Generalized Uncertainty Principle and Black Hole Thermodynamics *

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 uncertainty 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. ∑∞ 2i ∂p ′ 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 { ′} where the new coefficients of expansions ai are func- first-order correction, all terms regarding a′ should be in- { } ′ = 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 ′2 2 2 ( ) However, the term a (kBT/Mpc ) has been omitted in ∑∞ 2i 1 h¯ Lp∆p their treatment, perhaps because they consider this term ∆x∆p ⩾ a′ , (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 {a′} are all positive. ′ ′ / 2 4 ′2 / 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 ∑∞ 2i−1 Mp k T Sec. 2, and we restudy the final evolution behavior of M = a′ 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 ∑∞ 2i−2 kB ′ 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- ∫ ( ) ( )  2 2 2 ity. Finally, Sec. 5 provides some conclusions. CdT kB  Mpc ′ kBT S = =  + a  1 ln 2 T 16π kBT Mpc ( )  2 Reexamination of Banerjee-Ghosh's work ∑∞ 2(i−1) ′ (2i − 1) kBT  + a . (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 ′ ′ ′ 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 + a′ a′ +a′ B which is expressed as 2( 1 2 3) 2 ( ) Mpc ∑∞ 2i+1 ( ) 1 i Lp p 6 k = f (p) = a (−1) . (1) ′ ′ ′ 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 { } 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 ′ ( 1) ensures property 3), and we have a constraint for = + 2a . (9) π 1 → ∞ → 2 Mp kBT p , k L , i.e. p ( ) ′ ∑∞ 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

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( ) ( ) ( ) 2 2 2 2 8πM Mpc k T terms are recovered in the second-order correction, we = + a′ + a′2 B . (10) 2 1 1 2 have Mp kBT Mpc ( ) √ 2 ′ ′ kBT k T 1 2a Therefore, the term a 2 has been omitted in B = − 1 + B+ D 1 2 2 ′ Mpc Mpc 2 3a Banerjee-Ghosh’s treatment. Based on Eq. (9), Banerjee √ 2 ′ and Ghosh further obtained the result that the remnant 1 4a 2 × (8πM/Mp) − − 1 − B− D + √ , mass is greater than the critical mass when considering 2 3a′ ′ − ′ / ′ + + 2 a2 2a1 3a2 B D the first-order correction. (16) Based on Eq. (10), where the omitted term is recovered in the first-order correction, we can obtain where / ′ ′ ( ) 21 3(a 2 + 12a ) 2 B = ( 1√ 2) , (17) kBT 2 1/3 = √ . (11) ′ + 2 − 2 ( ) ( ) ( ) 3a F F G Mpc π 2 π π 2 2 8 M ′ 8 M 8 M ′ ( ) − a + − a √ 1/3 2 1 4 1 2 Mp Mp Mp F + F −G D = , (18) × 1/3 ′ Obviously, as a thermodynamic system, the black hole 3 2 a2 has a critical mass below (at) which the thermodynamic = ′3 − ′ ′ + ′ π / 2, entities become complex (ill-defined) [54, 68], and which F 2a1 72a1a2 27a2(8 M Mp) (19) is given by = ′2 + ′ 3, √ G 4(a1 12a2) (20) a′ = 1 . From Eq. (16), the critical mass below (at) which the Mcr π Mp (12) 4 thermodynamic entities become complex (ill-defined) can From (6) and (11), the heat capacity with the first-or- be determined by F2 −G ⩾ 0, that is der correction is given by √ √ ( ) ′ √ √ 8πM 1 2 a 3 ( ) ( ) ( ) cr = ′ − 1 + ′2 + ′ 3. π 2 π π 2 36a ′ (a 12a ) (21) 8 M ′ 8 M 8 M ′ M 3 3 1 a 1 2 [ − 2a + − 4a ] p 2 M 1 M M 1 = kB − p p p + ′ . From (6), the heat capacity with the second-order correc- C a1 8π 2 tion can now be written as (13)  ( ) ( )   2 2 2 kB  Mpc ′ ′ kBT  When the heat capacity becomes zero at the final stage of C = − + a + 3a . (22) 8π k T 1 2 M c2 black hole evaporation [58, 65, 66], the remnant mass is B p obtained as follows √ Then, at the final stage of black hole evaporation, the ′ remnant mass with the second-order correction is given a1 Mrem = Mp. (14) by 4π uv √ Thus, the remnant mass is equal to the critical mass when √ t ( ) ( ) − ′ + ′2 + ′ 3 2 8πM 1 2 a1 (a1 12a2) ′ kBT rem = a 2 ′ including the necessary term 1 2 for the first-or- Mpc Mp 3 3 a2 der correction term. ( √ ) × ′ + ′2 + ′ 3 Next, let us focus on the effects of the second-order 2a1 (a1 12a2) √ √ correction. The mass-temperature relation should be giv- ′ √ 1 2 a 3 en here, according to (8), as = 36a′ − 1 + (a′2 + 12a′ )3. (23) 3 3 1 a′ 1 2 ( ) ( ) ( ) 2 2 2 2 2 ( ) 8πM Mpc ′ ( ′ ′ ) kBT 4 = + 2a + a 2 + 2a ′ ′ kBT 1 1 2 2 It is clear that when the omitted terms 2a a and Mp kBT Mpc 1 2 2 ( ) ( ) ( ) Mpc 4 6 6 ′ kBT ′ ′ kBT ′2 kBT a 2 + 2a a + a . (15) 2 2 are recovered in the second-order correction, 1 2 M c2 2 M c2 Mpc p p the remnant mass is also equal to the critical mass. However, in Banerjee-Ghosh's treatment, the( contribu-) In this section, it is found that, in Banerjee-Ghosh's( ) 4 2 ′ ′ kBT k T tions of the correction terms 2a a and a′2 B 1 2 2 work, some necessary terms, e.g., the term 1 2 ( ) Mpc ( Mpc ) k T 6 k T 4 a′2 B were both omitted [68]. When the omitted a′ a′ B 2 2 for the first-order correction and the terms 2 1 2 2 Mpc Mpc

