# Generalized uncertainty principle and corpuscular gravity

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## 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 compute such characteristic thermodynamic quantities as the temperature and the evaporation rate of a black hole. By comparing the results obtained in the two scenarios, we are able to estimate the GUP deformation parameter \(\beta \), which turns out to be of order 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.

## 1 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.

^{1}[1],

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 [2] 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 [3].

*deformation parameter*\(\beta \), which is assumed to be of order unity in some models of quantum gravity, and in particular in string theory [8, 10]. 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. [28]). 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

^{2}Further confirmations were given starting from a non-commutative Schwarzschild geometry [28], or a deformed static Schwarzschild metric [32].

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. [33, 34], and then revisited from a complementary point of view in Refs. [35, 36] (see also Refs. [37, 38, 39, 40, 41, 42, 43] 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.

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. Specifically, by comparing the evaporation rate computed via the two different approaches, we infer that they are consistent, provided that \(\beta \) is of order unity, in agreement with some models of 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: Sect. 2 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 \(\beta \). In Sect. 3 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 Sect. 4. By requiring that the modified expressions of the evaporation rate derived in the two frameworks are equal (at the leading order), we are able to evaluate the GUP parameter \(\beta \). Finally, conclusions can be found in Sect. 5.

## 2 Uncertainty relations and black hole thermodynamics

*M*and Schwarzschild radius \(r_s=2MG\). Following the arguments of Refs. [13, 19], the Hawking temperature \(T_\mathrm{H}\) 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. \(\delta x\simeq \mu r_s\), where the constant \(\mu \) is of order of unity and will be fixed below. From Eq. (1), the corresponding momentum uncertainty is given by

*T*of the ensemble of photons,

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

*P*as

The above results have been derived starting from the HUP (1). Let us now perform similar calculations by resorting to the GUP in Eq. (3), so as to realize to what extent the GUP affects the black hole thermodynamics.

### 2.1 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, in contrast with predictions of ordinary black hole thermodynamics. Remarkably, one can verify that the evaporation process should stop at \(M\sim \sqrt{\beta }\,m_p\) by observing that the specific heat of the black hole *dM* / *dT* is negative under GUP modification and it vanishes as \(M\rightarrow \sqrt{\beta }\,m_p\) [45], thus leading to an *inert* remnant with finite temperature and size. On the other hand, different models predict a change of sign in the specific heat as the mass is wiped out; in that case black hole remnants may undergo a phase transition which renders them unstable. Notice also 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 [46].

Similarly, concerning the modified emission rate (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 \(M=0\).

### 2.2 GUP with \(\beta <0\)

Although the GUP with \(\beta >0\) cures the undesired infinite final temperature predicted by Hawking’s formula (7), it would create several complications, such as the entropy/information problem [47, 48], or the removal of the Chandrasekhar limit [30]. 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. [30], 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. [29] in the context of GUP in a crystal-like universe with lattice spacing of the order of Planck length.

*M*, in particular, the modified temperature in Eq. (10) can be approximated as

## 3 Corpuscular black holes

In the Corpuscular Gravity picture (which was first introduced in Refs. [33, 34] and then revised from a different perspective in Refs. [35, 36, 37, 38, 39, 40, 41, 42, 43]) black holes can be conceived as Bose–Einstein condensates of *N* interacting and non-propagating longitudinal gravitons, and thus as intrinsically quantum objects. Let us recall that a Bose–Einstein condensate of attractive bosons and their phase transitions were firstly studied in Ref. [49] in the context of condensed matter physics.

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.

### 3.1 Bose–Einstein condensate of gravitons

*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 [33]

*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 itself.

^{3}Hence, similarly to Eq. (13), it is possible to define a collective quantum coupling as

*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

^{4}

### 3.2 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 [33].

*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 [33]

*N*gravitons of wavelength \(\lambda \sim ~\sqrt{N}\ell _p\) 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 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 \(E_\mathrm{esc}\).

*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 higher-order corrections which scale as negative powers of the number of gravitons [33, 34, 40].

*dM*/

*dt*in order to make a quantitative comparison with the corresponding GUP result.

## 4 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.

However, the result (32) does not give any specific information about the sign of \(\beta \). Since a full-fledged analytic derivation of Eq. (31) 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 correction in Eq. (31) to contribute with the same sign as the zeroth-order term, since 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.

## 5 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. In particular, we have focused on the computation of the temperature and 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 \(\beta \). The obtained result shows that, in order for the GUP and CG predictions to be consistent, \(\beta \) must be of order unity.

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. [29], indeed, it was shown that a GUP with \(\beta <0\) can be derived assuming that the universe has an underlying crystal lattice-like structure. In Sect. 2 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 [54, 55], on the other hand their existence would be rather problematic for the entropy/information paradox [48].

