Abstract We use the generalized uncertainty principle to compute the first correction to the Hawking temperature associated to Hawking effect. From this value we obtain a new evaporation time and entropy of any Schwarzschild black hole analyzing their expressions and consequences. Keywords：Hawking effect;quantum fluctuations;uncertainty principle
Hawking radiation is a black-body radiation that appears near the event horizon of any black hole and was predicted by Hawking [1] in 1974 who provided theoretical arguments for its existence. The most important consequence of the existence of the Hawking radiation is the black hole evaporation of any isolated black hole, because the Hawking radiation reduces the mass and rotation energy of the black hole. Therefore an isolated black hole that cannot gain mass or energy is expected to be reduced and finally disappear. The expression of the classical Hawking temperature is given by$\begin{eqnarray}T=\displaystyle \frac{{\hslash }{c}^{3}}{8\pi {k}_{{\rm{B}}}{GM}},\end{eqnarray}$where the dependence respect to the parameters of the black hole is only through its mass M. If this mass is very small the temperature is bigger and the evaporation is faster. In fact micro black holes are predicted to be larger emitters of radiation and should evaporate very fast.
It is possible to compute the evaporation time of a black hole from the Hawking temperature (1) using the Stefan–Boltzmann power law that gives the total power radiation also called Bekenstein–Hawking luminosity of a black hole which is given by$\begin{eqnarray}P=A\sigma {T}^{4},\end{eqnarray}$where $A=4\pi {r}_{s}^{2}$ is the surface of the event horizon of Schwarzschild radius ${r}_{s}=2{GM}/{c}^{2}$ and σ is the Stefan–Boltzmann constant. Hence under the assumption of pure photon emission, substituting these values we get$\begin{eqnarray*}\begin{array}{rcl}P & = & \displaystyle \frac{16\pi {G}^{2}{M}^{2}}{{c}^{4}}\displaystyle \frac{{\pi }^{2}{k}_{{\rm{B}}}^{4}}{60{{\hslash }}^{3}{c}^{2}}{\left(\displaystyle \frac{{\hslash }{c}^{3}}{8\pi {k}_{{\rm{B}}}{GM}}\right)}^{4}\\ & = & \displaystyle \frac{{\hslash }{c}^{6}}{15360\pi {G}^{2}{M}^{2}}=\displaystyle \frac{\alpha }{{M}^{2}},\end{array}\end{eqnarray*}$where we have defined $\alpha ={\hslash }{c}^{6}/(15360\pi {G}^{2})$. Taking into account that P is also the rate of evaporation energy loss by the black hole we deduce$\begin{eqnarray*}P=-\displaystyle \frac{{\rm{d}}E}{{\rm{d}}t}=-{c}^{2}\displaystyle \frac{{\rm{d}}M}{{\rm{d}}t}=\displaystyle \frac{\alpha }{{M}^{2}}.\end{eqnarray*}$This differential equation established by the last equality is of separable variables and can be written as$\begin{eqnarray}{c}^{2}{M}^{2}{\rm{d}}M=-\alpha {\rm{d}}t.\end{eqnarray}$More complex models for the mass rate equation (3) are studied in [2] assuming that the Universe is pervaded by a quintessence field. Integrating equation (3) from the initial mass M0 of the black hole to zero which corresponds to the complete evaporation, and in the right hand side from zero to evaporation time tf, i.e.$\begin{eqnarray*}{c}^{2}{\int }_{{M}_{0}}^{0}{M}^{2}{\rm{d}}M=-{\int }_{0}^{{t}_{f}}\alpha {\rm{d}}t,\end{eqnarray*}$we obtain the classical evaporation time given by$\begin{eqnarray*}{t}_{f}=\displaystyle \frac{{c}^{2}{M}_{0}^{3}}{3\alpha }=\displaystyle \frac{5120\pi {G}^{2}{M}_{0}^{3}}{{\hslash }{c}^{4}}.\end{eqnarray*}$Nowadays all the existence candidates to be quantum gravity theories point out that there exists a minimal observable distance of Planck distance order, the existence of such a minimal length is due to the fact that, at Planck scale, the gravitational quantum fluctuations that must be taken into account. Of course these quantum fluctuations have an important role when we analyze the strong gravity field of a black hole. This minimal length leads to a modification of the uncertainty principle into a generalized uncertainty principle (GUP), see below. Therefore it is important to know how this GUP affects the properties and dynamics of any black hole.
