Chinese Physics C Vol. 44, No. 9 (2020) 095103 Generalized uncertainty principle and black hole thermodynamics * 1,2 2;1) 1;2) Jin Pu(蒲瑾) Qin-Bin Mao(毛勤斌) Qing-Quan Jiang(蒋青权) 1 1,3;3) Jing-Xia Yu(于景侠) Xiao-Tao Zu(祖小涛) 1School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China 2College of Physics and Space Science, China West Normal University, Nanchong 637002, China 3Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China Abstract: Banerjee-Ghosh's work shows that the singularity problem can be naturally avoided by the fact that black hole evaporation stops when the remnant mass is greater than the critical mass when including the generalized uncer- tainty principle (GUP) effects with first- and second-order corrections. In this paper, we first follow their steps to reexamine Banerjee-Ghosh's work, but we find an interesting result: the remnant mass is always equal to the critical mass at the final stage of black hole evaporation with the inclusion of the GUP effects. Then, we use Hossenfelder's GUP, i.e., another GUP model with higher-order corrections, to restudy the final evolution behavior of the black hole evaporation, and we confirm the intrinsic self-consistency between the black hole remnant and critical masses once more. In both cases, we also find that the thermodynamic quantities are not singular at the final stage of black hole evaporation. Keywords: black hole thermodynamics, generalized uncertainty principle, remnant mass, critical mass DOI: 10.1088/1674-1137/44/9/095103 1 Introduction potential [37]. As far as we are concerned, the GUP affects the well- known semi-classical laws of black hole thermodynam- The generalized uncertainty principle (GUP), which ics [38-68]. For example, the black hole entropy is no modifies the uncertainty principle to include a minimal length, has received increasing amounts of attention in longer proportional to the horizon area [51-59]; the black the past decade [1-4]. On the one hand, one may predict hole does not evaporate completely, but leaves a remnant the GUP effects through various experiments, such as the mass at the final stage of evaporation [57-65]; the rem- hydrogen Lamb shift [5-7], electron tunneling [5, 6], nant with the Planck scale can store information, which mechanical oscillators [8, 9], gravitational bar detectors gives a possible solution to the singularity problem [57- [10], ultra-cold atom experiments [11, 12], gravitational 65]; and there is a metastable remnant that asymptotes to wave experiments [13, 14], sub-kilogram acoustic reson- zero mass when considering the negative GUP correction ators [15], and large molecular wave-packets [16]. On the [66]. In [68], Banerjee and Ghosh intriguingly construc- other hand, one can also apply the GUP to study the ef- ted a GUP that contains the term predicted by string the- fects of quantum gravity on small- or large-scale physic- ory and a series of higher-order correction terms, and al systems. For example, the GUP effects have been stud- they studied the GUP effects on black hole thermody- ied with respect to the early Universe [17-21], compact namics. Their results show that, when considering the stellar objects [22-24], the Newtonian law of gravity [25], first- and second-order quantum corrections to black hole the equivalence principle [26-28], the entropic nature of thermodynamics, black hole evaporation always stops gravitational force [29-33], the Casimir effect [34, 35], when the remnant mass is greater than the critical mass, the Dirac δ-function potential [36], and post-Newtonian and the singularity problem in the semi-classical ap- Received 17 February 2020, Revised 26 April 2020, Published online 20 July 2020 * Supported by the Program for NCET-12-1060, by the Sichuan Youth Science and Technology Foundation (2011JQ0019), FANEDD (201319), the Innovative Research Team in College of Sichuan Province (13TD0003), Ten Thousand Talent Program of Sichuan Province, Sichuan Natural Science Foundation (16ZB0178) and Meritocracy Research Funds of China West Normal University (17YC513, 17C050) 1) E-mail: [email protected] 2) E-mail: [email protected] 3) E-mail: [email protected] ©2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 095103-1 Chinese Physics C Vol. 44, No. 9 (2020) 095103 proach is bypassed at the final stage of black hole evapor- From (1), we can