Generalized Uncertainty Principle and Corpuscular Gravity

TL;DR

This paper demonstrates that the generalized uncertainty principle (GUP) aligns with the corpuscular theory of gravity, linking quantum corrections to black hole radiation with a Bose-Einstein condensate model of gravitons, and estimates the GUP parameter.

Contribution

It establishes a connection between GUP effects and corpuscular gravity, providing an estimate for the GUP deformation parameter and exploring its sign within a quantum gravity framework.

Findings

GUP-induced temperature shifts correspond to non-thermal radiation corrections.

The GUP deformation parameter β is estimated to be around unity.

Discussion on the possibility of negative β in corpuscular gravity models.

Abstract

We show that the implications of the generalized uncertainty principle (GUP) in the black hole physics are 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 stuck at the critical point of a quantum phase transition. In particular, we prove that the GUP-induced shift of the Hawking temperature can be reinterpreted in terms of non-thermal corrections to the spectrum of the black hole radiation, in accordance with the corpuscular gravity picture. By comparing the two scenarios, we are able to estimate the GUP deformation parameter $\beta$, which turns out to be of the order of unity, in agreement with the expectations of some models of string theory. We also comment on the sign of $\beta$, exploring the possibility of having a negative deformation parameter when a corpuscular quantum description of the gravitational interaction is assumed to be valid.

Equations

$$\delta x\delta p\ge \frac{\hslash }{2},$$

$$[\hat{x},\hat{p}]=i\hslash .$$

$$\delta x\delta p\ge \frac{\hslash }{2}\pm 2|\beta |{\ell }_p^2\frac{\delta p^2}{\hslash }=\frac{\hslash }{2}\pm 2|\beta |\hslash \frac{\delta p^2}{m_p^2},$$

$$[\hat{x},\hat{p}]=i\hslash \left(1\pm |\beta |\left(\frac{{\hat{p}}^2}{m_p^2}\right)\right).$$

$$\delta p\simeq \frac{\hslash }{4\mu MG},$$

$$E=T\simeq \frac{\hslash }{4\mu MG}=\frac{m_p^2}{4\mu M}$$

$$T_{\mathrm{H}}=\frac{\hslash }{8\pi MG}\equiv \frac{m_p^2}{8\pi M}$$

$$P=A_s\epsilon \sigma T^4\simeq A_s\sigma T^4,$$

$$\frac{dM}{dt}=-P\simeq -\frac{1}{60{(16)}^2\pi \sqrt{\hslash G}}\frac{m_p^3}{M^2}=-\frac{1}{60{(16)}^2\pi }\frac{m_p^4}{\hslash M^2}.$$

$$T_{\mathrm{GUP}}=\pm \frac{M}{4\pi |\beta |}\left(1\pm \sqrt{1\mp |\beta |\frac{m_p^2}{M^2}}\right).$$

$${\left(\frac{dM}{dt}\right)}_{\mathrm{GUP}}\simeq \frac{-1}{60(16)\pi \hslash }\frac{M^6}{{|\beta |}^4{m}_p^4}{\left(1-\sqrt{1\mp |\beta |\frac{m_p^2}{M^2}}\right)}^4.$$

$$T_{\mathrm{GUP}}\simeq \frac{m_p}{4\pi \sqrt{|\beta |}}<\infty .$$

$${\mathcal{N}}_{\mathrm{H}}(\omega )=\frac{1}{e^{\hslash \omega /T_{\mathrm{H}}}-1}\underset{T\to T_{{}_{\mathrm{GUP}}}}{⟶}{\mathcal{N}}_{\mathrm{GUP}}(\omega )=\frac{1}{e^{\hslash \omega /T_{\mathrm{GUP}}}-1}.$$

$${\mathcal{N}}_{\mathrm{GUP}}(\omega )\simeq \frac{1}{e^{\hslash \omega /T_{\mathrm{H}}}-1}\left(1\mp |\beta |f(\omega )\frac{m_p^2}{M^2}\right),$$

$$f(\omega )\equiv \frac{\hslash \omega }{4T_{\mathrm{H}}}\frac{e^{\hslash \omega /T_{\mathrm{H}}}}{e^{\hslash \omega /T_{\mathrm{H}}}-1}.$$

$${\alpha }_g\equiv \frac{\hslash G}{{\lambda }^2}=\frac{{\ell }_p^2}{{\lambda }^2}.$$

$$N{\alpha }_g\equiv N\frac{\hslash G}{{\lambda }^2}\simeq N\frac{{\ell }_p^2}{R^2}.$$