In [3] it was given the modified Unruh temperature from the GUP and it was asked for an extension of the formalism presented for the Hawking effect. A heuristic derivation of the classical Hawking temperature has already given in [4] although some approximations were found before, see for instance [5]. In quantum information theory, the uncertainty principle is popularly formulized in terms of entropy. The effects of Hawking radiation on the entropic uncertainty in a Schwarzschild space-time are analyzed in [6] and the relationship between the entropic uncertainty and quantum coherence is obtained in [7]. Finally in [8] an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation is obtained.
In this work we present the correct derivation of the first correction of the Hawking temperature from the GUP and its consequences. Before that, a new rigorous mathematical derivation of the classical Hawking temperature made is given in the next section. Moreover it is computed also the correction to the evaporation time and the entropy taking into account the modified Hawking temperature.
2. The Hawking effect revisited
The quantum effects near the event horizon of the black hole produce a black-body radiation predicted by Hawking [1] and since then called the Hawking radiation. The quantum effects involved in the Hawking radiation are the quantum fluctuations that produce, attending to the uncertainty principle, the appearance of particle-antiparticle pairs close to the event horizon. One particle is captured by the black hole while the other escapes. The black hole loses mass because for an outside observer the black hole just emitted a particle. Consequently the captured particle has negative energy. The quantum fluctuations produce other macroscopic phenomena, see for instance [9–11].
Hawking [1] deduced the so-called Hawking temperature using the quantum field theory applied to the event horizon of the black hole. However a simple deduction is based on the uncertainty principle, see [4]. We reproduce and improve here the complete derivation. For the particle captured by the black hole the uncertainty in its position is given by the unique information that we known which is that the particle is inside the black hole. When a particle gets in a Schwarzschild black hole which we assume is like a sphere, its projection, in the plane defined by the point where the quantum fluctuation have appeared and the crossing point, is a disk. Then the one dimension uncertainty in position will be at least the diameter, i.e. ${\rm{\Delta }}{x}_{\min }=2{r}_{s}$ where rs is the Schwarzschild radius of a black hole. Of course any length greater than that is correct but one has to argue why it takes a greater specific length. Otherwise, physically one has only ${\rm{\Delta }}{x}_{\min }=2{r}_{s}$. However this position uncertainty accounts for the uncertainty in the position in a three-dimensional space inside the black hole. If we consider a point ${\text{}}{\boldsymbol{x}}=(x,y,z)$ in this three-dimensional space, the uncertainty of this point is given by ${\rm{\Delta }}{\boldsymbol{x}}=({\rm{\Delta }}x,{\rm{\Delta }}y,{\rm{\Delta }}z)$ and we can consider one-dimensional uncertainty in the position given by $| {\rm{\Delta }}{\boldsymbol{x}}| \,=\sqrt{{\rm{\Delta }}{x}^{2}+{\rm{\Delta }}{y}^{2}+{\rm{\Delta }}{z}^{2}}$. As the space inside the black hole is a sphere we take spherical coordinates $x=r\sin \theta \cos \varphi $, $y=r\sin \theta \sin \varphi $ and $z=r\cos \theta $. Consequently, we have$\begin{eqnarray}\begin{array}{rcl}| {\rm{\Delta }}{\boldsymbol{x}}| & = & \sqrt{{\rm{\Delta }}{x}^{2}+{\rm{\Delta }}{y}^{2}+{\rm{\Delta }}{z}^{2}}\\ & = & r\sqrt{{\rm{\Delta }}{\theta }^{2}+{\rm{\Delta }}{\varphi }^{2}{\sin }^{2}\theta }\geqslant r{\rm{\Delta }}\varphi | \sin \theta | .