# Black Hole Thermodynamics and Generalized Uncertainty Principle with Higher Order Terms in Momentum Uncertainty

Authors: Sunandan Gangopadhyay; Abhijit Dutta
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We study the modification of thermodynamic properties of Schwarzschild and Reissner-Nordström black hole in the framework of generalized uncertainty principle with correction terms up to fourth order in momentum uncertainty. The mass-temperature relation and the heat capacity for these black holes have been investigated. These have been used to obtain the critical and remnant masses. The entropy expression using this generalized uncertainty principle reveals the area law up to leading order logarithmic corrections and subleading corrections of the form1/An. The mass output and radiation rate using Stefan-Boltzmann law have been computed which show deviations from the standard case and the case with the simplest form for the generalized uncertainty principle.


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## Body

## 1. Introduction

The consistent unification of quantum mechanics (QM) with general relativity (GR) is one of the major tasks in theoretical physics. GR deals with the definition of world-lines of particles, which is in contradiction with QM since it does not allow the notion of trajectory due to the presence of an uncertainty in the determination of the momentum and position of a quantum particle. It has been the aim to unify these two theories into one theory known as quantum gravity. It is quite interesting that all approaches towards quantum gravity such as black hole physics [1–3], string theory [4, 5], or evenGedanken experiment [6] predict the existence of a minimum measurable length. The occurrence of such a minimal length also arises in various theories of quantum gravity phenomenology, namely, the generalized uncertainty principle (GUP) [5, 7, 8], modified dispersion relation (MDR) [9–12], and deformed special relativity (DSR) [13], to name a few. It is now widely accepted that Heisenberg uncertainty principle would involve corrections from gravity at energies close to the Planck scale. Thus, emergence of a minimal length seems to be inevitable when gravitational effects are taken into account. There has been a lot of work incorporating the existence of a minimal length scale in condensed matter and atomic physics experiments such as Lamb Shift, Landau levels and the scanning Tunneling Microscope [14–19], loop quantum gravity [20–22], noncommutative geometry [23], computing Planck scale corrections to the phenomena of superconductivity and quantum Hall effect [24], and understanding its consequences in cosmology [25, 26].The incorporation of the GUP to study black hole thermodynamics has been another interesting area of active research [27–38]. It has been observed that the GUP reveals a self-complete characteristic of gravity which basically amounts to hiding any curvature singularity behind an event horizon as a consequence of matter compression at the Planck scale [39–41]. Further, the effects of the GUP have also been considered in the tunneling formalism for Hawking radiation to evaluate the quantum-corrected Hawking temperature and entropy of a Schwarzschild black hole [42–46]. In our earlier findings [32–36], we have studied the modification of thermodynamic properties, namely, the temperature, heat capacity, and entropy of black holes due to the simplest form of the GUP. Interestingly the correction to the Schwarzschild black hole temperature due to quadratic and linear-quadratic GUP has also been compared with the corrections from the quantum Raychaudhuri equation [47]. Very recently the Lorentz-invariance-violating class of dispersion relations has been applied to study the thermodynamics of black holes [48]. It would therefore be interesting to compare these results with those coming from the GUP.The above studies motivate us to investigate the modification of thermodynamic properties for Schwarzschild and Reissner-Nordström (RN) black holes using the form of the GUP proposed in [30]. This GUP involves higher order terms in the momentum uncertainty. We compute the remnant and critical masses analytically for these black holes below where the temperature becomes ill-defined. We then use the Stefan-Boltzmann law to estimate the mass and the energy output as a function of time. We finally compute the entropy and obtain the well-known area theorem containing corrections from the GUP with higher order terms in momentum uncertainty.The paper is organized as follows. In Section2, we study the thermodynamics of Schwarzschild black hole taking into account the effect of the GUP, with higher order terms in momentum uncertainty. In Section 2.1, we also obtain the mass and radiation rate characteristics for the Schwarzschild black hole as a function of time by using the Stefan-Boltzmann law. In Section 3, we study the thermodynamics of Reissner-Nordström black holes taking into account the effect of the GUP. Finally, we conclude in Section 4.


