Generalized Uncertainty Principle and Black Hole Thermodynamics

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Jin Pu, Qin-Bin Mao, Qing-Quan Jiang, Jing-Xia Yu and Xiao-Tao Zu. Generalized Uncertainty Principle and Black Hole Thermodynamics[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/9/095103
Jin Pu, Qin-Bin Mao, Qing-Quan Jiang, Jing-Xia Yu and Xiao-Tao Zu. Generalized Uncertainty Principle and Black Hole Thermodynamics[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/9/095103 shu
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Generalized Uncertainty Principle and Black Hole Thermodynamics

  • 1. School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China
  • 2. College of Physics and Space Science, China West Normal University, Nanchong 637002, China
  • 3. Institute 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 the black hole evaporation stops at the remnant mass greater than the critical mass when including the GUP effects with the first- and second-order corrections. In this paper, we first follow their steps to reexamine the Banerjee-Ghosh's work, but find an interesting result that the remnant mass is always equal to the critical mass at the final stage of the black hole evaporation with the inclusion of the GUP effects. Then, we use the Hossenfelder's GUP, i.e. another GUP model with higher-order corrections, to restudy the final evolution behavior of the black hole evaporation, and confirm the intrinsic self-consistency between the black hole remnant and critical mass once more. In both cases, we also find that the thermodynamic quantities are not singular at the final stage of the black hole evaporation.

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    1.   Introduction
    • The generalized uncertainty principle (GUP) that modifies the uncertainty principle to include a minimal length has been received more and more attentions in the past decade [1-4]. On the one hand, one may predict the GUP effects through various experiments, such as hydrogen Lamb shift [5-7], electron tunneling [5, 6], mechanical oscillator [8, 9], gravitational bar detectors [10], ultra-cold atoms experiments [11, 12], gravitational wave experiments [13, 14], Sub-Kilogram Acoustic Resonators [15], large molecular wave-packets [16]. On the other hand, one can also apply the GUP to study the effects of quantum gravity on small- or large-scale physical systems. For example, the GUP effects have been studied on the early Universe [17-21], compact stellar objects [22-24], Newtonian law of gravity [25], equivalence principle [26-28], entropic nature of gravitational force [29-33], Casimir effect [34, 35], Dirac $ \delta $-function potential [36], post-Newtonian potential [37].

      As far as it is concerned by us, the GUP affects the well-known semiclassical laws of the black hole thermodynamics [38-68]. For example, the black hole entropy is no longer proportional to the horizon area [51-59], the black hole does not evaporate completely but leaves a remnant mass at the final stage of evaporation [57-65], the remnant with Planck scale can store information which gives a possible solution to the singularity problem [57-66], and there is a metastable remnant that asymptotes to zero mass when considering the negative GUP correction [67]. In [69], Banerjee and Ghosh intriguingly constructed a GUP, which contains the term predicted by string theory and a series of higher-order correction terms, and studied the GUP effects on the black hole thermodynamics. It is shown that, when considering the first- and second-order quantum corrections to the black hole thermodynamics, the black hole evaporation always stops at the remnant mass greater than the critical mass, and the singularity problem in the semiclassical approach is bypassed at the final stage of the black hole evaporation.

      However, the Banerjee-Ghosh's results in [69] are lack of creditability due to the fact that some necessary terms have been omitted in their treatments. For example, when dealing with the first-order correction, all terms about $ a'_1 $ should be included in the corrected Temperature-Mass relation. However, the term $ a'^2_1(k_BT/M_pc^2)^2 $ has been omitted in the their treatment, maybe they think this term is much smaller. On the other hand, when dealing with the second-order correction, Banerjee and Ghosh have omitted the necessary terms $ 2a'_1a'_2(k_BT/M_pc^2)^4 $ and $ a'^2_2(k_BT/M_pc^2)^6 $. If these omitted terms are recovered, the final evolution behavior of the black hole evaporation may evolve differently. Anyway, the Banerjee-Ghosh's work can not truly emerge the final evolution behavior of the black hole system with the inclusion of the GUP effects.

      In this paper, we first reexamine the Banerjee-Ghosh's work in Sec. 2, and restudy the final evolution behavior of the black hole evaporation when including the GUP effects with the first- and second-order corrections. In Sec. 3, we review the GUP proposed by Hossenfelder et al. in [69], i.e. another GUP model with higher-order corrections. In Sec. 4, we use the Hossenfelder's GUP to precisely study the first- and second-order quantum corrections to the black hole thermodynamics, and aim to discover the intrinsic self-consistency between the black hole remnant and critical mass when including the effects of quantum gravity. Sec. 5 ends up with some conclusions.