095103-3 Chinese Physics C Vol. 44, No. 9 (2020) 095103 ( ) k T 6 and a′2 B for the second-order correction, have According to the well-known commutation relation 2 M c2 p ∂p been omitted when considering the GUPeffects on the fi- [x ˆ, pˆ(kˆ)] = i , (27) ∂k nal evolution behavior of black hole evaporation. In fact, these omitted terms become necessary because without the uncertainty relation is given by 1 ⟨∂p⟩ them, the final evolution behavior of the black hole sys- ∆x∆p ⩾ | |. (28) tem cannot truly emerge with the inclusion of the GUP 2 ∂k effects, as discussed above. When these omitted terms are From (26) and (28), we can obtain recovered in the first- and second-order quantum correc-     tions, the black hole always stops evaporation when the h¯  ⟨pˆ2⟩ 1 ⟨pˆ4⟩ 2 ⟨pˆ6⟩  ∆x∆p ⩾ 1 + + + + ···. (29) remnant mass is equal to the critical mass. In the follow- 2 2 2 3 4 4 45 6 6 M f c M f c M f c ing section, we use another GUP model with higher-or- ⟨ 2i⟩ ⩾ ⟨ 2⟩i der corrections to restudy the final evolution behavior of Here, p p has been used. For a minimal position ⟨ ⟩ = black hole evaporation, and we attempt to confirm the in- uncertainty, we have p 0, so Hossenfelder's GUP is trinsic self-consistency between the black hole remnant given by and critical mass once again.  ( ) ( ) ( )  h¯  ∆p 2 f 4 ∆p 4 2 f 6 ∆p 6  ∆x∆p ⩾ 1 + f 2 + + + ···. 2 Mpc 3 Mpc 45 Mpc 3 New generalized uncertainty principle pro- (30) posed by Hossenfelder et al. Here, M f = Mp/ f , and Mp is the Planck mass. It is noteworthy that Hossenfelder's GUP contains not only To implement the notion of a minimal length L f , the term described in string theory [70, 71], but also high- Hossenfelder et al. have assumed that particles cannot er-order quantum corrections. However, the GUP deriva- possess arbitrarily small Compton wavelengths tions are phenomenological and normally take only the (λ = 2π/k); then, the vector k has an upper bound [69]. first-order or second-order correction as the subdominant This effect would show up when p approaches a certain one; there is nothing new about higher orders here. In the scale M f . To incorporate this behavior, they have as- next section, we will use Hossenfelder's GUP to pre- sumed that the relation k(p) between p and k is an un- cisely reexamine the first- and second-order corrections even function (because of parity) and asymptotically ap- to black hole thermodynamics. proaches 1/L f . Thus, Hossenfelder et al. have assumed the function behavior of k(p) is [69] [( )γ] 4 Black hole thermodynamics with correc- 1/γ p L f k(p) = tanh , (24) tions M f c γ where is a positive constant and L f and M f satisfy the Near the horizon of a Schwarzschild black hole, when = γ = relation L f M f c h¯ . For simplicity, we set 1. Expand- the production of a particle-antiparticle pair occurs as a ing the modified relation (24), there are two cases: (a) the result of quantum fluctuation in a vacuum, the particle regime of expanding tanh(x) for small arguments (i.e., π with negative energy falls into the horizon, and that with |x| < ); and (b) the high-energy limit p ≫ M . positive energy escapes to outside the horizon and is de- 2 f For case (a), its expanding expression is given by tected by an observer at infinity. For simplicity, we con- ( ) ∑∞ 2n 2n 2n−1 sider the emitted particle to be massless and its spectrum 1 2 (2 − 1)B L f p k(p) = 2n , (25) to be thermal. Therefore, we have [58] L (2n)! h¯ f n=1 kBT = △pc, (31) where B is the Bernoulli number. The expanding expression (25) can also be found in Banerjee-Ghosh's relation where kB is the Boltzmann constant, and the momentum (1). That is to say, Hossenfelder's relation (24) between