Black hole thermodynamics and generalized uncertainty principle with higher order terms in momentum uncertainty

In this paper we study the modification of thermodynamic properties of Schwarzschild and Reissner-Nordstr\"{o}m black hole in the framework of generalized uncertainty principle with correction terms upto 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 upto leading order logarithmic corrections and subleading corrections of the form $\frac{1}{A^n}$. 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.


Introduction
The consistent unification of quantum mechanics (QM) with general relativity (GR) is one of the major task 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 even Gedanken 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) [7]- [9], modified dispersion relation (MDR) [10]- [13], deformed special relativity (DSR) [14], 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 [15]- [20], loop quantum gravity [21]- [23], noncommutative geometry [24], computing Planck scale corrections to the phenomena of superconductivity and quantum Hall effect [25] and understanding its consequences in cosmology [26,27].
The incorporation of the GUP to study black hole thermodynamics has been another interesting area of active research [28]- [39]. 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 [40]- [42]. 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 [43]- [47]. In our earlier findings [33]- [37], 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 [48]. Very recently the Lorentz-invariance-violating class of dispersion relations have been applied to study the thermodynamics of black holes [49]. 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 [31]. This GUP involves higher order terms in the momentum uncertainty. We compute the remnant and critical masses analytically for these black holes below which 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 section 2, we study the thermodynamics of Schwarzschild black hole taking into account the effect of the GUP, with higher order terms in momentum uncertainty. In sub-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.

Thermodynamics of Schwarzschild black hole
In this paper we work with the following form of the GUP [50] ∆x∆p where l p is the Planck length (∼ 10 −35 m). Keeping terms upto fourth order in momentum uncertainty, we have 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 occur. 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 ∆p. Consequently where c is the speed of light and k B 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 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 eq.(s) (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 Solving this, we get where the positive sign before the square root has been taken so that the above result reduces to corresponding result when a 2 = 0 [34]. Finally we get the expression for T to be Now in terms of T , M the mass-temperature relation (8) can be represented as The remnant mass can now be obtained by substituting eq.(15) in eq. (16). This yields Reassuringly the above result reduces to the result in a 2 → 0 limit [34] .
Now for a 1 → 0 , a 2 = 0, the remnant mass is given by Also for a 1 → 0, the mass-temperature relation (16) reads The solution of this bi-quadratic equation in T yields The above relation readily implies the existence of a critical mass below which the temperature will be a complex quantity 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 (∆p) 8 in the momentum uncertainty. This is because it leads to an equation of the form a T 8 + b T 6 + c T 4 + d T 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 To obtain the entropy S in terms of the mass M of the black hole, we need to consider eq. (16) to obtain an expression for the temperature T in terms of mass M. Eq.(16) yields upto O(a 1 2 , a 2 2 , a 1 a 2 ) Now the entropy expression in terms of the mass can be written as 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 vs the horizon area for the GUP case and compare it with the standard case.

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 where σ is the Stefan-Boltzmann constant. In terms of Schwarzschild black hole mass M with the horizon area A = 4πr 2 where we have used T = k B T H Mpc 2 . 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 where we have set x = 8πM Mp and the characteristic time t ch is being defined as t ch = k B 4 2σMp 5 c 4 G 2 . If x i refers to the initial mass at time t = 0, the solution of the above equation yields the mass-time relation. Upto O(a 1 , a 2 ), we have 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.

Thermodynamics of Reissner-Nordström black hole
In this section, we consider the Reissner-Nordström (RN) black hole of mass M 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 where r h is the radius of the horizon of the RN black hole. Substituting the value of ∆p and ∆x from eq.(3) and eq.(32), the GUP (5) can be rewritten as Once again, in the absence of correction due to GUP, eq.(33) reduces to Comparing the above relation with the semi-classical Hawking temperature T = This finally fixes the form of the mass-charge-temperature relation (33) to be where the identity has been used. The heat capacity of the black hole can now be calculated using relation (9) and eq. (36): To express the heat capacity in terms of the mass, once again we make use of the relation (12) to recast eq.(36) in the form Now to find out the temperature where the radiation process stops, we set C = 0. Eq.(38) therefore yields from which solution of T reads where the positive sign before the square root has been taken to reproduce the result corresponding to the limit a 2 → 0 [34]. The remnant mass can now be computed by substituting eq.(41) in eq.(39). This would then give Thus we finally obtain the following cubic equation for the remnant mass which 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 [53], 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 non-thermal GUP correlations. This insight may be important to provide a solution for the well-known information loss paradox and is worth investigating in future.

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 (∆p) 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 upto leading order corrections from the GUP.