    2.   The reexamination of the Banerjee-Ghosh's work
    • In [69], Banerjee and Ghosh have assumed that the function relation between the wave vector k and the momentum p satisfies certain properties: $ 1) $ the function have to be an odd function to preserve parity; $ 2) $ the function should be chosen to satisfy $ p = \hbar k $ at small energy; $ 3) $ the wave vector k should be an upper bound $ 2\pi/L_p $. Thus, Banerjee and Ghosh have assumed an infinite order polynomial to satisfy these properties of the function, which is expressed as

      $ k = f(p) = \frac{1}{L_p}\sum\limits_{i = 0}^\infty{a_i(-1)^i\left(\frac{L_pp}{\hbar}\right)^{2i+1}}. $

      (1)

      Here, only odd powers of the momentum p appear in the polynomial because the function $ f(p) $ is an odd function to preserve parity. The coefficients $ \{a_i\} $ are all positive, and we have $ a_0 = 1 $ to recover $ p = \hbar k $ at small energy. The factor $ (-1)^i $ ensures the property $ 3) $, and we have a constraint for $ p\rightarrow\infty $, $ k\rightarrow\frac{2\pi}{L_p} $, i.e.

      $ \sum\limits_{i = 0}^\infty{a_i(-1)^i\left(\frac{L_pp}{\hbar}\right)^{2i+1}}\rightarrow 2\pi. $

      (2)

      From (1), we obtain

      $ \frac{\partial p}{\partial k} = \hbar\sum\limits_{i = 0}^\infty{a'_i\left(\frac{L_pp}{\hbar}\right)^{2i}}, $

      (3)

      where the new coefficients of expansions $ \{a'_i\} $ are functions of $ \{a_i\} $, and $ a'_0 = 1 $. Hence the form of the generalized uncertainty principle proposed by Banerjee and Ghosh is given by

      $ \Delta x\Delta p\geqslant\frac{\hbar}{2}\sum\limits_{i = 0}^\infty{a'_i\left(\frac{L_p\Delta p}{\hbar}\right)^{2i}}, $

      (4)

      where the coefficients $ \{a'_i\} $ are all positive.

      Subsequently, we use the GUP (4) to study the quantum-corrected thermodynamic entities of a Schwarzschild black hole and attempt to find relations among them. In [69], by comparing with the standard semiclassical Hawking temperature, the mass-temperature relationship of the Schwarzschild black hole is given by

      $ M = \frac{M_p}{8\pi}\sum\limits_{i = 0}^{\infty}a'_i\left(\frac{k_BT}{M_pc^2}\right)^{2i-1}. $

      (5)

      According to the definition of the heat capacity of the black hole $ C = c^2\dfrac{dM}{dT} $, we have

      $ C = \frac{k_B}{8\pi}\sum\limits_{i = 0}^{\infty}a'_i(2i-1)\left(\frac{k_BT}{M_pc^2}\right)^{2i-2}. $

      (6)

      And the entropy of the black hole is given by

      $ \begin{split} S =& \int\frac{CdT}{T} = \frac{k_B}{16\pi}\left[\left(\frac{M_pc^2}{k_BT}\right)^2+a'_1\ln\left(\frac{k_BT}{M_pc^2}\right)^2\right.\\ & \left.+\sum\limits_{i = 2}^{\infty}a'_i\frac{(2i-1)}{(i-1)}\left(\frac{k_BT}{M_pc^2}\right)^{2(i-1)}\right]. \end{split} $

      (7)

      In [69], due to express the heat capacity and the entropy in terms of the mass, the expression for $ T^2 $ in terms of M is given by

      $ \begin{split} \left(\frac{8\pi M}{M_p}\right)^2 =& \left(\frac{M_pc^2}{k_BT}\right)^2+2a'_1+(a'^2_1+2a'_2)\left(\frac{k_BT}{M_pc^2}\right)^2 \\ &\!+\!2(a'_1a'_2\!+\!a'_3)\left(\frac{k_BT}{M_pc^2}\right)^4\!\\&+\!(2a'_1a'_3\!+\!a'^2_2)\left(\frac{k_BT}{M_pc^2}\right)^6\!+\!\cdots \end{split} $

      (8)

      In the Banerjee-Ghosh’s treatment, when dealing with the first-order correction, they have obtained the corrected Mass-Temperature relation as

      $ \left(\frac{8\pi M}{M_p}\right)^2 = \left(\frac{M_pc^2}{k_BT}\right)^2+2a'_1, $