\end{array}\end{eqnarray}$Taking into account that θ ∈ [0, π] and φ ∈ [0, 2π] which implies that Δθ=π and Δφ=2π, the greatest value of $| {\rm{\Delta }}{\boldsymbol{x}}| $ satisfies $| {\rm{\Delta }}{\boldsymbol{x}}| \geqslant 2\pi r$. Hence $| {\rm{\Delta }}{\boldsymbol{x}}| \geqslant 2\pi {r}_{s}$ where rs is the Schwarzschild radius of a black hole. Since the Schwarzschild radius of a black hole is given by ${r}_{s}=2{GM}/{c}^{2}$ where M is the mass of the black hole, we obtain$\begin{eqnarray}| {\rm{\Delta }}{\boldsymbol{x}}| =\displaystyle \frac{4\pi {GM}}{{c}^{2}}.\end{eqnarray}$If we introduce this position uncertainty into the expression of the classical uncertainty principle$\begin{eqnarray}| {\rm{\Delta }}{\boldsymbol{p}}| | {\rm{\Delta }}{\boldsymbol{x}}| \simeq \displaystyle \frac{{\hslash }}{2},\end{eqnarray}$we obtain$\begin{eqnarray}| {\rm{\Delta }}{\boldsymbol{p}}| \,\displaystyle \frac{4\pi {GM}}{{c}^{2}}\simeq {\hslash }/2.\end{eqnarray}$From here we have$\begin{eqnarray}| {\rm{\Delta }}{\boldsymbol{p}}| \simeq \displaystyle \frac{{\hslash }{c}^{2}}{8\pi {GM}}.\end{eqnarray}$Since the energy of the photon is given by $E=| {\boldsymbol{p}}| c$, the uncertainty in the energy of a virtual photon is ${\rm{\Delta }}E=c| {\rm{\Delta }}{\boldsymbol{p}}| $ and equation (8) becomes$\begin{eqnarray}{\rm{\Delta }}E\simeq \displaystyle \frac{{\hslash }{c}^{3}}{8\pi {GM}}.\end{eqnarray}$Taking into account that the Hawking radiation is thermalised we have $E={k}_{{\rm{B}}}T$ where kB is the Boltzmann constant and equation (9) becomes$\begin{eqnarray}{\rm{\Delta }}T\simeq \displaystyle \frac{{\hslash }{c}^{3}}{8\pi {k}_{{\rm{B}}}{GM}}.\end{eqnarray}$We remind that the exact quantum-mechanical computation performed by Hawking provides exactly this same expression, see [1]. The natural question is: how is it possible that studying a single pair we can deduce the correct expression of the Hawking temperature? The answer is clear because we have deduced the average temperature of the photons that are emitted by the black hole. Is the similar case that in the Planck radiation of a blackbody and the relation of the peak radiation and the temperature of the blackbody radiator although the Hawking temperature is not a classical temperature.
Another form to deduce the Hawking temperature is using the Unruh temperature associated to any accelerated particle, see also [4]. This Unruh temperature is given by the expression$\begin{eqnarray}T=\displaystyle \frac{{\hslash }a}{2\pi {{ck}}_{{\rm{B}}}}.\end{eqnarray}$The gravitational acceleration that suffers a particle near to the event horizon of a black hole is$\begin{eqnarray}a=\displaystyle \frac{{GM}}{{r}_{s}^{2}}=\displaystyle \frac{{c}^{4}}{4{GM}},\end{eqnarray}$where rs is the Schwarzschild radius. Substituting this acceleration into the Unruh temperature (11) we get the correct expression of the Hawking temperature$\begin{eqnarray}T=\displaystyle \frac{{\hslash }{c}^{3}}{8\pi {k}_{{\rm{B}}}{GM}}.\end{eqnarray}$In fact the equivalence principle between an acceleration and a local gravitational field establish the link between both effects.
3. Modified Hawking effect
The most plausible quantum theories of gravity are the superstring theory and the loop quantum gravity, see [12, 13]. One of the consequence of these theories is a modification of the original uncertainty principle. The modified or GUP follows from the gravitational interaction of the photon and the particle being observed which modifies the uncertain principle with an additional term. Therefore in fact the GUP is a general consequence of any quantum gravity theory. Indeed it can be deduced from a dimensional analysis of the Newtonian theory and also from the general relativity theory, see [14]. From this seminal work [14] several authors have studied the possibility about a generalization of the Heisenberg uncertainty principle in order to take into account the gravitation. It is true that at quantum level gravity can be neglected if we compare it with the other fundamental forces. However at large scales like cosmic scales or near to a strong gravitational field, gravity has a fundamental role. In short gravity must affect the formulation of the Heisenberg’s principle and several proposals have been made, see for instance [3].
