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Black hole spectroscopy is an important content in the quantum properties of black holes. In this paper, we use the adiabatic invariants of black holes to investigate the entropy spectrum and area spectrum of the Kerr black hole in gravity's rainbow. Firstly, by considering the particles passing through the event horizon, the adiabatic invariance action for the modified Kerr black hole is calculated. Here, the Euclidean coordinate and the period of the Euclidean time of a loop about the event horizon are used. Combined the obtained adiabatic invariants with the Bohr-Sommerfen quantization condition, the equally spaced entropy spectra that are the same as the original Beckenstein spectra are derived. The entropy spectrum of the gravity's rainbow is independent of the test particle energy. Next, using the first law of the black hole thermodynamics and the black hole entropy spectrum, the area spectrum of the modified Kerr black hole is studied. Due to the quantum gravity effect of the gravity's rainbow, the obtained area spectrum is different from the original Beckenstein spectrum. The present area spectrum is non-equidistant and dependent on the horizon area of the black hole. With the decrease of black hole area, the area space gradually turns smaller. When the black hole reaches the minimum area on a Planck scale, the area quantum is zero. Thus the black hole area no longer decrease and a remnant of the black hole radiation appears. In the case of a large black hole, the correction of the area spectrum to the equally spaced spectra can be ignored, and the area spectrum of the Kerr black hole in gravity's rainbow can return to the original Beckenstein spectrum. It is also shown that like the entropy spectrum, the area spectrum of the gravity's rainbow does not depend on the energy of the test particles either. In addition, the entropy of the modified Kerr black hole in gravity's rainbow is discussed by using the first law of the black hole thermodynamics. The black hole entropy with quantum correction items as the area reciprocal to the Beckenstein-Hawking entropy is derived and the relation between the quantum correction items and the area is discussed. In addition, the consistency between the entropy correction and the area correction for the modified black hole is analyzed. The current research supports that in different spacetimes including quantum corrected spacetimes, the black hole entropy spectrum has the universality, but the black hole area spectrum is dependent on the area due to the spacetime quantum properties.
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Keywords:
- gravity's rainbow black hole /
- action invariance /
- quantum effect
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[44] Chen N S, Zhang J Y 2015 Chin. Phys. B 24 020401
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[47] Amelino-Camelia G 2002 Int. J. Mod. Phys. D 11 35
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[57] Altamirano N, Kubiznak D, Mann R B, Sherkatghanad Z 2014 Galaxies 2 89
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[59] Ling Y, Hu B, Li X 2006 Phys. Rev. D 73 087702
[60] Liu C Z, Zhu J Y 2008 Gen. Relat. Gravit. 40 1899
[61] Zhang J Y, Zhao Z 2005 Mod. Phys. Lett. A 20 1673
[62] Jiang Q Q, Wu S Q, Cai X 2006 Phys. Rev. D 73 064003
[63] Gibbons G W, Hawking S W 1977 Phys. Rev. D 15 2752
[64] Adler R J, Chen P, Santiago D I 2001 Gen. Rel. Grav. 33 2101
[65] Amelino-Camelia G, Arzano M, Procaccini A 2004 Phys. Rev. D 70 107501
[66] Ali A F, Mohammed M F, Khalil M 2015 Nucl. Phys. B 894 341
[67] Ali A F 2014 Phys. Rev. D 89 104040
[68] Kaul R K, Majumder P 2000 Phys. Rev. Lett. 84 5255
[69] Don N 2005 Page, New. J. Phys. 7 203
[70] Jing J L, Yan M L 1999 Phys.Rev. D 60 084015
[71] Carlip S 2000 Class. Quant. Grav. 17 4175
[72] Jing J L, Yan M L 2000 Phys. Rev. D 63 024003