      (9)

      In fact, all terms about $ a'_1 $ should be included in the first-order correction, so the corrected Mass-Temperature relation should be written as

      $ \left(\frac{8\pi M}{M_p}\right)^2 = \left(\frac{M_pc^2}{k_BT}\right)^2+2a'_1+a'^2_1\left(\frac{k_BT}{M_pc^2}\right)^2. $

      (10)

      So, the term $ a'^2_1\left(\dfrac{k_BT}{M_pc^2}\right)^2 $ has been omitted in the Banerjee-Ghosh’s treatment. Basing on Eq.(9), Banerjee and Ghosh further obtained the result that the remnant mass is greater than the critical mass when considering the first-order correction.

      Basing on Eq.(10) where the omitted term is recovered in the first-order correction, we can obtain

      $ \left(\frac{k_BT}{M_pc^2}\right)^2 = \frac{2}{\left(\dfrac{8\pi M}{M_p}\right)^2-2a'_1+\left(\dfrac{8\pi M}{M_p}\right)\sqrt{\left(\dfrac{8\pi M}{M_p}\right)^2-4a'_1}}. $

      (11)

      Obviously, as a thermodynamic system, the black hole has a critical mass below (at) which the thermodynamic entities become complex (ill defined) [54, 69], and which is given by

      $ {M}_{cr} = \frac{\sqrt{a'_1}}{4\pi}M_p. $

      (12)

      From (6) and (11), the heat capacity with the first-order correction is given by

      $ C = \dfrac{k_B}{8\pi}\left[-\frac{\left(\dfrac{8\pi M}{M_p}\right)^2-2a'_1+\left(\dfrac{8\pi M}{M_p}\right)\sqrt{\left(\dfrac{8\pi M}{M_p}\right)^2-4a'_1}}{2}+a'_1\right]. $

      (13)

      When the heat capacity becomes zero at the final stage of the black hole evaporation [58, 66, 67], the remnant mass is obtained

      $ {M}_{rem} = \frac{\sqrt{a'_1}}{4\pi}M_p. $

      (14)

      Thus, the remnant mass is equal to the critical mass when including the necessary term $ a'^2_1\left(\dfrac{k_BT}{M_pc^2}\right)^2 $ for the first-order correction term.

      Next, let's focus on the effects of the second-order correction. The Mass-Temperature relation should be here given, according to (8), by

      $ \begin{split} \left(\frac{8\pi M}{M_p}\right)^2 =& \left(\frac{M_pc^2}{k_BT}\right)^2+2a'_1+\big(a'^2_1+2a'_2\big)\left(\frac{k_BT}{M_pc^2}\right)^2 \\& +2a'_1a'_2\left(\frac{k_BT}{M_pc^2}\right)^4+a'^2_2\left(\frac{k_BT}{M_pc^2}\right)^6. \end{split} $

      (15)

      However, in the Banerjee-Ghosh's treatment, the contribution of the correction terms $ 2a'_1a'_2\left(\dfrac{k_BT}{M_pc^2}\right)^4 $ and $ a'^2_2\left(\dfrac{k_BT}{M_pc^2}\right)^6 $ have been both omitted in [69]. When the omitted terms are both recovered in the second-order correction, we have

      $ \begin{split} \frac{k_BT}{M_pc^2} =& \frac{1}{2}\sqrt{-\frac{2a'_1}{3a'_2}+B+D}\\ & -\frac{1}{2}\sqrt{-\frac{4a'_1}{3a'_2}-B-D+\frac{2\times(8\pi M/M_p)}{a'_2\sqrt{-2a'_1/3a'_2+B+D}}}, \end{split} $

      (16)

      where

      $ B = \frac{2^{1/3}(a'^2_1+12a'_2)}{3a'_2\left(F+\sqrt{F^2-G}\right)^{1/3}}, $

      (17)

      $D = \frac{\left(F+\sqrt{F^2-G}\right)^{1/3}}{3\times 2^{1/3}a'_2}, $

      (18)

      $ F = 2a'^3_1-72a'_1a'_2+27a'_2(8\pi M/M_p)^2, $

      (19)

      $ G = 4(a'^2_1+12a'_2)^3,$

      (20)

      From Eq. (16), the critical mass below (at) which the thermodynamic entities become complex (ill defined) can be determined by $ F^2-G\geqslant 0 $, that is