## 2. Thermodynamics of Schwarzschild Black Hole

In this paper, we work with the following form of the GUP [49]:(1)ΔxΔp≥ħ2∑i=0∞ailpΔpħ2i;a0=1;ai>0,i=1,2,…where lp is the Planck length (~10-35m). Keeping terms up to fourth order in momentum uncertainty, we have(2)ΔxΔp≥ħ2a0+a1lpΔpħ2+a2lpΔpħ4.We now consider a Schwarzschild black hole of mass M. In the vicinity of the event horizon of the black hole, let a pair (particle-antiparticle) production occurs. For simplicity we consider the particle to be massless. The particle with negative energy falls inside the horizon and that with positive energy escapes outside the horizon and gets observed by some observer at infinity. The momentum of the emitted particle (p), which also characterizes the temperature (T), is of the order of its uncertainty in momentum Δp. Consequently(3)T=ΔpckBwhere c is the speed of light and kB is the Boltzmann constant.The Hawking temperature of the black hole will be equal to the temperature of the particle when thermodynamic equilibrium is reached. The uncertainty in the position of a particle near the event horizon of the Schwarzschild black hole will be of the order of the Schwarzschild radius of the black hole(4)Δx=ϵrs;rs=2GMc2where ϵ is a calibration factor, rs is the Schwarzschild radius, and G is Newton’s universal gravitational constant.To relate the Hawking temperature of the black hole with the mass of the black hole, we consider the saturated form of the GUP (2)(5)ΔxΔp=ħ2a0+a1lpΔpħ2+a2lpΔpħ4.Substituting (3) and (4) in (5) gives(6)M=Mp2c24ϵkBT1+a1kB2T2Mp2c4+a2kB4T4Mp4c8where the relations cħ/lp=Mpc2 and Mp=c2lp/G (Mp being the Planck mass) have been used. In the absence of corrections due to GUP, (6) reduces to(7)M=Mp2c24ϵkBT.The value of ϵ now gets fixed to 2π by comparing this expression with the semiclassical Hawking temperature T=Mp2c2/8πMkB [50, 51]. The mass-temperature relation (6) finally takes the form(8)M=Mp2c28π1kBT+a1kBTMp2c4+a2kB3T3Mp4c8.The heat capacity of the black hole therefore reads(9)C=c2dMdT(10)=Mp2c48π-1kBT2+a1kBMp2c4+3a2kB3T2Mp4c8=kB8π-Mpc2kBT2+a1+3a2kBTMpc22.The above expression takes the form(11)C=kB8π-1T′2+a1+3a2T′2after the following notations are introduced(12)M′=8πMMp;T′=kBTMpc2.The mass of the black hole decreases due to radiation from the black hole. This leads to an increase in the temperature of the black hole. It can be observed from (7) and (10) that there exists a finite temperature at which the heat capacity vanishes. To find out this temperature, we set C = 0. This gives(13)3a2T′4+a1T′2-1=0.Solving this, we get(14)T′2=16a2-a1+a12+12a2where the positive sign before the square root has been taken so that the above result reduces to corresponding result when a2=0 [33].Finally we get the expression forT′ to be(15)T′=16a2-a1+a12+12a21/2.Now in terms of T′, M′ the mass-temperature relation (8) can be represented as(16)a2T′4+a1T′2-M′T′+1=0.The remnant mass can now be obtained by substituting (15) in (16). This yields(17)Mrem′=1T′a2T′4+a1T′2+1⟹Mrem=Mp8π6a29a2-a1+a12+12a21/2-a12+a1a12+12a2+12a2.Reassuringly the above result reduces to the result in a2→0 limit [33](18)Mrem=Mpa14π.Now for a1→0, a2≠0, the remnant mass is given by(19)Mrem=Mp6π3a21/4.Also for a1→0, the mass-temperature relation (16) reads(20)T′4-M′a2T′+1a2=0.The solution of this biquadratic equation in T′ yields(21)T′=M′216a22+M′4256a24-127a231/3+M′216a22-M′4256a24-127a231/3.The above relation readily implies the existence of a critical mass below which the temperature will be a complex quantity(22)Mcr=Mp6π3a21/4.This demonstrates that the remnant and critical masses are equal.At this point, we would like to make a comment. It can be observed from the above analysis that analytical expressions for the remnant and critical masses can be obtained even if one retains terms of order of(Δp)8 in the momentum uncertainty. This is because it leads to an equation of the form aT′8+bT′6+cT′4+dT′2+e=0 when the condition C=0 is imposed. This equation can be solved analytically to obtain the remnant mass. If we keep terms beyond this order in momentum uncertainty, analytical expressions for the remnant and critical masses can not be obtained.The black hole entropy from the first law of black hole thermodynamics is given by(23)S=∫c2dMT=∫CdTT=∫kB8π-Mpc2kBT2+a1+3a2kBTMpc22dTT=kB8π12Mpc2kBT2+a1ln⁡kBTMpc2+3a22Mpc2kBT=kB8π12T′2+a1log⁡T′+32a2T′2.To obtain the entropy S in terms of the mass M of the black hole, we need to consider (16) to obtain an expression for the temperature T in terms of mass M.Equation (16) yields up to O(a12,a22,a1a2)(24)T′=1M′+a1M′3+a2M′5+2a12M′5+6a1a2M′7+4a22M′9.Now the entropy expression in terms of the mass can be written as(25)SkB=18π12M′2-a1-a1log⁡M′+a122M′2+a22M′2+a1a2M′4+a222M′6+Oa12a2,a1a22,a13,a23=SBHkB-a116πlog⁡SBHkB-a116πlog⁡16π-a18π+a12+a216π2SBHkB-1+2a1a216π3SBHkB-2+a2216π4SBHkB-3+Oa12a2,a1a22,a13,a23where SBH/kB=4πM2/Mp2 is the semiclassical Bekenstein-Hawking entropy for the Schwarzschild black hole. In terms of the black hole horizon area A=4πrs2=16πG2M2/c4=4lp2SBH/kB, the above entropy expression can be written as(26)SkB=A4lp2-a116πlog⁡A4lp2-a116πlog⁡16π-a18π+a12+a216π2A4lp2-1+2a1a216π3A4lp2-2+a2216π4A4lp2-3+Oa12a2,a1a22,a13,a23.This completes our discussion of the effect of the GUP on the thermodynamic properties of the Schwarzschild black hole. In Figure 1, we present the plot of the entropy of the black hole versus the horizon area for the GUP case and compare it with the standard case.Figure 1
The entropy versus area plot. Here the solid line (lower curve) represents the GUP case (considering only the first two terms in the right hand side of (26)), and the dashed line (upper curve) represents the standard case.