$$E_g\simeq \frac{G{M}^2}{R}.$$

$$E_g\simeq N\frac{\hslash }{\lambda }\simeq N\frac{\hslash }{R}.$$

$$M\simeq \sqrt{N}{m}_p,$$

$$r_s\simeq \sqrt{N}{\ell }_p.$$

$$N{\alpha }_g\simeq 1.$$

$$\hslash N{\alpha }_g\equiv \hslash N\frac{L_p^2}{{\lambda }^2},$$

$$E_{\mathrm{esc}}=\frac{\hslash N{\alpha }_g}{R},$$

$$E_{\mathrm{esc}}=\frac{\hslash }{{\lambda }_{\mathrm{esc}}}.$$

$$E_{\mathrm{esc}}\simeq \frac{\hslash }{\sqrt{N}{\ell }_p},{\lambda }_{\mathrm{esc}}\simeq \sqrt{N}{\ell }_p.$$

$$\Gamma \simeq {\alpha }_g^2{N}^2{E}_{\mathrm{esc}}\simeq \frac{\hslash }{\sqrt{N}{\ell }_p}.$$

$$\frac{dM}{dt}=-\frac{\Gamma }{{\lambda }_{\mathrm{esc}}}\simeq -\frac{\hslash }{N{\ell }_p^2}\simeq -\frac{m_p^4}{\hslash M^2},$$

$$\frac{dN}{dt}\simeq -\frac{1}{\sqrt{N}{\ell }_p}.$$

$$T_{\mathrm{H}}\simeq \frac{\hslash }{\sqrt{N}{\ell }_p}\simeq \frac{m_p^2}{M}.$$

Full text

I Introduction

One of the most difficult and stimulating challenges the physics community has been struggling with for a long time is to understand whether the gravitational interaction has an intrinsic quantum nature and, if so, how to formulate a thorough quantum theory of gravity which avoids conceptual problems and is able to make successful predictions at any energy scale. In pursuing the aim of combining both gravitational and quantum effects, the question inevitably arises as to whether the basic principles of quantum mechanics need to be revised in the quantum gravity realm.

It is well known that one of the fundaments of quantum mechanics is the Heisenberg Uncertainty Principle (HUP)444Throughout the paper, we work with the units $c=k_{\rm B}=1$, where $k_{\rm B}$ is the Boltzmann constant, and $\hbar\neq 1$. The Planck length is defined as $\ell_{p}=\sqrt{\hbar\,G}$, while the Planck mass as $m_{p}=\hbar/\ell_{p}$. aHeisenberg:1927zz ,

[TABLE]

which can be derived from the non–vanishing commutator between the position and momentum operators $\hat{x}$ and $\hat{p}$, respectively,

[TABLE]

The above inequality asserts that, in the quantum regime, the more precisely the position (momentum) of a particle is known, the less precisely one can say what its momentum (position) is. This implies the existence of a region $\delta x\delta p$ of size $\hbar$ in the phase space, in which any physical prediction cannot be tested. In spite of this, however, no quantum limit on sharp measurements of either position or momentum separately is fixed a priori: in other terms, arbitrarily short distances may in principle be detected via arbitrarily high energy probes, and vice–versa.

The situation becomes more subtle if one tries to merge quantum and gravity effects within the same framework. In that case, indeed, several models of quantum gravity propose the existence of a minimum length at Planck scale Maggiore:1993zu that accounts for a limited resolution of spacetime. The Planck length $\ell_{p}$ thus appears as a natural threshold beyond which spacetime would no longer be smooth, but rather it would have a foamy structure due to inherent quantum fluctuations Wheeler:1955zz .

Along this line, many studies Snyder:1946qz ; Yang:1947ud ; maed ; Karolyhazy:1966zz ; Amati:1987wq ; Gross:1987kza ; Amati:1988tn ; Konishi:1989wk ; Maggiore:1993kv ; Scardigli:1995qd ; FS ; Capozziello:1999wx ; Bojowald:2011jd ; Scardigli:2003kr ; Adler:1999bu ; Adler ; Casadio:2009jc ; Casadio:2013aua ; Scardigli:2016pjs ; VAF ; Lambiase:2017adh ; Scardigli:2018jlm ; Luciano2019 have converged on the idea that the HUP should be properly modified at the quantum gravity scale, in order to accommodate the existence of such a fundamental length. In this sense, one of the most adopted generalizations of the uncertainty principle (GUP) reads