      $ \left(\frac{8\pi M_{cr}}{M_p}\right) = \frac{1}{3}\sqrt{\frac{2}{3}}\sqrt{36a'_1-\frac{a'^3_1}{a'_2}+\sqrt{(a'^2_1+12a'_2)^3}}. $

      (21)

      From (6), the heat capacity with the second-order correction can now be written as

      $ C = \frac{k_B}{8\pi}\left[-\left(\frac{M_pc^2}{k_BT}\right)^2+a'_1+3a'_2\left(\frac{k_BT}{M_pc^2}\right)^2\right]. $

      (22)

      Then, at the final stage of the black hole evaporation, the remnant mass with the second-order correction is given by

      $ \begin{split} \left(\frac{8\pi M_{rem}}{M_p}\right) =& \frac{1}{3}\sqrt{\frac{2}{3}}\sqrt{\frac{-a'_1+\sqrt{(a'^2_1+12a'_2)^3}}{a'_2}}\\& \times\Big(2a'_1+\sqrt{(a'^2_1+12a'_2)^3}\Big) \\ =& \frac{1}{3}\sqrt{\frac{2}{3}}\sqrt{36a'_1-\frac{a'^3_1}{a'_2}+\sqrt{(a'^2_1+12a'_2)^3}}. \end{split} $

      (23)

      It is obvious that when the omitted terms $ 2a'_1a'_2\left(\dfrac{k_BT}{M_pc^2}\right)^4 $ and $ a'^2_2\left(\dfrac{k_BT}{M_pc^2}\right)^6 $ are recovered in the second-order correction, the remnant mass is also equal to the critical mass.

      In this section, it is found that, in the Banerjee-Ghosh's work, some necessary terms, e.g. the term $ a'^2_1\left(\dfrac{k_BT}{M_pc^2}\right)^2 $ for the first-order correction, and the terms $ 2a'_1a'_2\left(\dfrac{k_BT}{M_pc^2}\right)^4 $ and $ a'^2_2\left(\dfrac{k_BT}{M_pc^2}\right)^6 $ for the second-order correction, have been omitted when considering the GUP effects on the final evolution behavior of the black hole evaporation. In fact, these omitted terms become necessary because lacking of them can not truly emerge the final evolution behavior of the black hole system with the inclusion of the GUP effects as discussed above. When these omitted terms are recovered in the first- and second-order quantum corrections, the black hole always stops evaporation at the remnant mass equal to the critical mass. In the following section, we use another GUP model with higher-order corrections to restudy the final evolution behavior of the black hole evaporation, and attempt to confirm the intrinsic self-consistency between the black hole remnant and critical mass once more.

    3.   The new generalized uncertainty principle by Hossenfelder et al.
    • In order to implement the notion of a minimal length $ L_f $, Hossenfelder et al. have assumed that particles can not possess arbitrarily small Compton wavelengths ($ \lambda = 2\pi/k $), and then the vector k has an upper bound [70]. This effect would show up when p approaches a certain scale $ M_f $. To incorporate this behavior, they have assumed a relation $ k(p) $ between p and k is an uneven function (because of parity) and which asymptotically approaches $ 1/L_f $. Thus, Hossenfelder et al. have assumed the function behaviour of $ k(p) $ is [70]

      $ L_fk(p) = \tanh^{1/\gamma}\left[\left(\frac{p}{M_fc}\right)^\gamma\right], $

      (24)

      where $ \gamma $ is a positive constant, $ L_f $ and $ M_f $ satisfy the relation $ L_fM_fc = \hbar $. For simplicity, we set $ \gamma = 1 $. Expanding the modified relation (24) has two cases: (a)the regime of expanding $ \tanh(x) $ for small arguments (i.e. $ |x|<\dfrac{\pi}{2} $); (b) the high energy limit $ p\gg M_f $.

      So, for case (a), its expanding expression is given by

      $ k(p) = \frac{1}{L_f}\sum\limits_{n = 1}^{\infty}\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\left(\frac{L_fp}{\hbar}\right)^{2n-1}, $

      (25)

      where B is the Bernoulli number. The expanding expression (25) can also be found in the Banerjee-Ghosh's relation (1). This is to say, the Hossenfelder's relation (24) between the wave vector and the momentum exhibits much more physics, because its expanding expression not only contains the regime of the Banerjee-Ghosh's relation, but also includes the high energy limit $ p\gg M_f $.