### 2.1. Energy Output as a Function of Time

Due to radiation of the black hole, the mass of the black hole reduces, while its temperature keeps on increasing. If one assumes that the energy loss is dominated by photons, then one can apply the Stefan-Boltzmann law to estimate the energy radiated as a function of time(27)dMdt=-σATH4where σ is the Stefan-Boltzmann constant. In terms of Schwarzschild black hole mass M with the horizon area A=4πrs2=16πG2M2/c4, the above equation implies(28)ddt8πMMp=-σ2G2c4Mp5kB48πMMp2T′4where we have used T′=kBTH/Mpc2.We now write the above equation taking into account the effect of the GUP. Thus, considering the mass-temperature relation (24), the radiation rate takes the following form:(29)dxdt=-1tchx21x+a1x3+a2x5+2a12x5+6a1a2x7+4a22x94where we have set x=8πM/Mp and the characteristic time tch is being defined as tch=kB4/2σMp5c4G2. If xi refers to the initial mass at time t=0, the solution of the above equation yields the mass-time relation. Up to O(a1,a2), we have(30)x=-3ttch+xi3-12a1xi+12a2xi+12a1xi3-3ttch1/3-12a2xi3-3ttch-1/31/3where(31)ttch=xi33-4a1xi+4a2xi.In Figures 2 and 3, we have plotted the mass of the black hole as a function of time and the radiation rate as a function of time.Figure 2
The mass of the black hole versus time. The mass is in units of Planck mass and the time is in units of characteristic time. Here thin line (lower curve) represents GUP case (considering botha2 and a1), the dashed line (middle curve) represents GUP case (considering only a1), and thick line (upper curve) represents the standard case.Figure 3
The radiation rate of the black hole versus time. The rate is in units of Planck mass per characteristic time and the time is in units of characteristic time. Here thin line (upper curve) represents GUP case (considering botha2 and a1), the dashed line (middle curve) represents GUP case (considering only a1), and thick line (lower curve) represents the standard case.