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 \(\delta x\delta p=0\), i.e. the quantum uncertainty would be completely erased. Therefore, it would seem that physics can become classical again at Planck scale. The possibility of a classical Planckian regime has been already discussed in Ref. [29] in connection with the doubly special relativity model [56, 57, 58, 59] and with the deterministic interpretation of quantum mechanics [60]. Further arguments have been given in Ref. [61] 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. [62] as condition for black hole complementarity principle to always hold. A similar result has been recently recovered in Ref. [23], 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 \(\beta \).

## Footnotes

- 1.
Throughout the paper, we work with the units \(c=k_\mathrm{B}=1\), where \(k_\mathrm{B}\) is the Boltzmann constant, and \(\hbar \ne 1\). The Planck length is defined as \(\ell _p=\sqrt{\hbar \,G}\), while the Planck mass as \(m_p=\hbar /\ell _p\).

- 2.
It must be pointed out, however, that in Ref. [31] it was argued that this issue can be fixed also for the case \(\beta >0\), provided that a non-vanishing cosmological constant is included in the treatment.

- 3.
More 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 [33]. - 4.
Note that, in Eq. (16), 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.

## Notes

### Acknowledgements

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

## References

- 1.W. Heisenberg, Z. Phys.
**43**, 172 (1927)CrossRefADSGoogle Scholar - 2.M. Maggiore, Phys. Rev. D
**49**, 5182 (1994)CrossRefADSMathSciNetGoogle Scholar - 3.J.A. Wheeler, Phys. Rev.
**97**, 511 (1955)CrossRefADSMathSciNetGoogle Scholar - 4.H.S. Snyder, Phys. Rev.
**71**, 38 (1947)CrossRefADSMathSciNetGoogle Scholar - 5.C.N. Yang, Phys. Rev.
**72**, 874 (1947)CrossRefADSMathSciNetGoogle Scholar - 6.C.A. Mead, Phys. Rev.
**135**, B849 (1964)CrossRefADSGoogle Scholar - 7.F. Karolyhazy, Nuovo Cim. A
**42**, 390 (1966)CrossRefADSGoogle Scholar - 8.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B
**197**, 81 (1987)CrossRefADSGoogle Scholar - 9.D.J. Gross, P.F. Mende, Phys. Lett. B
**197**, 129 (1987)CrossRefADSMathSciNetGoogle Scholar - 10.D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B
**216**, 41 (1989)CrossRefADSGoogle Scholar - 11.K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B
**234**, 276 (1990)CrossRefADSMathSciNetGoogle Scholar - 12.M. Maggiore, Phys. Lett. B
**319**, 83 (1993)CrossRefADSMathSciNetGoogle Scholar - 13.F. Scardigli, Nuovo Cim. B
**110**, 1029 (1995)CrossRefADSGoogle Scholar - 14.F. Scardigli, Phys. Lett. B
**452**, 39 (1999)CrossRefADSGoogle Scholar - 15.S. Capozziello, G. Lambiase, G. Scarpetta, Int. J. Theor. Phys.
**39**, 15 (2000)CrossRefGoogle Scholar - 16.M. Bojowald, A. Kempf, Phys. Rev. D
**86**, 085017 (2012)CrossRefADSGoogle Scholar - 17.F. Scardigli, R. Casadio, Class. Quant. Grav.
**20**, 3915 (2003)CrossRefADSGoogle Scholar - 18.R.J. Adler, D.I. Santiago, Mod. Phys. Lett. A
**14**, 1371 (1999)CrossRefADSGoogle Scholar - 19.R.J. Adler, P. Chen, D.I. Santiago, Gen. Rel. Grav.
**33**, 2101 (2001)CrossRefADSGoogle Scholar - 20.R. Casadio, R. Garattini, F. Scardigli, Phys. Lett. B
**679**, 156 (2009)CrossRefADSMathSciNetGoogle Scholar - 21.R. Casadio, F. Scardigli, Eur. Phys. J. C
**74**, 2685 (2014)CrossRefADSGoogle Scholar - 22.F. Scardigli, G. Lambiase, E. Vagenas, Phys. Lett. B
**767**, 242 (2017)CrossRefADSMathSciNetGoogle Scholar - 23.E.C. Vagenas, S.M. Alsaleh, A. Farag, EPL
**120**, 40001 (2017)CrossRefADSGoogle Scholar - 24.G. Lambiase, F. Scardigli, Phys. Rev. D
**97**, 075003 (2018)CrossRefADSMathSciNetGoogle Scholar - 25.F. Scardigli, M. Blasone, G. Luciano, R. Casadio, Eur. Phys. J. C
**78**, 728 (2018)CrossRefADSGoogle Scholar - 26.G.G. Luciano, L. Petruzziello, Eur. Phys. J. C
**79**, 283 (2019)CrossRefADSGoogle Scholar - 27.M. Blasone, G. Lambiase, G.G. Luciano, L. Petruzziello, F. Scardigli. arXiv:1902.02414 [hep-th]
- 28.T. Kanazawa, G. Lambiase, G. Vilasi, A. Yoshioka, Eur. Phys. J. C