The most used deformation of the Heisenberg uncertainty and the form of the GUP we are going to use is$\begin{eqnarray}{\rm{\Delta }}p{\rm{\Delta }}x\geqslant \displaystyle \frac{{\hslash }}{2}\left(1+\beta \displaystyle \frac{4\,{{\ell }}_{p}^{2}{\rm{\Delta }}{p}^{2}}{{{\hslash }}^{2}}\right)=\displaystyle \frac{{\hslash }}{2}\left[1+\beta {\left(\displaystyle \frac{{\rm{\Delta }}p}{{m}_{p}c}\right)}^{2}\right],\end{eqnarray}$where ${{\ell }}_{p}{m}_{p}={\hslash }/(2c)$ where mp is the Planck mass given by ${m}_{p}={\varepsilon }_{p}/{c}^{2}\approx {10}^{-8}\,{\rm{kg}}$, where the Planck energy ϵp satisfies ${\varepsilon }_{p}{{\ell }}_{p}={\hslash }c/2$ and the Planck length is ${{\ell }}_{p}=\sqrt{G{\hslash }/{c}^{3}}$. The dimensionless parameter β is assumed to be of order one. For the deforming parameter β are known some bounds for its possible value, see [3]. The inequality (14) for symmetric states with $\langle \hat{p}\rangle =0$ is equivalent to the commutator$\begin{eqnarray}\left[\hat{x},\hat{p}\right]={\rm{i}}{\hslash }\left[1+\beta {\left(\displaystyle \frac{\hat{p}}{{m}_{p}c}\right)}^{2}\right].\end{eqnarray}$In fact the GUP is based on the idea that we must add a term taking into the presence of a strong gravitational field. Hence the position uncertainty is given by a first term (the classical one) that says that smallest is the detail of an object large energies of photons are required to explore it plus a term taking into account that at high scattering energies we must take into account the possible creation of micro black holes with gravitational radius rs=rs(E) which are proportional to the scattering energy, see for instance [15], so we have$\begin{eqnarray}{\rm{\Delta }}x\geqslant \displaystyle \frac{{\hslash }}{2E}+\beta {r}_{s},\end{eqnarray}$where ${r}_{s}=2{GE}/{c}^{4}=2{{\ell }}_{p}^{2}E/({\hslash }c)$. Hence we modify the uncertainty principle as follows$\begin{eqnarray}{\rm{\Delta }}p{\rm{\Delta }}x\geqslant \displaystyle \frac{{\hslash }}{2}+\beta {r}_{s}{\rm{\Delta }}p.\end{eqnarray}$Equation (17) can be written into the form (14) taking into account that the beam of photons energy is given by $E=c{\rm{\Delta }}p$.