## 2.1. Energy Output as a Function of Time

Due to radiation of the black hole, the mass of the black hole reduces, while its temperature keeps on increasing. If one assumes that the energy loss is dominated by photons, then one can apply the Stefan-Boltzmann law to estimate the energy radiated as a function of time(27)dMdt=-σATH4where σ is the Stefan-Boltzmann constant. In terms of Schwarzschild black hole mass M with the horizon area A=4πrs2=16πG2M2/c4, the above equation implies(28)ddt8πMMp=-σ2G2c4Mp5kB48πMMp2T′4where we have used T′=kBTH/Mpc2.We now write the above equation taking into account the effect of the GUP. Thus, considering the mass-temperature relation (24), the radiation rate takes the following form:(29)dxdt=-1tchx21x+a1x3+a2x5+2a12x5+6a1a2x7+4a22x94where we have set x=8πM/Mp and the characteristic time tch is being defined as tch=kB4/2σMp5c4G2. If xi refers to the initial mass at time t=0, the solution of the above equation yields the mass-time relation. Up to O(a1,a2), we have(30)x=-3ttch+xi3-12a1xi+12a2xi+12a1xi3-3ttch1/3-12a2xi3-3ttch-1/31/3where(31)ttch=xi33-4a1xi+4a2xi.In Figures 2 and 3, we have plotted the mass of the black hole as a function of time and the radiation rate as a function of time.Figure 2
The mass of the black hole versus time. The mass is in units of Planck mass and the time is in units of characteristic time. Here thin line (lower curve) represents GUP case (considering botha2 and a1), the dashed line (middle curve) represents GUP case (considering only a1), and thick line (upper curve) represents the standard case.Figure 3
The radiation rate of the black hole versus time. The rate is in units of Planck mass per characteristic time and the time is in units of characteristic time. Here thin line (upper curve) represents GUP case (considering botha2 and a1), the dashed line (middle curve) represents GUP case (considering only a1), and thick line (lower curve) represents the standard case.