of the emitted particle p is on the order of its momentum △ the wave vector and the momentum exhibits much more uncertainty p. For thermodynamic equilibrium, the tem- physics because its expanding expression not only con- perature of the emitted particle is identified with the tem- tains the regime of Banerjee-Ghosh's relation, but also in- perature of the black hole itself. Moreover, near the horizon of a Schwarzschild black hole, the position uncer- cludes the high-energy limit p ≫ M f . According to Eq. (25), we have tainty of a particle will be on the order of the Schwarz- schild radius for the black hole [58, 59]. Consequently, ( ) ( ) ( ) 1 ∂p p 2 1 p 4 2 p 6 = 1 + + + + ··· . (26) 2GM ∂ ∆x = rs = , (32) h¯ k M f c 3 M f c 45 M f c c2

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where rs is the Schwarzschild radius, G is Newton's grav- area when considering the effects of quantum gravity [55- itational constant, c is the speed of light, and M is the 59], and the leading-order correction has a logarithmic mass of a Schwarzschild black hole. form similar to that obtained by other methods, such as We can now relate the temperature T with the mass M field theory [73], quantum geometry [74], string theory of the black hole by recasting Hossenfelder's GUP (30) in [75], and loop quantum gravity [76, 77]. terms of T and M. Hossenfelder's GUP (30) with first-or- Using Hossenfelder's GUP, we derive the corrected der and second-order corrections is given by black hole thermodynamics with the first- and second-or- [ ( ) ( ) ] 2 4 der corrections, which leads to some interesting proper- h¯ ∆p f 4 ∆p ∆x∆p = ϵ 1 + f 2 + , (33) ties that may solve certain puzzles in the semi-classical 2 M c 3 M c p p case. In the following section, we will focus on the first- where the parameter ϵ is a scale factor saturating the un- and second-order corrections to black hole thermodynam- certainty relation. It should be noted that we take only the ics without loss of generality to precisely reexamine the first-order or second-order correction as the subdominant final evolution behavior of the black hole system from one because there is nothing new about higher orders different physical perspectives. here. Substituting Eqs. (31) and (32) into Eq. (33), Hossenfelder's GUP in terms of T and M is recast as 4.1 First-order correction [( ) ( ) ( ) ] From (35), we can obtain the mass-temperature rela- 2 4 3 Mp Mpc k T f k T M = ϵ + f 2 B + B , (34) tion with the first-order correction, given by 2 2 4 kBT Mpc 3 Mpc M = 1 + 2T , 2 2 f (38) where Mp = Lpc /G and ch¯/Lp = Mpc have been used. If T = Hossenfelder's GUP effects are not considered (i.e., f 0 where we introduce the notations M = 8πM/Mp and 2 in (34)), the semi-classical mass-temperature relation is T = kBT/(Mpc ) for convenience. Thus, the temperature = 2 2/ π reproduced by T Mpc (8 kBM) [72]. We thus fix the T in terms of the mass M can be written as calibration factor ϵ = 1/2π. Therefore, the corrected mass- 2 T = √ . temperature relation with the first- and second-order cor- (39) M M2 − 4 f 2 rections is given by [( ) ( ) ( ) ] Here, only the (+) sign is acceptable, and the (−) sign is 2 4 3 Mp Mpc k T f k T physically problematic because it cannot recover the clas- M = + f 2 B + B . (35) 2 2 = 8π kBT Mpc 3 Mpc sical result if f 0. Obviously, as a thermodynamic system, the black hole has a lower limit for the mass to guar- The heat capacity of a black hole is defined as antee a meaningful range of the thermodynamic temperat- C = c2 dM , so the heat capacity with the first- and second- dT ure with the first-order correction. During the evapora- order corrections is obtained as [ ( ) ( ) ] tion process, if the black hole mass exceeds the lower 2 2 2 k Mpc k T limit, the thermodynamic temperature becomes complex, C = B − + f 2 + f 4 B . (36) 2 and the thermodynamic system cannot be well described 8π kBT Mpc here. Therefore, the lower limit of the black hole mass is = Obviously, in the semi-classical case (i.e., f 0 in (36)), usually called the critical mass [54, 58, 68], which is giv- a Schwarzschild black hole always possesses a negative en, from (39), by heat capacity, which means that a black hole is an un- f stable system that loses its mass