[TABLE]

where the sign $\pm$ refers to positive/negative values of the dimensionless deformation parameter $\beta$, which is assumed to be of order unity in some models of quantum gravity, and in particular in string theory Amati:1987wq ; Amati:1988tn . However, one could also investigate Eq. (3) from a phenomenological point of view, and seek experimental bounds on $\beta$ (for a recent review of the various phenomenological approaches, see, for example, Ref. Kanazawa:2019llj ). Clearly, for $\beta\hbar/m_{p}^{2}\rightarrow 0$, standard quantum mechanics is recovered, so that modifications to the HUP only become relevant at the Planck scale, as expected. Moreover, for mirror–symmetric states (i.e. $\left\langle\hat{p}\right\rangle=0$), Eq. (3) can be deduced from the modified commutator

[TABLE]

If, on the one hand, the assumption that $\beta\sim\mathcal{O}(1)$ is quite generally accepted and it has also been confirmed by achievements in contexts other than string theory Scardigli:2016pjs ; VAF ; Lambiase:2017adh ; Capozziello:1999wx ; Luciano2019 , on the other hand the problem of the sign of $\beta$ is much more debated. Although in various derivations and gedanken experiments on GUP it seems more reasonable to have a positive parameter, arguments in favor of the opposite choice are not lacking. For instance, in Ref. Jizba:2009qf it was emphasized that the GUP with $\beta<0$ would be consistent with a description in which the universe has an underlying crystal lattice–like structure. Similarly, in Ref. Ong:2018zqn a negative GUP parameter was proved to be the only setting compatible with the Chandrasekhar limit for white dwarfs. A further confirmation was given in Ref. Kanazawa:2019llj in the context of non–commutative Schwarzschild geometry.

In this connection, a clue to an answer for such a dichotomy may be found within the framework of Corpuscular Gravity (CG) originally formulated in Refs. Dvali:2011aa ; Dvali:2012en , and then revisited from a complementary point of view in Refs. Casadio:2013ulk ; Casadio:2014vja (see also Refs. Dvali:2013vxa ; Dvali:2013eja ; Dvali:2014ila ; Dvali:2015aja ; Casadio:2015bna ; Casadio:2016aum ; Casadio:2016zpl ; giusti-tesi for further developments). According to this model, black holes can be understood as a Bose–Einstein condensate of weakly interacting gravitons at the critical point of a quantum phase transition. As a consequence of this portrait, black hole characteristics get non–trivially modified; for instance, one can show the evaporation rate gains a correction of order $1/N^{3/2}$ with respect to the standard semiclassical expression, where $N\sim{(M/m_{p})}^{2}$ is the number of constituents (gravitons).

To the best of our knowledge, the GUP and CG approaches to black hole physics have been regarded as completely unrelated treatments to date; indeed, whilst corrections arising in the former case are typically assumed to be thermal, and, therefore, can be cast in the form of a shift of the Hawking temperature Adler , the effects induced by the CG description do spoil the thermality of the black hole radiation Dvali:2011aa , giving rise to seemingly different pictures.

Starting from the outlined scenario, in the present paper we provide a link between the GUP black hole thermodynamics and the corpuscular model of gravity, showing that GUP corrections to the Hawking temperature can be equivalently rephrased in terms of a non–thermal distortion of the spectrum. This allows for a straightforward comparison of GUP and CG predictions. In particular, we infer that the two approaches are consistent, provided that $\beta$ is of order unity, in agreement with string theory. We further comment on the sign of $\beta$, speculating on the possibility to obtain a negative value.

The paper is organized as follows: Section II is devoted to a review of black hole thermodynamics in the context of the GUP. We compute corrections to the Hawking temperature and the evaporation rate, both for positive and negative values of the deformation parameter $\beta$. Then, we show how to recast the ensuing shift of the temperature in terms on a non–thermal spectrum. In Section III we discuss some fundamental aspects of CG, focusing on the derivation of the formula for the emission rate. GUP and CG approaches are then compared in Section IV. By requiring that the modified expressions of the evaporation rate derived in the two frameworks are equal (at the first order), we are able to evaluate the GUP parameter $\beta$. Finally, conclusions and discussion can be found in Section V.

II Uncertainty relations and black hole thermodynamics

Let us consider a spherically symmetric black hole with mass $M$ and Schwarzschild radius $r_{s}=2MG$. Following the arguments of Refs. Scardigli:1995qd ; Adler , the Hawking temperature $T_{\rm H}$ can be derived in a heuristic way by using the HUP (1) and general properties of black holes. To this aim, let us observe that, just outside the event horizon, the position uncertainty of photons emitted by the black hole is of the order of its Schwarzschild radius, i.e. $\delta x\simeq\mu\hskip 0.56905ptr_{s}$, where the constant $\mu$ is of order of unity and will be fixed below. From Eq. (1), the corresponding momentum uncertainty is given by

[TABLE]

which also represents the characteristic energy of the emitted photons, since $\delta p\simeq p=E$. According to the equipartition theorem, this can be now identified with the temperature $T$ of the ensemble of photons,

[TABLE]

which agrees with the Hawking temperature,

[TABLE]

provided that $\mu=2\pi$.