From the uncertainty relation (14) we can arrive to first correction to the Hawking radiation due to the gravitational interaction. In order to do that we solve from (14) the momentum uncertainty in terms of the distance uncertainty first dividing by Δx. In fact we have a quadratic equation respect to Δp that gives a concave parabola which is positive between its roots. However only the biggest root has physical meaning. Hence we take is as a bound for the value of Δp, so we get$\begin{eqnarray}{\rm{\Delta }}p\geqslant \displaystyle \frac{{\hslash }{\rm{\Delta }}x}{4\beta {{\ell }}_{p}^{2}}\left[1\mp \sqrt{1-\beta \displaystyle \frac{4{{\ell }}_{p}^{2}}{{\rm{\Delta }}{x}^{2}}}\ \right],\end{eqnarray}$where for the case ${\rm{\Delta }}x\gg {{\ell }}_{p}$ we reobtain the classical uncertainty (6) if the negative sign is chosen. Moreover for $\beta 4{{\ell }}_{p}^{2}/({\rm{\Delta }}{x}^{2})\gt 1$ the expression has not physical meaning. The expression (18) was obtained in [16] but for the particular case β=1 (which must be determined by the accepted quantum gravity theory) and with a mistake as a straightforward dimensional analysis reveals. If we substitute the expression (5) in (18) with the correct sign we obtain$\begin{eqnarray}{\rm{\Delta }}p\geqslant \displaystyle \frac{\pi {cM}}{\beta }\left[1-\sqrt{1-\beta \displaystyle \frac{{m}_{p}^{2}}{{\pi }^{2}{M}^{2}}}\ \right].\end{eqnarray}$From (19) we find the expression of the temperature associated to the GUP which taking into account that ${\rm{\Delta }}E=c{\rm{\Delta }}p$ and ${\rm{\Delta }}T={\rm{\Delta }}E/{k}_{{\rm{B}}}$ it is given by$\begin{eqnarray}{\rm{\Delta }}T\geqslant \displaystyle \frac{\pi {c}^{2}M}{\beta {k}_{{\rm{B}}}}\left[1-\sqrt{1-\beta \displaystyle \frac{{m}_{p}^{2}}{{\pi }^{2}{M}^{2}}}\ \right].\end{eqnarray}$We remark that the temperature becomes complex and without physical meaning for values $\beta {m}_{p}^{2}/({\pi }^{2}{M}^{2})\gt 1$. Moreover recalling that for x ≪ 1 we have $\sqrt{1-x}\approx 1-x/2\,-{x}^{2}/8+{ \mathcal O }({x}^{3})$ then for ${m}_{p}/M\ll 1$ equation (20) becomes$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}T & \gtrsim & \displaystyle \frac{{c}^{3}{\hslash }}{8\pi {k}_{{\rm{B}}}{GM}}\left[1+\beta {\left(\displaystyle \frac{{m}_{p}}{2\pi M}\right)}^{2}\right]\\ & = & \displaystyle \frac{{c}^{3}{\hslash }}{8\pi {k}_{{\rm{B}}}{GM}}\left[1+\beta \displaystyle \frac{c{\hslash }}{16{\pi }^{2}{{GM}}^{2}}\right],\end{array}\end{eqnarray}$where the correction only depends on the mass of the Schwarzschild black hole. The same expression (21) was found in [17] (see also [18]) but here in a straightforward way and using the same method that for the derivation of the classical Hawking temperature of the previous section. In fact in all previous works [16–18] the authors use a calibration factor in order to determine a free parameter to recover the limit of the classical Hawking radiation.
4. The modified evaporation time
In this section we compute the evaporation time of any black hole attending to the modified Hawking temperature using the new expression of the temperature (20). From the Stefan–Boltzmann power law (2) we have$\begin{eqnarray*}P=\displaystyle \frac{4{c}^{2}{G}^{2}{M}^{6}{\pi }^{7}}{15{{\hslash }}^{3}{\beta }^{4}}{\left[1-\sqrt{1-\beta \displaystyle \frac{{m}_{p}^{2}}{{\pi }^{2}{M}^{2}}}\right]}^{4}.\end{eqnarray*}$Since P is the rate of evaporation energy loss by the black hole we have$\begin{eqnarray*}P=-\displaystyle \frac{{\rm{d}}E}{{\rm{d}}t}=-{c}^{2}\displaystyle \frac{{\rm{d}}M}{{\rm{d}}t}.\end{eqnarray*}$Hence to compute the evaporation time we integrate this new differential equation which is of separable variables and we have$\begin{eqnarray*}{t}_{f}={\int }_{0}^{{t}_{f}}{\rm{d}}t=-{c}^{2}{\int }_{{M}_{0}}^{{M}_{f}}\displaystyle \frac{{\rm{d}}M}{P}.\end{eqnarray*}$Here we introduce a value Mf in order to have an integrant well-defined and in order to avoid complex numbers. This implies that a small Schwarzschild with mass $M\lt {m}_{p}\sqrt{\beta }/(\pi )$ cannot radiate further. We will see the implications of this fact in the conclusions. Substituting the value of P we obtain$\begin{eqnarray*}\begin{array}{rcl}{t}_{f} & = & \left[\displaystyle \frac{5{{\hslash }}^{3}}{4{G}^{2}{{Mm}}_{p}^{8}{\pi }^{3}}\left(3{m}_{p}^{4}{\beta }^{2}+4{M}^{2}{m}_{p}^{2}\pi \beta \right.