      According to Eq.(25), we have

      $ \frac{1}{\hbar}\frac{\partial p}{\partial k} = 1+\Big(\frac{p}{M_fc}\Big)^2+\frac{1}{3}\Big(\frac{p}{M_fc}\Big)^4+\frac{2}{45}\Big(\frac{p}{M_fc}\Big)^6+\cdots. $

      (26)

      According to the well-known commutation relation

      $ [\hat{x},\hat{p}(\hat{k})] = i\frac{\partial p}{\partial k}, $

      (27)

      the uncertainty relation is given by

      $ \Delta x\Delta p\geqslant \frac{1}{2}|\Big\langle\frac{\partial p}{\partial k}\Big\rangle|. $

      (28)

      From (26) and (28), we can obtain

      $ \Delta x\Delta p\geqslant\frac{\hbar}{2}\left(1+\frac{\langle\hat{p}^2\rangle}{M^2_fc^2}+\frac{1}{3}\frac{\langle\hat{p}^4\rangle}{M^4_fc^4}+\frac{2}{45}\frac{\langle\hat{p}^6\rangle}{M^6_fc^6}+\cdots\right). $

      (29)

      Here, $ \langle p^{2i}\rangle\geqslant\langle p^2\rangle^i $ has been used. For a minimal position uncertainty, we have $ \langle p\rangle = 0 $, so the Hossenfelder's GUP is given by

      $ \Delta x\Delta p\geqslant\frac{\hbar}{2}\left[1+f^2\left(\frac{\Delta p}{M_pc}\right)^2+\frac{f^4}{3}\left(\frac{\Delta p}{M_pc}\right)^4+\frac{2f^6}{45}\left(\frac{\Delta p}{M_pc}\right)^6+\cdots\right]. $

      (30)

      Here, $ M_f = M_p/f $, $ M_p $ is the Planck mass. It is noteworthy that the Hossenfelder's GUP contains not only the term as described in string theory [71, 72], but also higher-order quantum corrections. However, the GUP derivations are phenomenological and normally the take only the first-order or second-order correction as the subdominant one, there is nothing new about higher-orders here. In the next section, we will use the Hossenfelder's GUP to precisely reexamine the first- and second-order correction to the black hole thermodynamics.

    4.   The black hole thermodynamics with corrections
    • Near the horizon of the Schwarzschild black hole, when the production of a particle-antiparticle pair occurs due to quantum fluctuation in vacuum, the particle with negative energy falls into the horizon and that with positive energy escapes outside the horizon and is detected by observer at infinity. For simplicity, we consider the emitted particle is massless and its spectrum is thermal. So, we have [58]

      $ k_BT = \triangle pc, $

      (31)

      where $ k_B $ is the Boltzmann constant, and the momentum of the emitted particle $ p $ is of the order of its momentum uncertainty $ \triangle p $. For thermodynamic equilibrium, the temperature of the emitted particle is identified with the temperature of the black hole itself. Also, near the horizon of the Schwarzschild black hole, the position uncertainty of a particle will be of the order of the Schwarzschild radius for the black hole [58, 59]. Consequently,

      $ \Delta x = r_s = \frac{2GM}{c^2}, $

      (32)

      where $ r_s $ is the Schwarzschild radius, G is the Newton's gravitational constant, c is the speed of light and M is the mass of the Schwarzschild black hole.

      We can now relate the temperature T with the mass M of the black hole by recasting the Hossenfelder's GUP (30) in terms of T and M. The Hossenfelder's GUP (30) with the first-order and second-order corrections is given by

      $ \Delta x\Delta p = \epsilon\frac{\hbar}{2}\Bigg[1+f^2\left(\frac{\Delta p}{M_pc}\right)^2+\frac{f^4}{3}\left(\frac{\Delta p}{M_pc}\right)^4\Bigg], $

      (33)

      where the parameter $ \epsilon $ is a scale factor saturating the uncertainty relation. It should be noted that we take only the first-order or second-order correction as the subdominant one, because there is nothing new about higher-orders here. Substituting Eqs.(31) and (32) into Eq.(33), the Hossenfelder's GUP in terms of T and M is recast as

      $ M = \epsilon\frac{M_p}{4}\Bigg[\left(\frac{M_pc^2}{k_BT}\right)+f^2\left(\frac{k_BT}{M_pc^2}\right)+\frac{f^4}{3}\left(\frac{k_BT}{M_pc^2}\right)^3\Bigg], $

      (34)

      where $ M_p = L_pc^2/G $, and $ c\hbar/L_p = M_pc^2 $ have been used. If the Hossenfelder's GUP effects are not considered(i.e. $ f = 0 $ in (34)), the semiclassical mass-temperature relation could be reproduced by $ T = M^2_pc^2/(8\pi k_BM) $ [73]. We thus fix the calibration factor $ \epsilon = 1/2\pi $. Therefore, the corrected mass-temperature relation with the first- and second-order correction is given by