Therefore, on the basis of the HUP and thermodynamic consistency, we have recovered the standard Hawking formula Eq. (7) for the temperature of the radiation emitted by the black hole.

Now, it is well known that black holes with temperature greater than the background temperature (about $2.7\,\mathrm{K}$ for the present universe) shrink over time by radiating energy in the form of photons and other ordinary particles. In certain conditions Page:1976df , however, it is reasonable to assume that the evaporation is dominated by photon emission. In this case, we can exploit the Stefan–Boltzmann law to estimate the radiated power $P$ as

[TABLE]

where $A_{s}=4\pi\hskip 0.56905ptr_{s}^{2}$ is the black hole sphere surface area at Schwarzschild radius $r_{s}$, $\sigma=\pi^{2}/60\hbar^{3}$ is the Stefan–Boltzmann constant, and we have assumed for simplicity the black hole to be a perfect blackbody, i.e. $\varepsilon\simeq 1$.

Using Eqs. (6) and (8), the black hole energy loss can be easily evaluated as a function of time, yielding

[TABLE]

Therefore, the evaporation process leads black holes to vanish entirely with both the temperature (6) and emission rate (9) blowing up as the mass decreases.

The above results have been derived starting from the HUP in Eq. (1). We now wish to follow a similar procedure by resorting to Eq. (3), so as to realize to what extent the GUP affects the black hole thermodynamics. In this case, solving Eq. (3) with respect to the momentum uncertainty $\delta p$ and setting again $\delta x$ of the order of the Schwarzschild radius, we obtain the following expression for the modified Hawking temperature

[TABLE]

In the semiclassical limit $\sqrt{|\beta|}m_{p}/M\ll 1$, this agrees with the standard Hawking result in Eq. (6), provided that the negative sign in front of the square root is chosen, whereas the positive sign has no physical meaning. Similarly, the emission rate in Eq. (9) is modified as

[TABLE]

In what follows, the implications of Eqs. (10) and (11) will be discussed separately for the cases of $\beta>0$ and $\beta<0$.

II.1 GUP with $\beta>0$

Let us start by analyzing the most common setting of GUP with positive deformation parameter. In this case, from Eq. (10) it is easy to see that the GUP naturally introduces a minimum size allowed for black holes: for $M<\sqrt{\beta}\,m_{p}$ (i.e. $r_{s}<2\sqrt{\beta}\,\ell_{p}$), indeed, the temperature would become complex. This means that the evaporation process should stop at $M\sim\sqrt{\beta}\,m_{p}$ ($r_{s}\sim\sqrt{\beta}\,\ell_{p}$), thus leading to an inert remnant with finite temperature and size Adler , in contrast with predictions of ordinary black hole thermodynamics. We remark that the idea of black hole remnants dates back to Aharonov–Casher–Nussinov, who first addressed the issue in the context of the black hole unitarity puzzle Aharonov:1987tp .

Similarly, concerning the modified emission rate Eq. (11), we find that it is finite at the endpoint of black hole evaporation $M\sim\sqrt{\beta}\,m_{p}$, whereas the corresponding HUP result (9) diverges at the endpoint when $M=0$. Again, we stress that the standard behavior in Eq. (9) is recovered for $\sqrt{\beta}m_{p}/M\ll 1$.

II.2 GUP with $\beta<0$

Although the GUP with $\beta>0$ cures the undesired infinite final temperature predicted by Hawking’s formula (7) giving rise to black hole remnants, it would create several complications, such as the entropy/information problem Mathur:2009hf ; Bekenstein:1993dz , or the removal of the Chandrasekhar limit Ong:2018zqn . The latter prediction, in particular, would allow white dwarfs to be arbitrarily large, a result that is at odds with astrophysical observations. An elegant way to overcome these ambiguities was proposed in Ref. Ong:2018zqn , where it was shown that both the infinities in black hole and white dwarf physics can be avoided by choosing a negative deformation parameter in Eq. (3). A similar scenario had previously been encountered in Ref. Jizba:2009qf in the context of GUP in a crystal-like universe with lattice spacing of the order of Planck length.