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[1] Bekenstein J D 1972 Lett. Nuovo. Cim. 4 737
[2] Bekenstein J D 1973 Phys. Rev. D 7 2333
[3] Bekenstein J D 1974 Lett. Nuovo Cim. 11 467
[4] Bekenstein J D 1998 arXiv:gr-qc/9808028
[5] Kunstatter G 2003 Phys. Rev. Lett. 90 161301
[6] Nollert H P 1999 Class. Quant. Grav. 16 R159
[7] Hod S 1998 Phys. Rev. Lett. 81 4293
[8] Hod S 1998 Phys. Rev. D 59 024014
[9] Maggiore M 2008 Phys. Rev. Lett. 100 141301
[10] Wang B, Lin C Y, Molina C 2004 Phys. Rev. D 70 064025
[11] Medved A J M 2008 Class. Quantum Grav. 25 205014
[12] Vagenas E C 2008 JHEP 2008 073
[13] Ropotenko K 2010 Phys. Rev. D 82 044037
[14] Kothawala D, Padmanabhan T, Sarkar S 2008 Phys. Rev. D 78 104018
[15] Wei S W, Li R, Liu Y X, Ren J R 2009 JHEP 2009 076
[16] Li W B, Xu L X, Lu J B 2009 Phys. Lett. B 676 177
[17] Jing J L, Ding C K 2008 Chin. Phys. Lett. 25 858
[18] Pan Q Y, Jing J L 2005 Chin. Phys. B 14 268
[19] Chen J H, Wang Y J 2010 Chin. Phys. B 19 060401
[20] Wei S W, Liu Y X, Yang K, Zhong Y 2010 Phys. Rev. D 81 104042
[21] Liu C Z 2012 Eur. Phys. J. C 72 2009
[22] Barvinsky A, Das S, Kunstatter G 2001 Class. Quant. Grav. 18 4845
[23] Barvinsky A, Das S, Kunstatter G 2002 Found. Phys. 32 1851
[24] Ropotenko K 2009 Phys. Rev. D 80 044022
[25] Kwon Y, Nam S 2010 Class. Quant. Grav. 27 125007
[26] Louko J, Makela J 1996 Phys. Rev. D 54 4982
[27] Majhi B R, Vagenas E C 2011 Phys. Lett. B 701 623
[28] Liu C Z 2012 Chin. Phys. B 21 070401
[29] Li L 2012 Int. J. Ther. Phys. 51 1924
[30] Liu C Z 2012 Mod. Phys. Lett. A 27 1250139
[31] Zeng X X, Liu W B 2012 Eur. Phys. J. C 72 1987
[32] Qi D J 2014 Astrophys. Space. Sci. 349 33
[33] Garay L J 1995 Int. J. Mod. Phys. A 10 145
[34] Gross D J, Mende P F 1988 Nucl. Phys. B 303 407
[35] Witten E 1997 Phys. Today 49 24
[36] Smolin L 2004 arXiv:hep-th.0408048
[37] Ali A F, Faizal M, Khalil M M 2014 JHEP 2014 159
[38] Ali A F, Faizal M, Khalil M M 2015 Phys. Lett. B 743 295
[39] Gangopadhyay S, Dutta A, Saha A 2014 Gen. Rel. Grav. 46 1661
[40] Dutta A, Gangopadhyay S 2014 Gen. Rel. Grav. 46 1747
[41] Gangopadhyay S, Dutta A, Faizal M 2015 Euro. Phys. Lett. 112 20006
[42] Dutta A, Gangopadhyay S 2016 Int. J. Theo. Phys. 55 2746
[43] Ma H, Li J 2017 Chin. Phys. B 26 60401
[44] Chen N S, Zhang J Y 2015 Chin. Phys. B 24 020401
[45] Ibungochouba S T 2015 Chin. Phys. B 24 70401
[46] Ye B B, Chen J H, Wang Y J 2017 Chin. Phys. B 26 90202
[47] Amelino-Camelia G 2002 Int. J. Mod. Phys. D 11 35
[48] Amelino-Camelia G 2001 Phys. Lett. B 510 255
[49] Kowalski-Glikman J 2001 Phys. Lett. A 286 391
[50] Magueijo J, Smolin L 2002 Phys. Rev. Lett. 88 190403
[51] Magueijo J, Smolin L 2003 Phys. Rev. D 67 044017
[52] Kimberly D, Magueijo J, Medeiros J 2004 Phys. Rev. D 70 084007
[53] Magueijo J, Smolin L 2004 Class. Quant. Grav. 21 1725
[54] Heuson C 2006 arXiv:gr-qc/0606124
[55] Amelino-Camalia G, Ellis N E, Mavromatos D V 1997 Int. J. Mod. Phys. A 12 607
[56] Amelino-Camalia G 2013 Living. Rev. Rel. 16 5
[57] Altamirano N, Kubiznak D, Mann R B, Sherkatghanad Z 2014 Galaxies 2 89
[58] Ling Y, Li X, Hu B 2007 Mod. Phys. Lett. A 22 2749
[59] Ling Y, Hu B, Li X 2006 Phys. Rev. D 73 087702
[60] Liu C Z, Zhu J Y 2008 Gen. Relat. Gravit. 40 1899
[61] Zhang J Y, Zhao Z 2005 Mod. Phys. Lett. A 20 1673
[62] Jiang Q Q, Wu S Q, Cai X 2006 Phys. Rev. D 73 064003
[63] Gibbons G W, Hawking S W 1977 Phys. Rev. D 15 2752
[64] Adler R J, Chen P, Santiago D I 2001 Gen. Rel. Grav. 33 2101
[65] Amelino-Camelia G, Arzano M, Procaccini A 2004 Phys. Rev. D 70 107501
[66] Ali A F, Mohammed M F, Khalil M 2015 Nucl. Phys. B 894 341
[67] Ali A F 2014 Phys. Rev. D 89 104040
[68] Kaul R K, Majumder P 2000 Phys. Rev. Lett. 84 5255
[69] Don N 2005 Page, New. J. Phys. 7 203
[70] Jing J L, Yan M L 1999 Phys.Rev. D 60 084015
[71] Carlip S 2000 Class. Quant. Grav. 17 4175
[72] Jing J L, Yan M L 2000 Phys. Rev. D 63 024003
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