      $ M = \frac{M_p}{8\pi}\Bigg[\left(\frac{M_pc^2}{k_BT}\right)+f^2\left(\frac{k_BT}{M_pc^2}\right)+\frac{f^4}{3}\left(\frac{k_BT}{M_pc^2}\right)^3\Bigg]. $

      (35)

      The heat capacity of the black hole is defined as $ C = c^2\frac{dM}{dT} $, and then the heat capacity with the first- and second-order correction is obtained as

      $ C = \frac{k_B}{8\pi}\Bigg[-\left(\frac{M_pc^2}{k_BT}\right)^2+f^2+f^4\left(\frac{k_BT}{M_pc^2}\right)^2\Bigg]. $

      (36)

      Obviously, in the semiclassical case(i.e. $ f = 0 $ in (36)), the Schwarzschild black hole always possesses a negative heat capacity, which means the black hole is an unstable system that loses its mass with an increase of its temperature during the evaporation process. When including the effects of quantum gravity, the corrected heat capacity (36) has some positive corrections which cause the heat capacity to monotonically increase as the black hole temperature is gradually increased during the evaporation process.

      According to the first law of the black hole thermodynamics $ S = \int\frac{c^2dM}{T} $, the black hole entropy with the first- and second-order correction is given by

      $ S = \frac{k_B}{16\pi}\Bigg[\bigg(\frac{M_pc^2}{k_BT}\bigg)^2+f^2\ln{\bigg(\frac{k_BT}{M_pc^2}\bigg)^2}+f^4\bigg(\frac{k_BT}{M_pc^2}\bigg)^2\Bigg]. $

      (37)

      We interestingly find that the corrected entropy (37) of the black hole is no longer proportional to the horizon area when considering the effects of quantum gravity [55-59], and the leading-order correction has a logarithmic form similar to that obtained by other methods such as field theory [74], quantum geometry [75], string theory [76], loop quantum gravity [77, 78].

      By using the Hossenfelder's GUP, we derive the corrected black hole thermodynamics with the first- and second-order corrections, which emerge some interesting properties that may solve the puzzles in the semiclassical case. In the following, we will focus on the first- and second-order corrections to the black hole thermodynamics without loss of generality to precisely reexamine the final evolution behavior of the black hole system from different physical perspectives.

    • 4.1.   First-order correction

    • From (35), we obtain the mass-temperature relation with the first-order correction, which is given by

      $ {\cal M} = \frac{1}{{\cal T}}+f^2{\cal T}, $

      (38)

      where we introduce the notations as $ {\cal M} = 8\pi M/M_p $ and $ {\cal T} = k_BT/(M_pc^2) $ for convenience. Thus, the temperature $ {\cal T} $ in terms of the mass $ {\cal M} $ can be written as

      $ {\cal T} = \frac{2}{{\cal M}\pm\sqrt{{\cal M}^2-4f^2}}. $

      (39)

      Here, only the ($ + $) sign is acceptable, and the ($ - $) sign is physically problematic because it can not recover the classical result if we put $ f = 0 $. Obviously, as a thermodynamic system, the black hole has a lower limit of the mass to guarantee the meaningful range of the thermodynamic temperature with the first-order correction. During the evaporation process, if the black hole mass exceeds the lower limit, the thermodynamic temperature becomes complex, and the thermodynamic system can not be well described here. Therefore, the lower limit of the black hole mass is usually called as the critical mass [54, 58, 69], which is given, from (39), by

      $ {M}_{cr} = \frac{f}{4\pi}M_p. $

      (40)

      Next, we start from the corrected thermodynamic entropy to reexamine the lower limit of the black hole mass. By substituting (39) into (37), the thermodynamic entropy with the first-order correction is given by

      $ \frac{S}{k_B} = \frac{{\cal A}}{4L^2_p}-\frac{f^2}{16\pi}\ln{\bigg[\frac{{\cal A}}{4L^2_p}\bigg]}-\frac{f^2}{16\pi}\ln{[16\pi]}, $

      (41)

      where $ A = 16\pi G^2M^2c^{-4} $ is the semiclassical area of the black hole horizon, and

      $ {\cal A} = \Bigg[\frac{\sqrt{A}}{2}+\sqrt{\frac{A}{4}-\frac{f^2}{4\pi}L^2_p}\Bigg]^2, $

      (42)

      is the reduced area which is introduced to emerge the area theorem as a tractable form, and the semiclassical area A is reproduced for $ f = 0 $ in (42). Obviously, the entropy is explicitly expressed as a function of the reduced area $ {\cal A} $ rather than the actual area A. From (42), we can also see there is a lower limit of the black hole mass below which the reduced area becomes a complex quantity, which is given by

      $ {M}_{cr} = \frac{f}{4\pi}M_p. $

      (43)

      The critical mass is correctly reproduced again by analysing the black hole thermodynamic entropy and reduced area.