Let us then consider the case $\beta<0$. With this setting, from Eqs. (9) and (11) we obtain that both the modified temperature and emission rate are well defined even for $M<\sqrt{|\beta|}\,m_{p}$. For a sufficiently small $M$, in particular, the modified temperature in Eq. (10) can be approximated as

[TABLE]

Although no lower bound on the black hole size arises in this framework, the Hawking temperature remains finite as the black hole evaporates to zero mass. From Eq. (12) we also deduce that the bound on the Hawking temperature is independent of the initial black hole mass.

II.3 GUP induced non–thermal corrections

So far, we have assumed that the correction induced by the GUP has a thermal character, and, therefore, it can be cast in the form of a shift of the Hawking temperature which does not affect the overall thermality of the Planck spectrum of the black hole, i.e.

[TABLE]

Here, we show that such a viewpoint can be safely reversed. In other terms, starting from the above results, we reinterpret GUP effects on the black hole radiation as corrections that spoil the thermal nature of the Planck distribution, while leaving the Hawking formula (7) unchanged.

To this aim, let us observe that, by inserting the modified Hawking temperature (10) in Eq. (13) and expanding for $\sqrt{|\beta|}m_{p}/M\ll 1,$ it follows that

[TABLE]

where $\mp$ corresponds to positive or negative deformation parameters, respectively, and we have defined

[TABLE]

Equation (14) does provide the expected result, since it states that the thermal distribution (13) with a shifted temperature $T=T_{GUP}$ can be rewritten as the standard Planck factor with $T=T_{\rm H}$, plus non-thermal corrections depending on the GUP parameter $\beta$.

In what follows, we shall essentially refer to this alternative interpretation of the GUP-induced correction. Notice that, although it is quite unusual in the context of black hole physics, a similar result has been recently derived for the Unruh radiation within the framework of QFT with modified commutation relations (see Ref. Scardigli:2018jlm for details). In light of this, it is thus reasonable to expect that a non-thermal distortion of the spectrum may be also derived via a more rigorous QFT treatment of the black hole thermodynamics in the presence of the GUP. We further remark that, by considering the non–thermal distribution (14) for the black hole radiation, the Stephan–Boltzmann law (8) should be accordingly modified, and this must be consistent with the relation (11) for the evaporation rate.

In the next Section, it will be shown how the above interpretation is conceptually fundamental to establish a correspondence between the GUP and CG approaches to black hole physics.

III Corpuscular black holes

It is well known that, in general relativity, black holes are solutions of the Einstein field equations that are fully characterized by means of three parameters only (no–hair theorem) hawk : mass, charge and angular momentum. This suggests that, classically, a black hole can carry a little amount of information. Conversely, from the quantum mechanical point of view, a black hole possesses a huge number of states due to its extremely large entropy, and this is the cause that allows for the emergence of the information paradox Mathur:2009hf . Indeed, if we perform the classical limit starting from such a quantum picture, it appears that the entropy is infinite (i.e. the number of states is infinite), but this is in contradiction with the fact that a classical black hole is featureless, according to what discussed above Dvali:2018tno .

The aforementioned paradox can be elegantly solved within the framework of Corpuscular Gravity (CG), which was introduced in Refs. Dvali:2011aa ; Dvali:2012en , and revised from a different perspective in Refs. Casadio:2013ulk ; Casadio:2014vja ; Dvali:2013vxa ; Dvali:2013eja ; Dvali:2014ila ; Dvali:2015aja ; Casadio:2015bna ; Casadio:2016aum ; Casadio:2016zpl ; giusti-tesi . According to this picture, indeed, black holes can be conceived as Bose–Einstein condensates of $N$ interacting and non–propagating longitudinal gravitons, and thus as intrinsically quantum objects555Let us recall that a Bose–Einstein condensate of attractive bosons and their phase transitions were firstly studied in Ref. Bogolyubov:1947zz in the context of condensed matter physics.. As argued in Ref. Dvali:2015aja , the corpuscular nature of the gravitational interaction induces non–thermal corrections to the black hole radiation, which scale as $1/N.$ The thermal Hawking radiation of the semiclassical picture is then consistently recovered in the limit $N\rightarrow\infty.$

In order to better understand such a black hole quantum’s portrait and make a comparison with the GUP predictions, let us review the most relevant properties of the corpuscular theory of gravity.