\right.\\ & & \times \left(6\pi +5\sqrt{{\pi }^{2}-\displaystyle \frac{{m}_{p}^{2}\beta }{{M}^{2}}}\,\right)\\ & & -{M}^{4}\left(8{\pi }^{4}+8{\pi }^{3}\sqrt{{\pi }^{2}-\displaystyle \frac{{m}_{p}^{2}\beta }{{M}^{2}}}\,\right)\\ & & {\left.\left.+12{{Mm}}_{p}^{3}\pi {\beta }^{3/2}\arcsin \left(\displaystyle \frac{{m}_{p}\sqrt{\beta }}{M\pi }\right)\right)\right]}_{{M}_{0}}^{{M}_{f}}.\end{array}\end{eqnarray*}$We remark that tf is positive taking into account that M0>Mf and that the limit β tending to zero has no sense because β must considered different from zero to obtain equation (18), the same happens in the expression of ${\rm{\Delta }}T$. In fact the most important case is when $M\gg {m}_{p}$. In this case we use equation (21). Then substituting into the Stefan–Boltzmann power law (2) we get$\begin{eqnarray*}P=\displaystyle \frac{{\hslash }{c}^{6}}{15360\pi {G}^{2}{M}^{2}}{\left(1+\beta \displaystyle \frac{c{\hslash }}{16{\pi }^{2}{{GM}}^{2}}\right)}^{4}=\displaystyle \frac{\alpha }{{M}^{2}}{\left(1+\displaystyle \frac{\gamma }{{M}^{2}}\right)}^{4},\end{eqnarray*}$where $\alpha ={\hslash }{c}^{6}/(15360\pi {G}^{2})$ and $\gamma =\beta c{\hslash }/(16{\pi }^{2}G)$. As before since P is the rate of evaporation energy loss by the black hole we have$\begin{eqnarray*}P=-\displaystyle \frac{{\rm{d}}E}{{\rm{d}}t}=-{c}^{2}\displaystyle \frac{{\rm{d}}M}{{\rm{d}}t}=\displaystyle \frac{\alpha }{{M}^{2}}{\left(1+\displaystyle \frac{\gamma }{{M}^{2}}\right)}^{4}.\end{eqnarray*}$Hence to determine the evaporation time in this case we compute$\begin{eqnarray}{t}_{f}={\int }_{0}^{{t}_{f}}{\rm{d}}t=-{c}^{2}{\int }_{{M}_{0}}^{{M}_{f}}\displaystyle \frac{{M}^{2}}{\alpha {\left(1+\tfrac{\gamma }{{M}^{2}}\right)}^{4}}{\rm{d}}M.\end{eqnarray}$Also as before we introduce a positive small value Mf but not zero in order that the integrant will be well defined in the limit of integration. Taking into account that $\gamma /{M}^{2}\ll 1$, we can be developed at first order in terms of γ/M2 and we have$\begin{eqnarray}{t}_{f}={\int }_{0}^{{t}_{f}}{\rm{d}}t\approx -{c}^{2}{\int }_{{M}_{0}}^{{M}_{f}}\displaystyle \frac{{M}^{2}}{\alpha }\left(1-\displaystyle \frac{4\gamma }{{M}^{2}}\right){\rm{d}}M.\end{eqnarray}$This integration gives the value of the evaporation time$\begin{eqnarray*}{t}_{f}\approx \displaystyle \frac{{c}^{2}({M}_{0}^{3}-{M}_{f}^{3})}{3\alpha }-\displaystyle \frac{4{c}^{2}\gamma ({M}_{0}-{M}_{f})}{\alpha },\end{eqnarray*}$where we have that ${M}_{0}\gg {M}_{f}$. It is clear that if Mf is very small then the reduction of tf respect to the classical result is very important. We recall that there is a limit given ${M}_{f}\,\gt {m}_{p}\sqrt{\beta }/(\pi )$ to have radiation for the case β>0. Nevertheless it is also worth mention that in the case β<0 this limit does not appears and in [19] it was shown that a GUP with β<0 can be derived assuming that the Universe has an underlying crystal lattice-like structure. In [20] it was found that this choice of β is the only compatible with the Chandrasekhar limit, that is, a positive β would allow arbitrary large white dwarfs, which is not observed in the current astrophysical observations. In fact the implications of GUP with a negative deformation parameter β have been studied in [21] around the context of black hole physics. In that work is also discussed the proposed quantum picture of black holes based on a corpuscular gravity (CG). In particular it is shown that GUP in the black hole physics is consistent with the predictions of the corpuscular theory of gravity, in which a black hole is conceived as a Bose–Einstein condensate of weakly interacting gravitons. The characteristic thermodynamic quantities as the temperature and the evaporation rate of a black hole are also computed and compared the results obtained in the two scenarios, giving estimations of the value of the deformation parameter β.