      By substituting (39) into (36), the heat capacity in terms of the mass $ {\cal M} $ is then given by

      $ C = \frac{k_B}{8\pi}\left[-\frac{{{\cal M}}^2+{\cal M}\sqrt{{{\cal M}}^2-4f^2}-2f^2}{2}+f^2\right]. $

      (44)

      Obviously, there is a positive correction to the semiclassical heat capacity when including the first-order quantum correction. In the semiclassical case, the black hole with a negative heat capacity could evaporate completely in late evolution. In the presence of the first-order correction, the positive correction is emerged to prevent further evaporation at zero heat capacity ($ C = c^2\dfrac{dM}{dT} = 0 $) when the black hole mass no longer changes with the black hole temperature [58, 66, 67, 69]. It means that the black hole evaporation stops at a finite mass when including the first-order correction, which is called the remnant mass given by

      $ {M}_{rem} = \frac{f}{4\pi}M_p. $

      (45)

      The remnant mass can also be obtained by minimising the entropy (41), $ \dfrac{dS}{dM} = 0 $, and looking at the second derivative ($ \dfrac{d^2S}{dM^2}>0 $). Obviously, the black hole stops evaporation at the remnant mass equal to the critical mass. It reveals the intrinsic self-consistency between the black hole remnant and critical mass when including the first-order correction. In addition, we can easily find that the thermodynamic temperature (39) and reduced area (42) are always positive even at the final stage of the black hole evaporation, and therefore there is no singularity in the thermodynamic temperature (39) and entropy (41) when including the first-order quantum correction.

    • 4.2.   Second-order correction

    • Next, we will continue this issue by considering the effects of the second-order correction. From (35), the mass-temperature relation with the second-order correction is given by

      $ {\cal M} = \frac{1}{{\cal T}}+f^2{\cal T}+\frac{f^4}{3}{\cal T}^3. $

      (46)

      Then, through complicated calculation, the temperature in terms of the mass can be expressed as

      $ \begin{split} \frac{1}{{\cal T}} =& \frac{{\cal M}}{4}+\frac{1}{2}\sqrt{\frac{{\cal M}^2}{4}+H}+\frac{1}{2}\left[\frac{{\cal M}^2}{2}-H-2f^2\right. \\ & +\left.\frac{1}{4}\big(\frac{{\cal M}^2}{4}+H\big)^{-\frac{1}{2}}\big({\cal M}^3-4f^2{\cal M}\big)\right]^{\frac{1}{2}}, \end{split} $

      (47)

      where

      $ H = \frac{5}{3}f^4K^{-1}+\frac{1}{3}K-\frac{2}{3}f^2, $

      (48)

      $ K = 2^{-\frac{1}{3}}\Big(9f^4{\cal M}^2+F-22f^6\Big)^{\frac{1}{3}}, $

      (49)

      $ F = \sqrt{81f^8{\cal M}^4-396f^{10}{\cal M}^{2}-16f^{12}}. $

      (50)

      The semiclassical Hawking temperature can be well reproduced by the quantum-corrected temperature (47) at $ f = 0 $ when $ F = K = H = 0 $. From (47), there is a lower limit of the black hole mass below which the quantum-corrected temperature becomes a complex quantity, which is determined by $ 81f^8{\cal M}^4-396f^{10}{\cal M}^{2}-16f^{12}\geqslant 0 $. So, the lower limit of the black hole mass(i.e. the critical mass) is given by

      $ M_{cr} = \frac{\left(22+10\sqrt{5}\right)^{\frac{1}{2}}}{24\pi}fM_p. $

      (51)

      Substituting (47) into (37), the entropy with the second-order correction is given by

      $ \frac{S}{k_B} = \frac{{\cal A}}{4L^2_p}-\frac{f^2}{16\pi}\ln\bigg[\frac{{\cal A}}{4L^2_p}\bigg]-\frac{f^2}{16\pi}\ln[16\pi]+\frac{f^4L^2_p}{64\pi^2{\cal A}}, $