## 3. Thermodynamics of Reissner-Nordström Black Hole

In this section, we consider the Reissner-Nordström (RN) black hole of massM and charge Q. In this case, near the horizon of the black hole, the position uncertainty of a particle will be of the order of the RN radius of the black hole(32)Δx=ϵrhrh=Gr0c2r0=M+M2-Q2where rh is the radius of the horizon of the RN black hole. Substituting the value of Δp and Δx from (3) and (32), the GUP (5) can be rewritten as(33)r0=ħc32ϵGkBT1+a1lpΔpħ2+a2lpΔpħ4.Once again, in the absence of correction due to GUP, (33) reduces to(34)r0=Mp2c22ϵkBT.Comparing the above relation with the semiclassical Hawking temperature T=Mp2c2(Mr0-Q2)/2πkBr03 yields the value of ϵ to be(35)ϵ=πr02Mr0-Q2.This finally fixes the form of the mass-charge-temperature relation (33) to be(36)r02r0-M=Mp2πMpc2kBT+a1kBTMpc2+a2kBTMpc23where the identity(37)r0Mr0-Q2=1r0-Mhas been used.The heat capacity of the black hole can now be calculated using relation (9) and equation (36):(38)C=kBr0-M32πr022r0-3M-Mpc2kBT2+a1+3a2kBTMpc22.To express the heat capacity in terms of the mass, once again we make use of relation (12) to recast (36) in the form(39)a2T′4+a1T′2-gr0MpT′+1=0where(40)gr0=2πr02r0-M.Now to find out the temperature where the radiation process stops, we set C = 0. Equation (38) therefore yields(41)3a2T′4+a1T′2-1=0from which solution of T′ reads(42)T′=16a2-a1+a12+12a21/2where the positive sign before the square root has been taken to reproduce the result corresponding to the limit a2→0 [33].The remnant mass can now be computed by substituting (42) in (39). This would then give(43)gr0Mp=6a29a2-a1+a12+12a21/2-a12+a1a12+12a2+12a2.Thus, we finally obtain the following cubic equation for the remnant mass:(44)2MpZπMrem3-MpZ2π2Mrem2-2MpZπQ2Mrem+Q4+MpZ2π2Q2=0where(45)Z=6a29a2-a1+a12+12a21/2-a12+a1a12+12a2+12a2.Solution of (44) gives us the exact expression of remnant mass for the RN black hole. This yields(46)Mrem=MpZ24π1+Mp2Z2/4π2+48Q2B/21/3+4π2Mp2Z2B21/3where(47)B=Mp6Z632π6-18Mp4Z4π4Q2-108Mp2Z2π2Q4+-2716Mp10Z10π10Q2-35027Mp8Z8π8Q4-4968Mp6Z6π6Q6+11664Mp4Z4π4Q8.The above expression for the remnant mass reduces to the remnant mass for the Schwarzschild black hole (17) in the Q→0 limit.Finally, we proceed to compute the entropy of the RN black hole. To do that, we first obtain an expression ofT′(T) from (39) in terms of the mass and the charge of the RN black hole. This gives up to O(a12,a1a2,a22)(48)T′=Mpgr01+a1Mpgr02+a2Mpgr04+2a12Mpgr04+6a1a2Mpgr06+4a22Mpgr08.From this one now can calculate the entropy for the RN black hole using (38) and (48):(49)SkB=πrh2lp2-a116πln⁡πrh2lp2-a1Q223Mp2-a228π2πrh2lp2-1-a227πMp2-a1πQ425Mp5πrh2lp2-2-a2Q427Mp4πrh2lp2-3-a2πQ628Mp6πrh2lp2-4-15a2π2Q828Mp8πrh2lp2-5+Oa12,a1a2,a22=SBHkB-a116πln⁡SBHkB-a1Q223Mp2-a228π2SBHkB-1-a227πMp2-a1πQ425Mp5SBHkB-2-a2Q427Mp4SBHkB-3-a2πQ628Mp6SBHkB-4-15a2π2Q828Mp8SBHkB-5+Oa12,a1a2,a22where SBH/kB=πrh2/lp2 is the semiclassical Bekenstein-Hawking entropy for the RN black hole. In terms of the area of the horizon A=4πrh2=4lp2SBH/kB, the above equation can be written as (50)SkB=A4lp2-a116πln⁡A4lp2-a1Q223Mp2-a228π2A4lp2-1-a227πMp2-a1πQ425Mp5A4lp2-2-a2Q427Mp4A4lp2-3-a2πQ628Mp6A4lp2-4-15a2π2Q828Mp8A4lp2-5+Oa12,a1a2,a22which is the area theorem for the RN black hole with corrections from the GUP containing higher order terms in the momentum uncertainty.We would like to conclude this section by mentioning that, in [52], it has been pointed out that there is a part of the information (leaking out of the black hole due to Hawking radiation) related to nonthermal GUP correlations. This insight may be important to provide a solution for the well-known information loss paradox and is worth investigating in future.


## 4. Conclusions

In this paper, we have investigated the modifications of the various thermodynamic properties of Schwarzschild and Reissner-Nordström black holes using higher order momentum uncertainty terms in the GUP. We obtain the GUP modified mass-temperature relation. This then leads to the existence of a remnant mass thereby preventing the complete evaporation of the black hole. The expression for the remnant and critical masses have been obtained analytically. In this regard, we observe that analytical expressions for these masses can be obtained even if we keep terms of the order of(Δp)8 in the momentum uncertainty. Beyond this, it is no longer possible to obtain analytical expression for the critical and remnant masses. We also compute the mass and energy outputs as functions of time using the Stefan-Boltzman law. We observe that these expressions get modified from the standard case as well as the case where the simplest form of the GUP is used. The expression for the entropy exhibits the well-known area theorem in terms of the horizon area in both cases up to leading order corrections from the GUP.



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