One of the main consequences of the CG is studied in [22] where is argued that gravity at quantum level may induce non-thermal corrections to the black hole radiation. In the GUP framework studied in the present paper is assumed that quantum corrections manifest as a shift of the Hawking temperature, without affecting the thermality of the spectrum. The non-thermal corrections of CG, that do not appear at the first order as it is shown in [21], and their interface with the GUP corrections can be object of a future work. Recently, a corpuscular interaction gravity has been obtained in the context of the CG giving the exact form of the Newton gravity law at macroscopic scales, see [23].
5. The modified black hole entropy
In this section we compute the black hole entropy from the modified Hawking temperature given in (20). Solving equation (20) equation with respect to the mass M we obtain$\begin{eqnarray}M=\displaystyle \frac{{c}^{3}{\hslash }}{8{{Gk}}_{{\rm{B}}}\pi T}+\displaystyle \frac{\beta {k}_{{\rm{B}}}T}{2\pi {c}^{2}}.\end{eqnarray}$It is important to note that this equation (24) is also obtained in [18] with some missprints. The dimensional analysis highlights the mistakes.
Now we compute the heat capacity of the black hole that is defined by$\begin{eqnarray}C(T)={c}^{2}\displaystyle \frac{{\rm{d}}M}{{\rm{d}}T}=-\displaystyle \frac{{c}^{5}{\hslash }}{8{{Gk}}_{{\rm{B}}}\pi {T}^{2}}+\displaystyle \frac{\beta {k}_{{\rm{B}}}}{2\pi }.\end{eqnarray}$Next we can determine the black hole entropy from the first law of the black hole thermodynamics given by$\begin{eqnarray}S={c}^{2}{\int }_{{T}_{0}}^{T}\displaystyle \frac{{\rm{d}}M}{T}={\int }_{{T}_{0}}^{T}\displaystyle \frac{C(T)}{T}{\rm{d}}T,\end{eqnarray}$where T0 is a minimum value for the temperature because if the temperature is zero the integrant is not well-defined. In our case we obtain$\begin{eqnarray}S=\displaystyle \frac{{c}^{5}{\hslash }}{16{{Gk}}_{{\rm{B}}}\pi }\left(\displaystyle \frac{1}{{T}^{2}}-\displaystyle \frac{1}{{T}_{0}^{2}}\right)+\displaystyle \frac{\beta {k}_{{\rm{B}}}}{2\pi }\mathrm{log}\left(\displaystyle \frac{T}{{T}_{0}}\right),\end{eqnarray}$where we have normalized the modified entropy to zero at T0. Therefore we have a lower bound in the temperature given by T0. Equation (24) has a minimum of mass (in relation with the Planck mass) coming from the minimal length suggested by uncertainty relation (14) which implies also the existence of a maximum in the temperature, see [16]. The minimum of temperature T0 corresponds to the temperature of the black hole at the initial time that by normalization corresponds to the value of the entropy equals zero. Any black hole has a temperature greater that the ambient temperature which is equal to 2.7 K for the present Universe.
6. Conclusion
First we revisit the simple derivation of the Hawking temperature based on the vacuum fluctuations and the classical uncertainty principle. The derivations of the first correction to the Hawking temperature due to the gravitational interactions is obtained based on the GUP. The computation of the modified Hawking temperature and the entropy of a Schwarzschild black hole can be found in [24, 25]. However here they are deduced in a straightforward way using the same method that for the derivation of the classical Hawking temperature given in section 2 and without a calibration factor in order to recover the limit of the classical Hawking radiation. Moreover we also compute the evaporation time of any Schwarzschild black hole from the modified Hawking temperature. We observe two phenomena.
