A 38 L631
Google Scholar
[34] Gruppuso A 2005 J. Phys. A 38 2039
Google Scholar
[35] Cai R G, Wang A 2010 Phys. Lett. B 686 166
Google Scholar
[36] Nicolini P, Smailagic A, Spallucci E 2006 ESA Spec. Publ. 637 547
[37] Nozari K, Fazlpour B 2007 Mod. Phys. Lett. A 22 2917
Google Scholar
[38] Banerjee R, Majhi B R, Samanta S 2008 Phys. Rev. D 77 124035
Google Scholar
[39] Baneriee R, Majhi B R, Modak S K 2009 Class. Quant. Grav. 26 085010
Google Scholar
[40] Nozari K, Mehdipour S H 2008 Class. Quant. Grav. 25 175015
Google Scholar
[41] Myung Y S, Kim Y W, Park Y J 2007 JHEP 0702 012
[42] Kim W, Son Edwin J, Yoon M 2008 JHEP 0804 042
[43] Vakili B, Khosravi N, Sepangi H R 2009 Int. J. Mod. Phys. D 18 159
Google Scholar
[44] Buric M, Madore J 2008 Eur. Phys. J. C 58 347
Google Scholar
[45] Huang W H, Huang K W 2009 Phys. Lett. B 670 416
Google Scholar
[46] Nozari K, Mehdipour S H 2009 JHEP 0903 061
[47] Nozari K, Islamzadeh S 2013 Astrophys. Space. Sci. 347 299
Google Scholar
[48] Mehdipour S H, Keshavarz A 2012 Europhys. Lett. 9 10002
[49] Sharif M, Javed W 2011 Can. J. Phys. 89 1027
Google Scholar
[50] Faizal M, Amorim R G G, Ulhoa S C 2015 Int. J. Mod. Phys. D 27 1850053
[51] Ma M S, Zhao R 2014 Class. Quantum. Grav. 31 24
[52] Casadio R, Micu O, Nicolini P 2015 High. Energy Phys. 178 293
[53] Ali A F, Nafie H, Shalaby M 2012 Europhys. Lett. 100 20004
Google Scholar
[54] Liu Z Y, Ren J R 2014 Commun. Theor. Phys. 62 819
Google Scholar
[55] Mu B, Wang P, Yang H 2015 Adv. High. Energy Phys. 2015 1
[56] Ma H, Li J 2017 Chin. Phys. B 26 60401
Google Scholar
[57] Chen N S, Zhang J Y 2015 Chin. Phys. B 24 020401
Google Scholar
[58] Ibungochouba S T 2015 Chin. Phys. B 24 70401
Google Scholar
[59] Ye B B, Chen J H, Wang Y J 2017 Chin. Phys. B 26 90202
Google Scholar
[60] 刘成周, 邓岳君, 骆叶成 2018 物理学报 67 060401
Google Scholar
Liu C Z, Deng Y J, Luo Y C 2018 Acta Phys. Sin. 67 060401
Google Scholar
[61] 吴迪, 朱晓丹, 吴双清 2014 物理学报 63 180401
Google Scholar
Wu D, Zhu X D, Wu S Q 2014 Acta Phys. Sin. 63 180401
Google Scholar
[62] Parikh M K, Wilczek F 2000 Phys. Rev. Lett. 85 5042
Google Scholar
[63] Parikh M K 2004 Int. J. Mod. Phys. D 13 2351
Google Scholar
[64] Painleve P 1921 C. R. Acad. Sci. (Paris)
173 677 [65] 刘辽, 赵峥 2004 广义相对论(北京: 高等教育出版社) 第109页
Liu L, Zhao Z 2004 General Relativity (Beijing: Higher Education Press) p109 (in Chinese)
[66] Camellia G A, Arzano M, Procaccini A 2014 Phys. Rev. D 70 107501
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