III.1 Bose-Einstein condensate of gravitons

Let us consider a Bose–Einstein condensate of total mass $M$ and radius $R$, which is made up of $N$ weakly interacting gravitons. At low energy, we can define a quantum gravitational self–coupling for each single graviton of wavelength $\lambda$ as follows Dvali:2011aa

[TABLE]

One of the main features of a Bose–Einstein condensate is that, due to the interaction, its constituents acquire a collective behavior, so that their wavelengths get increasingly larger and their masses smaller; strictly speaking, the constituents become softer bosons. In particular, most of the gravitons composing the gravitational system will have a wavelength of the order $\lambda\sim R$, namely of the order of the size of the system itself666More precisely, the condensate is also made up of gravitons with shorter wavelengths, $\lambda<R.$ However, all physical quantities will be mostly determined by those gravitons that possess a wavelength of the order $\lambda\sim R$, since the contribution arising from hard gravitons turns out to be exponentially suppressed and can thus be neglected Dvali:2011aa .. Hence, similarly to Eq. (16), it is possible to define a collective quantum coupling as

[TABLE]

We now seek the relation that links the total mass of a Bose–Einstein condensate and its radius to the number $N$ of quanta composing the system. By performing a standard computation, one can show that the gravitational binding energy of the system is given by

[TABLE]

On the other hand, from a purely quantum point of view, the binding energy can be expressed as the sum of the energies associated to each single graviton, i.e.777Note that, in Eq. (19), we are allowed to employ the usual de Broglie relation because we are considering low energy weakly interacting gravitons with large wavelengths. Quantum gravity effects only emerge as a collective feature of the whole gravitational system.

[TABLE]

Therefore, by comparing Eqs. (18) and (19), we obtain

[TABLE]

which also implies for the Schwarzschild radius

[TABLE]

By assuming that the size of the condensate is $R\sim r_{s}$ (i.e. the overall gravitational system is a black hole) and using the expression in Eq. (21) for the Schwarzschild radius, we notice that the collective quantum coupling defined in Eq. (17) is always of order unity in the case of a black hole

[TABLE]

In condensed matter physics, it is well known that the inequality $N\alpha_{g}<1$ corresponds to a phase in which a Bose–Einstein condensate is weakly interacting. On the other hand, the equality $N\alpha_{g}=1$ represents a critical point at which a phase transition occurs, thus letting the condensate become strongly interacting, whereas for $N\alpha_{g}>1$ it is possible to observe only a strongly interacting phase Bogolyubov:1947zz . Thus, in this quantum corpuscular picture, a black hole can be defined as a Bose–Einstein condensate of gravitons stuck at the critical point of a quantum phase transition Dvali:2012en . This is a crucial property which ensures that a gravitational bound system can exist for any $N$ or, in other words, that such a graviton condensate is self–sustained.

III.2 Thermodynamic properties of corpuscular black holes

We now analyze some thermodynamic aspects of quantum corpuscular black holes, and in particular we show that gravitons can escape from the considered system. Such a phenomenon represents the corpuscular counterpart of the black hole radiation emission Dvali:2011aa .

First of all, we need to compute the probability for a graviton to escape from a gravitational bound state, namely we have to determine the so–called escape energy and escape wavelength of a single graviton. To this aim, observe that, for $N$ weakly interacting quanta composing a condensate of radius $R$ and mass $M$, a quantum gravitational interaction strength can be defined as Dvali:2011aa

[TABLE]

so that each graviton is subject to the following binding potential

[TABLE]

which is the threshold to exceed in order to escape. The corresponding escape wavelength is defined as

[TABLE]

If we now employ Eqs. (21) and (22) for the case of a black hole, we obtain

[TABLE]

This means that, although $N$ gravitons of wavelength $\lambda\sim\leavevmode\nobreak\ \sqrt{N}\ell_{p}$ can form a gravitational bound state, at the same time a depletion process is present, which is traduced in a leakage of the constituents of the condensate for any $N$. Clearly, this is related to the fact that $\lambda_{\mathrm{esc}}$ coincides with the wavelength of each graviton belonging to the condensate, that is, $\sqrt{N}\ell_{p}.$ In this sense, a black hole is a leaky Bose–Einstein condensate stuck at its critical point.

In terms of scattering amplitudes, the above picture can be regarded as a $2\rightarrow 2$ scattering process, in which one of the two gravitons is energetic enough to be able to exceed the threshold given by $E_{\rm esc}$.

We can also obtain an estimation for the depletion rate $\Gamma$ of such a process. As usual, this should be given by a product involving the squared coupling constant $\alpha_{g}^{2}$, the characteristic energy scale of the process $E_{\rm esc}$ and a combinatoric factor $N(N-1)$, which can be approximated by $N^{2}$ for a very large number of constituents Dvali:2011aa , i.e.