The first is that the black hole does not evaporate at all and a small remnant Mf appears that does not evaporate otherwise the integral has a singularity. Consequently the GUP implies that any Schwarzschild black hole becomes, at the end of its evaporation stage, a remnant that only possesses gravitational interactions that does not have a black hole horizon. This phenomenon is not new and was mentioned in the seminal paper [16]. In that paper, assuming that the energy loss is dominated by photons is estimate the mass and energy output as a function of the temperature as well as the evaporation time at which the evaporation stops and the black hole remnant is shaped. Here we found this evaporation time in the general case and in the case that ${m}_{p}/M\ll 1$. Such remnants have been appearing throughout all the history of the Universe and are candidates to be the not found dark matter of the Universe, see for instance [26] and references therein.
The second is that the evaporation time is less than for the classical case when we use the classical Hawking temperature. This fact can has implications at cosmologic scales about the Universe evolution model. Especially at the formation epoch of primordial black holes and mini primordial black holes in the early Universe during their evaporation stage and in the last epoch of accelerated expansion when the Universe ends in a heat death. This happens after an extremely long time during which the matter will collapse into black holes which will then evaporate via Hawking radiation or the matter would run beyond the cosmological horizon before having time to plunge into a black hole. The fact that the evaporation time will be less implies that the heat death of the Universe occurs earlier.
Moreover for micro black holes the consequence is that they should evaporate even faster than with the classical Hawking temperature but not at all and they become remnants that modify the gravitational field. The possible formation of micro black holes in high energy scatterings can produce a great number of remnants.
Finally we also compute the new black hole entropy associated to this modified Hawking temperature. To avoid unphysical predictions we must to introduce a lower bound in the temperature. On the other hand as we know that in the evaporation process appears a remnant we know that there is also a maximum in the temperature of the Schwarzschild black hole and consequently a minimum in its entropy.
We have focused in the case of the existence of a minimal length that is given by the GUP. Another possibility is to assume the existence of a minimal uncertainty in momentum. Using this assumption it is possible to obtain a modified uncertainty principle which is called extended uncertainty principle studied in [18] and references therein.
Acknowledgments
The author is grateful to the referees for their valuable comments and suggestions to improve this paper. The author is partially supported by a MINECO/ FEDER Grant Number 2017-84383-P and an AGAUR (Generalitat de Catalunya) Grant Number 2017SGR 1276.
GinéJPeralta-SalasD2012 A note on the Liouvillian integrability and the qualitative properties of the mass rate equation for black holes J. Math. Phys. 53 062701 DOI:10.1063/1.4724173 [Cited within: 1]
ScardigliF1995 Some heuristic semi-classical derivations of the Planck length, the Hawking effect and the Unruh effect Nuovo Cimento B 110 10291034 DOI:10.1007/BF02726152 [Cited within: 1]
WangD et al. 2018 Effects of Hawking radiation on the entropic uncertainty in a Schwarzschild space-time Ann. Phys. 530 1800080 DOI:10.1002/andp.201800080 [Cited within: 1]
AdlerR JChenPSantiagoD I2001 The generalized uncertainty principle and black hole remnants Gen. Relativ. Gravit. 33 21012108 DOI:10.1023/A:1015281430411 [Cited within: 4]
JizbaPKleinertHScardigliF2010 Uncertainty relation on a world crystal and its applications to micro black holes Phys. Rev. D 81 084030 DOI:10.1103/PhysRevD.81.084030 [Cited within: 1]
OngY C2018 Generalized uncertainty principle, black holes, and white dwarfs: a tale of two infinities J. Cosmol. Astropart. Phys. JCAP09(2018)015 DOI:10.1088/1475-7516/2018/09/015 [Cited within: 1]
DalianisITringasG2019 Primordial black hole remnants as dark matter produced in thermal, matter, and runaway-quintessence postinflationary scenarios Phys. Rev. D 100 083512 DOI:10.1103/PhysRevD.100.083512 [Cited within: 1]
,Departament de Matemàtica, 