with an increase in its M = M . (40) cr 4π p temperature during the evaporation process. When including the effects of quantum gravity, the corrected heat Next, we start from the corrected thermodynamic en- capacity (36) has some positive corrections that cause the tropy to reexamine the lower limit of the black hole mass. heat capacity to monotonically increase as the black hole By substituting (39) into (37), the thermodynamic entropy with the first-order correction is given by temperature gradually increases during the evaporation [ ] process. S A f 2 A f 2 = − ln − ln[16π], (41) 2 π 2 π According to∫ the first law of black hole thermody- kB 4Lp 16 4Lp 16 2 namics S = c dM , the black hole entropy with the first- − T where A = 16πG2 M2c 4 is the semi-classical area of the and second-order corrections is given by black hole horizon, and [ ] [ √ √ ] ( 2 )2 ( )2 ( )2 2 k Mpc k T k T A A f 2 S = B + f 2 B + f 4 B . (37) A = + − 2 , ln 2 2 Lp (42) 16π kBT Mpc Mpc 2 4 4π Interestingly, we find that the corrected entropy (37) of is the reduced area, which is introduced to produce the the black hole is no longer proportional to the horizon area theorem in tractable form, and the semi-classical

095103-5 Chinese Physics C Vol. 44, No. 9 (2020) 095103 area A is reproduced for f = 0 in (42). Obviously, the en- Then, through complicated calculation, the temperature in tropy is explicitly expressed as a function of the reduced terms of the mass can be expressed as area A rather than the actual area A. From (42), we can √ [ 1 M 1 M2 1 M2 also see that there is a lower limit of the black hole mass = + + H + − H − 2 f 2 below which the reduced area becomes a complex quant- T 4 2 4 2 2 ] 1 ( ) 1 ity, which is given by 2 − 2 1 M 2 ( ) f + + H M3 − 4 f 2M , (47) M = M . (43) 4 4 cr 4π p The critical mass is correctly reproduced again by ana- where 5 1 2 lyzing the black hole thermodynamic entropy and re- H = f 4K−1 + K − f 2, (48) duced area. 3 3 3 ( ) 1 1 By substituting (39) into (36), the heat capacity in − 4 2 6 3 K = 2 3 9 f M + F − 22 f , (49) terms of the mass M is then given by  √  √  2 2 2 2  kB  M + M M − 4 f − 2 f  = 8M4 − 10M2 − 12. C = − + f 2. (44) F 81 f 396 f 16 f (50) 8π 2 The semi-classical Hawking temperature can be well re- Clearly, there is a positive correction to the semi-classic- produced by the quantum-corrected temperature (47) at al heat capacity when including the first-order quantum f = 0 when F = K = H = 0. From (47), there is a lower correction. In the semi-classical case, the black hole with limit of the black hole mass below which the quantum- a negative heat capacity could evaporate completely in corrected temperature becomes a complex quantity, late evolution. In the presence of the first-order correc- which is determined by 81 f 8M4 − 396 f 10M2 − 16 f 12 ⩾ 0. tion, the positive correction emerges to prevent further Therefore, the lower limit of the black hole mass (i.e., the dM evaporation at zero heat capacity ( C = c2 = 0) when critical mass) is given by dT ( √ ) 1 the black hole mass no longer changes with the black 22 + 10 5 2 hole temperature [58, 65, 66, 68]. This result means that Mcr = f Mp. (51) 24π black hole evaporation stops at a finite mass when includ- Substituting (47) into (37), the entropy with the ing the first-order correction, which is called the remnant second-order correction is given by mass, given by f 2 [ ] 2 4 2 = . S A f A f f Lp Mrem Mp (45) = − ln − ln[16π] + , (52) 4π 2 π 2 π π2A kB 4Lp 16 4Lp 16 64 The remnant mass can also be obtained by minimizing dS where A is the reduced area, given by the entropy (41), = 0, and looking at the second de-  √ dM  √ [ 2  2 d S  A 1 A HL2 1 A HL rivative ( > 0). Obviously, the black hole stops evap- A = + + p + − p dM2  π π oration when the remnant mass is equal to the critical  4 2 4 4 2 2 4  mass. This reveals the intrinsic self-consistency between ] 1 ( ) 1 ( ) 2 2 2 √ 2 − 2 2 2  the black hole remnant and critical masses when includ- f Lp A HLp 2 A f Lp  − + A + −  . (53) ing the first-order correction. In addition, we can easily 2π 4 4π 4 4π  find that the thermodynamic temperature (39) and re- which is introduced to produce the area theorem in tract- duced area (42) are always positive, even at the final able form. A is the usual area of the black hole horizon, stage of black hole evaporation, and there is therefore no and singularity in the thermodynamic temperature (39) and 5 1 2 entropy (41) when including the first-order quantum cor- H = f 4K−1 + K − f 2, (54) rection. 