[TABLE]

From the above relation, we can easily obtain the corresponding time scale of the considered process, which is given by $\Delta t=\hbar/\Gamma\simeq\sqrt{N}\ell_{p}.$

On the other hand, Eq. (27) allows us to infer the mass decrease over time of the condensate, i.e.

[TABLE]

which can be cast in terms of the rate of emitted gravitons by use of Eq. (20),

[TABLE]

We stress that, up to the factor $1/[60(16)^{2}\pi]$, Eq. (28) reproduces the thermal evaporation rate of a black hole in Eq. (9), assuming the Hawking temperature in the corpuscular model to be given by Dvali:2011aa

[TABLE]

We can also estimate the lifetime $\tau$ of a quantum black hole; indeed, by imposing $dN/dt\sim-N/\tau,$ we get

[TABLE]

For the sake of completeness, we also mention that the gravitational entropy $S$ of a corpuscular black hole takes the simple form $S\sim N.$ As a consequence, the number of states a black hole can use to store information is exponentially large, and it is given by $e^{N}$ Dvali:2011aa .

III.3 Non-thermal features of corpuscular black holes

We have seen that the black hole quantum $N$-portrait manages to reproduce the semiclassical result, according to which a black hole emits a thermal radiation with temperature given by the Hawking formula (7). However, from a more scrupulous investigation, one can see that such a result holds true only to the leading order, since in general there will be non–thermal corrections to the spectrum which scale as negative powers of the number of gravitons (see Refs. Dvali:2011aa ; Dvali:2012en ; Dvali:2015aja for a more detailed discussion on the non-thermal nature of the black hole radiation in CG and the ensuing resolution of the information loss paradox).

In this connection, notice that, in the computation of the depletion rate $\Gamma$ in Eq. (27), we have only considered the simplest kind of interaction (i.e. a tree-level scattering diagram with two vertices); nevertheless, one expects that even higher-order processes provide $\Gamma$ with contributions that induce gravitons to escape. For instance, the next relevant $2\rightarrow 2$ scattering process would possess three vertices, thus contributing with terms proportional to $\alpha_{g}^{3}.$ Therefore, up to first order corrections, the depletion rate would take the form

[TABLE]

As for the leading order in Eq. (28), the mass decrease of the black–hole can be now estimated from the modified depletion rate (32), obtaining Dvali:2012en

[TABLE]

These last findings strengthen the awareness that, in the CG theory, black holes can be uniquely described through the variable $N$. Indeed, once the number of gravitons composing the condensate is known, we can determine all the macroscopic physical quantities (i.e. the mass, the radius, the evaporation rate, etc.). Clearly, this feature appears to hold also for the corrections to the semiclassical formulas, as Eqs. (32) and (33) denote.

IV Consistency between GUP and corpuscular gravity

In the previous Sections, the evaporation rate of a black hole has been computed within both the GUP and CG frameworks. Here, we compare the two expressions: as it will be shown, this allows us to set the value of the GUP deformation parameter $\beta$ for which the GUP and CG treatments are consistent.

For this purpose, we consider the GUP–modified expressions of the emission rate in Eq. (11) expanded up to the order $\mathcal{O}(1/M^{4})$, and the CG outcome in Eq. (33) with the proper coefficients restored. This yields

[TABLE]

and

[TABLE]

where we recall that the sign $\pm$ in Eq. (34) corresponds to a positive/negative value of the deformation parameter $\beta$. Note that, at least up to the leading order, the GUP- and CG-induced corrections turn out to have the same functional dependence on the black hole mass. Furthermore, since the coefficient in front of the correction in Eq. (35) is predicted to be of order unity Dvali:2011aa ; Dvali:2012en , numerical consistency between the two expressions automatically leads to

[TABLE]

which is in agreement with the predictions of other models of quantum gravity, and in particular with string theory Amati:1987wq ; Amati:1988tn . Therefore, in spite of their completely different underlying backgrounds, the GUP and CG approaches are found to be compatible with each other.

However, the result (36) does not give any specific information about the sign of $\beta$. Since a full-fledged analytic derivation of Eq. (35) including also higher-order scattering processes is still lacking, a definitive conclusion on this issue cannot be reached. On the one hand, relying on basic considerations on the nature of the scattering amplitudes, we would naively expect the second term in Eq. (35) to contribute with the same sign as the leading order term, because we are only adding higher order diagrams describing the probability for a graviton to escape from the condensate. This would yield a positive value for the deformation parameter.