obtain ! ation. X1 2i @p 0 Lp p However, Banerjee-Ghosh's results in [68] lack cred- = h¯ a ; (3) @k i h¯ ibility because some necessary terms have been omitted i=0 in their treatment. For example, when dealing with the f 0g where the new coefficients of expansions ai are func- first-order correction, all terms regarding a0 should be in- f g 0 = 1 tions of ai , and a0 1. Hence, the form of the GUP pro- cluded in the corrected temperature-mass relation. posed by Banerjee and Ghosh is given by 02 2 2 ! However, the term a (kBT=Mpc ) has been omitted in X1 2i 1 h¯ Lp∆p their treatment, perhaps because they consider this term ∆x∆p > a0 ; (4) 2 i h¯ to be negligible. When dealing with the second-order cor- i=0 rection, Banerjee and Ghosh have omitted the necessary where the coefficients fa0g are all positive. 0 0 = 2 4 02 = 2 6 i terms 2a1a2(kBT Mpc ) and a2 (kBT Mpc ) . If these Subsequently, we use the GUP (4) to study the omitted terms are recovered, the final evolution behavior quantum-corrected thermodynamic entities of a Schwarz- of black hole evaporation may be different. Anyway, schild black hole and attempt to find relationships among Banerjee-Ghosh's work cannot truly demonstrate the fi- them. In [68], by comparison with the standard semi-clas- nal evolution behavior of a black hole system with the in- sical Hawking temperature, the mass-temperature rela- clusion of the GUP effects. tionship of a Schwarzschild black hole is given by ! In this paper, we reexamine Banerjee-Ghosh's work in X1 2i−1 Mp k T Sec. 2, and we restudy the final evolution behavior of M = a0 B : (5) 8π i M c2 black hole evaporation when including the GUP effects i=0 p with first- and second-order corrections. In Sec. 3, we re- According to the definition of the heat capacity of a black view the GUP proposed by Hossenfelder et al. in [69], M = 2 d i.e., another GUP model with higher-order corrections. In hole C c , we have dT ! Sec. 4, we use Hossenfelder's GUP to precisely study X1 2i−2 kB 0 kBT first- and second-order quantum corrections to black hole C = a (2i − 1) : (6) 8π i M c2 thermodynamics, and we aim to discover the intrinsic i=0 p self-consistency between the black hole remnant and crit- The entropy of a black hole is given by ical masses when including the effects of quantum grav- Z 2 ! ! 6 2 2 2 ity. Finally, Sec. 5 provides some conclusions. CdT kB 6 Mpc 0 kBT S = = 6 + a 4 1 ln 2 T 16π kBT Mpc ! 3 2 Reexamination of Banerjee-Ghosh's work X1 2(i−1)7 0 (2i − 1) kBT 7 + a 57: (7) i − 2 = (i 1) Mpc In [68], Banerjee and Ghosh have assumed that the i 2 function relation between the wave vector k and the mo- In [68], because the heat capacity and entropy are ex- mentum p satisfies certain properties: 1) the function has pressed in terms of the mass, the expression for T 2 in to be an odd function to preserve parity; 2) the function terms of M is given by should be chosen to satisfy p = hk¯ at small energy; and 3) ! ! ! 2 2 2 2 π/ 8πM Mpc 0 0 0 k T the wave vector k should have an upper bound of 2 Lp . = + a + a 2 + a B 2 1 ( 1 2 2) 2 Thus, Banerjee and Ghosh have assumed an infinite-or- Mp kBT Mpc ! der polynomial to satisfy these properties of the function, k T 4 +2(a0 a0 +a0 ) B which is expressed as 1 2 3 M c2 1 ! p X 2i+1 ! 1 i Lp p 6 k = f (p) = a (−1) : (1) 0 0 0 kBT i +(2a a +a 2) +··· : (8) Lp = h¯ 1 3 2 2 i 0 Mpc Here, only odd powers of the momentum p appear in the In Banerjee-Ghosh’s treatment, when dealing with the polynomial because the function f (p) is an odd function first-order correction, they have obtained the corrected f g to preserve parity. The coefficients ai are all positive, mass-temperature relation as and a = 1 to recover p = hk¯ at small energy. The factor ! ! 0 2 2 2 − i 8πM Mpc 0 ( 1) ensures property 3), and we have a constraint for = + 2a : (9) π 1 ! 1 ! 2 Mp kBT p , k L , i.e. p ! 0 X1 2i+1 In fact, all terms concerning a should be included in the i Lp p 1 ai(−1) ! 2π. (2) first-order correction, so the corrected Mass-Temperature h¯ i=0 relation should be written as 095103-2 Chinese Physics C Vol. 44, No. 9 (2020) 095103 ! ! ! 2 2 2 2 8πM Mpc k T terms are recovered in the second-order correction, we = + a0 + a02 B : (10) 2 1 1 2 have Mp kBT Mpc ! s 2 0 0 kBT k T 1 2a Therefore, the term a 2 has been omitted in B = − 1 + B+ D 1 2 2 0 Mpc Mpc 2 3a Banerjee-Ghosh’s treatment.