3 3 3 ( ) 1 1 − 4 −2 6 3 = 3 π + − , 4.2 Second-order correction K 2 36 f Lp A F 22 f (55) Next, we will continue this issue by considering the √ = π2 8 −4 2 − π 10 −2 − 12. effects of the second-order correction. From (35), the F 1296 f Lp A 1584 f Lp A 16 f (56) mass-temperature relation with the second-order correc- To guarantee an effective range of the reduced area (53), tion is given by the lower limit of the black hole mass is given by 1 f 4 π2 8 −4 2 − π 10 −2 − 12 = M = + f 2T + T 3. (46) 1296 f Lp A 1584 f Lp A 16 f 0. In this case, T 3 the critical mass with the second-order correction is giv-

095103-6 Chinese Physics C Vol. 44, No. 9 (2020) 095103 en by tropy, and the reduced area go beyond its effective range ( ) √ 1 [54, 58, 69], and the remnant mass is determined by the + 2 22 10 5 zero heat capacity C = 0 or by minimizing the entropy Mcr = f Mp. (57) 24π [58, 66, 67, 69]. Although the remnant and critical The critical mass is consistently obtained by guarantee- masses are respectively determined from different physic- ing the effective ranges of the black hole thermodynamic al perspectives with respect to the final evolution behavi- temperature (47) and reduced area (53). or of the black hole system, they are equal to each other From (36), the heat capacity with the second-order when including the first- and second-order corrections. In addition, we can easily find that the thermodynamic correction can be written as( ) quantities, e.g., the thermodynamic temperature and en- kB 1 C = − + f 2 + f 4T 2 . (58) tropy, are not singular at the final stage of black hole 8π T 2 evaporation with the inclusion of the first- and second-or- There are two positive corrections in the corrected heat der corrections. capacity (58), enabling the black hole evaporation to stop at zero heat capacity easier than that with the first-order 5 Conclusions correction, where there is only one positive correction. At the final stage of black hole evaporation, when the black hole mass no longer changes with the black hole temper- In this paper, we reveal the intrinsic self-consistency dM between the remnant and critical masses during the final ature (i.e., C = c2 = 0), the remnant mass with the dT stage of black hole evaporation when including the ef- second-order correction is obtained as fects of quantum gravity. When including all the neces- ( √ ) 1 2 sary terms in the first- and second-order corrections, we 22 + 10 5 Mrem = f Mp. (59) first reexamine Banerjee-Ghosh's work and find that the 24π black hole stops evaporation when the remnant mass is Obviously, the black hole always stops evaporation when equal to the critical mass with the inclusion of the GUP the remnant mass is equal to the critical mass. This res- effects. Then, we use another GUP model with higher-or- ult reveals again the intrinsic self-consistency between der corrections proposed by Hossenfelder et al. to re- the black hole remnant and critical masses when includ- study the final evolution behavior of the black hole evap- ing the second-order correction. In addition, the thermo- oration, and we again reveal the intrinsic self-consistency dynamic temperature (47) and reduced area (53) are al- between the black hole remnant and critical masses. In ways positive, even during the final evolution of the addition, we can easily find that the thermodynamic black hole system, so there is no singularity in the ther- quantities, e.g., the thermodynamic temperature and en- modynamic temperature (47) and entropy (52) when in- tropy, are not singular at the final stage of black hole cluding the second-order correction. evaporation with the inclusion of the first- and second-or- The critical mass is the lower limit of the black hole der corrections; therefore, the singularity problem in the mass below which the thermodynamic temperature, en- semi-classical approach can be naturally bypassed here.

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