On the other hand, there are different claims which assert that the first order correction in the depletion rate (32) should be opposite to the leading order term, in such a way to slightly decrease the evaporation rate of the black hole given in Eq. (35). This was shown, for example, within the framework of Horizon Quantum Mechanics horizon-QM , where the depletion rate up to first order correction takes the form Casadio:2015bna

[TABLE]

which would imply the following formula for the evaporation rate

[TABLE]

where $N_{H}\equiv\sqrt{3}/\sqrt{\pi^{2}-6\zeta(3)}\simeq 1.06$ and $\zeta(x)$ is the Riemann zeta function. Note also that the constant factor in Eq. (38) is given by $3\hbar\gamma^{2}N_{H}^{2}\left(6\zeta(3)-\pi^{4}/15\right)\simeq 2.7\gamma^{2}$ and $\gamma$ may be of order one Casadio:2015bna . With such a setting, the comparison of Eqs. (34) and (38) would further confirm the result in Eq. (36), but it would lead to a negative value for the deformation parameter,

[TABLE]

Moreover, we remark that positive corrections to the evaporation rate of a black hole are required by the principle of energy conservation Casadio:1997yv ; Alberghi:2006km , thus enforcing the validity of Eq. (39). A more detailed discussion on the physical meaning and implications of such a result can be found in the Conclusions.

V Conclusions

In this paper, we have analyzed to what extent the black hole thermodynamics gets modified both in the presence of a generalized uncertainty principle (GUP) and in the corpuscular gravity (CG) theory. After remarking that, in both contexts, corrections to the standard semiclassical results can be viewed as originating from non-thermal deviations of the Hawking radiation, we have focused on the computation of the evaporation rate of a black hole. By comparing the expressions derived within the two frameworks, we have finally managed to estimate the GUP deformation parameter $\beta$. Specifically, in order for the GUP and CG predictions to be consistent, we have found that $\beta$ must be of order unity. This is certainly a non-trivial result, since it states that the GUP and CG approaches to black hole physics are consistent with each other, and also with some other models of quantum gravity such as string theory.

Furthermore, we have speculated on the sign of $\beta$. Although on this matter we are still far from the definitive solution, a preliminary analytic evaluation of the evaporation rate within the framework of Horizon Quantum Mechanics and some considerations related to the conservation of energy, suggest that the most plausible picture is the one with a negative deformation parameter, $\beta<0$.

In this connection, we emphasize that a similar result would not be surprising in the context of a corpuscular (i.e discrete) description of black holes; in Ref. Jizba:2009qf , indeed, it was shown that a GUP with $\beta<0$ can be derived assuming that the universe has an underlying crystal lattice-like structure. Moreover, in Ref. Ong:2018zqn it was found that $\beta<0$ is the only choice compatible with the Chandrasekhar limit; in other terms, the GUP with positive deformation parameter would allow arbitrarily large white dwarfs to exist, a result that clashes with current astrophysical observations. In Section II we have also seen that, in the case $\beta>0$, black holes would not entirely evaporate. However, if on the one hand black hole remnants may be viewed as potential candidates for dark matter Chen:2004rha , on the other hand their existence would be rather problematic for the entropy/information paradox Bekenstein:1993dz .

Apart from these supporting arguments, it also worth remarking that, if $\beta<0$, then from Eq. (3) there would be a maximum value of $\delta p$ around the Planck scale for which $\delta x\hskip 0.85358pt\delta p=0$, i.e. the quantum uncertainty would be completely erased. Therefore, it would seem that physics would can become classical again at Planck scale. The possibility of a classical Planckian regime has been already addressed in Ref. Hossenfelder:2012uy , by regarding $\hbar$ as a dynamical field that vanishes in the Planckian limit.

Given the absolute lack of knowledge about physics at Planck scale, it is clear that in principle all possible scenarios should be contemplated in order to achieve a better understanding of how gravity and quantum effects behave when combined together. For example, one cannot exclude a priori the possibility that $\beta$ is a function rather than a pure number. This has been originally suggested in Ref. Chen:2014bva as condition for black hole complementarity principle to always hold. A similar result has been recently recovered in Ref. Vagenas:2018zoz , where, by conjecturing the equality between the GUP–deformed black hole temperature of a Schwarzschild black hole and the modified Hawking temperature of a quantum-corrected Schwarzschild black hole, it has been obtained a GUP parameter depending on the ratio $m_{p}/M$.

In light of the above discussion, the present investigation should thus be regarded as a further attempt to gain information about the GUP black hole physics through a connection with an intrinsically corpuscular description of gravity. More work is inevitably required to provide a definite answer about the sign of the GUP deformation parameter $\beta$ inprepa .

Acknowledgements.

The authors would like to thank Roberto Casadio and Andrea Giusti for helpful conversations.