      (52)

      where $ {\cal A} $ is the reduced area given by

      $ \begin{split} {\cal A} =& \left\{\frac{\sqrt{A}}{4}+\frac{1}{2}\sqrt{\frac{A}{4}+\frac{HL^2_p}{4\pi}}+\frac{1}{2}\Bigg[\frac{A}{2}-\frac{HL^2_p}{4\pi}\right. \\ &-\left.\frac{f^2L^2_p}{2\pi}+\sqrt{A}\bigg(\frac{A}{4}+\frac{HL^2_p}{4\pi}\bigg)^{-\frac{1}{2}}\bigg(\frac{A}{4}-\frac{f^2L^2_p}{4\pi}\bigg)\Bigg]^{\frac{1}{2}}\right\}^2. \end{split} $

      (53)

      which is introduced to emerge the area theorem as a tractable form, and A is the usual area of the black hole horizon, and

      $ H = \frac{5}{3}f^4K^{-1}+\frac{1}{3}K-\frac{2}{3}f^2, $

      (54)

      $ K = 2^{-\frac{1}{3}}\Big(36\pi f^4L^{-2}_pA+F-22f^6\Big)^{\frac{1}{3}}, $

      (55)

      $ F = \sqrt{1296\pi^2f^8L^{-4}_pA^2-1584\pi f^{10}L^{-2}_pA-16f^{12}}. $

      (56)

      To guarantee an effective range of the reduced area (53), the lower limit of the black hole mass is given by $ 1296\pi^2f^8 $ $ L^{-4}_pA^2-1584\pi f^{10}L^{-2}_pA-16f^{12} = 0 $. In this case, the critical mass with the second-order correction is given by

      $ M_{cr} = \frac{\left(22+10\sqrt{5}\right)^{\frac{1}{2}}}{24\pi}fM_p. $

      (57)

      The critical mass is consistently obtained by guarantying the effective ranges of the black hole thermodynamic temperature (47) and reduced area (53).

      From (36), the heat capacity with the second-order correction is written as

      $ C = \frac{k_B}{8\pi}\left(-\frac{1}{{\cal T}^2}+f^2+f^4{\cal T}^2\right). $

      (58)

      There are two positive corrections in the corrected heat capacity (58), enabling the black hole evaporation to stop at zero heat capacity easier than that with the first-order correction where there is only one positive correction. At the final stage of the black hole evaporation when the black hole mass no longer changes with the black hole temperature (i.e. $ C = c^2\dfrac{dM}{dT} = 0 $), the remnant mass with the second-order correction is obtained as

      $ M_{rem} = \frac{\left(22+10\sqrt{5}\right)^{\frac{1}{2}}}{24\pi}fM_p. $

      (59)

      Obviously, the black hole always stops evaporation at the remnant mass equal to the critical mass. It reveals again the intrinsic self-consistency between the black hole remnant and critical mass when including the second-order correction. In addition, the thermodynamic temperature (47) and reduced area (53) are always positive even at the final evolution behavior of the black hole system, so there is no singularity in the thermodynamic temperature (47) and entropy (52) when including the second-order correction.

      Anyway, the critical mass is the lower limit of the black hole mass below which the thermodynamic temperature, entropy and the reduced area go beyond its effective range [54, 58, 69], and the remnant mass is determined by the zero heat capacity $ C = 0 $ or minimising the entropy [58, 66, 67, 69]. Although the remnant and critical mass are respectively determined from different physical perspectives for the final evolution behavior of the black hole system, they are equal to each other when including the first- and second-order corrections. In addition, we can easily find that the thermodynamic quantities, e.g. the thermodynamic temperature and entropy, are not singular at the final stage of the black hole evaporation with the inclusion of the first- and second-order corrections.

    5.   Conclusions
    • In this paper, we reveal the intrinsic self-consistency between the remnant and critical mass at the final stage of the black hole evaporation when including the effects of quantum gravity. When including all the necessary terms in the first- and second-order corrections, we first reexamine the Banerjee-Ghosh's work, and find that the black hole stops evaporation at a remnant mass equal to the critical mass with the inclusion of the GUP effects. Then, we use another GUP model with higher-order corrections proposed by Hossenfelder et al. to restudy the final evolution behavior of the black hole evaporation, and again reveal the intrinsic self-consistency between the black hole remnant and critical mass. In addition, we can easily find that the thermodynamic quantities, e.g. the thermodynamic temperature and entropy, are not singular at the final stage of the black hole evaporation with the inclusion of the first- and second-order corrections, and therefore the singularity problem in the semiclassical approach may be naturally bypassed here.

Reference (80)

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