**79**, 95 (2019)CrossRefADSGoogle Scholar - 29.P. Jizba, H. Kleinert, F. Scardigli, Phys. Rev. D
**81**, 084030 (2010)CrossRefADSGoogle Scholar - 30.Y.C. Ong, JCAP
**1809**, 015 (2018)CrossRefADSGoogle Scholar - 31.Y.C. Ong, Y. Yao, Phys. Rev. D
**98**, 126018 (2018)CrossRefADSGoogle Scholar - 32.F. Scardigli, R. Casadio, Eur. Phys. J. C
**75**, 425 (2015)CrossRefADSGoogle Scholar - 33.G. Dvali, C. Gomez, Fortsch. Phys.
**61**, 742 (2013)CrossRefADSGoogle Scholar - 34.G. Dvali, C. Gomez, Eur. Phys. J. C
**74**, 2752 (2014)CrossRefADSGoogle Scholar - 35.R. Casadio, A. Orlandi, JHEP
**1308**, 025 (2013)CrossRefADSGoogle Scholar - 36.R. Casadio, A. Giugno, O. Micu, A. Orlandi, Phys. Rev. D
**90**, 084040 (2014)CrossRefADSGoogle Scholar - 37.G. Dvali, D. Flassig, C. Gomez, A. Pritzel, N. Wintergerst, Phys. Rev. D
**88**, 124041 (2013)CrossRefADSGoogle Scholar - 38.G. Dvali, C. Gomez, JCAP
**1401**, 023 (2014)CrossRefADSGoogle Scholar - 39.G. Dvali, C. Gomez, R.S. Isermann, D. Lüst, S. Stieberger, Nucl. Phys. B
**893**, 187 (2015)CrossRefADSGoogle Scholar - 40.G. Dvali, Fortsch. Phys.
**64**, 106 (2016)CrossRefADSMathSciNetGoogle Scholar - 41.R. Casadio, A. Giugno, A. Orlandi, Phys. Rev. D
**91**, 124069 (2015)CrossRefADSGoogle Scholar - 42.R. Casadio, A. Giugno, A. Giusti, Phys. Lett. B
**763**, 337 (2016)CrossRefADSGoogle Scholar - 43.R. Casadio, R. da Rocha, Phys. Lett. B
**763**, 434 (2016)CrossRefADSGoogle Scholar - 44.D.N. Page, Phys. Rev. D
**13**, 198 (1976)CrossRefADSGoogle Scholar - 45.P. Chen, Y.C. Ong, Dh Yeom, Phys. Rept
**603**, 1 (2015)CrossRefADSGoogle Scholar - 46.Y. Aharonov, A. Casher, S. Nussinov, Phys. Lett. B
**191**, 51 (1987)CrossRefADSGoogle Scholar - 47.S.D. Mathur, Class. Quant. Grav.
**26**, 224001 (2009)CrossRefADSGoogle Scholar - 48.J.D. Bekenstein, Phys. Rev. D
**49**, 1912 (1994)CrossRefADSGoogle Scholar - 49.N.N. Bogolyubov, J. Phys. (USSR)
**11**, 23 (1947) [Izv. Akad. Nauk Ser. Fiz.**11**, 77 (1947)]Google Scholar - 50.R. Casadio. arXiv:1305.3195 [gr-qc]
- 51.R. Casadio, A. Giugno, A. Giusti, Gen. Rel. Grav.
**49**, 32 (2017)CrossRefADSGoogle Scholar - 52.R. Casadio, B. Harms, Y. Leblanc, Phys. Rev. D
**58**, 044014 (1998)CrossRefADSMathSciNetGoogle Scholar - 53.G.L. Alberghi, R. Casadio, A. Tronconi, J. Phys. G
**34**, 767 (2007)CrossRefADSGoogle Scholar - 54.P. Chen, Mod. Phys. Lett. A
**19**, 1047 (2004)CrossRefADSGoogle Scholar - 55.P. Chen, New Astron. Rev
**49**, 233 (2005)CrossRefADSGoogle Scholar - 56.G. Amelino-Camelia, Int. J. Mod. Phys. D
**11**, 35 (2002)CrossRefADSMathSciNetGoogle Scholar - 57.J. Magueijo, L. Smolin, Phys. Rev. Lett.
**88**, 190403 (2002)CrossRefADSGoogle Scholar - 58.G. Amelino-Camelia, Nature
**418**, 34 (2002)CrossRefADSGoogle Scholar - 59.J. Magueijo, L. Smolin, Phys. Rev. D
**67**, 044017 (2003)CrossRefADSMathSciNetGoogle Scholar - 60.G. ’t Hooft, The cellular automaton interpretation of quantum mechanics. Fundam. Theor. Phys.
**185**, 49–55 (2006)CrossRefGoogle Scholar - 61.S. Hossenfelder, Phys. Lett. B
**725**, 473 (2013)CrossRefADSMathSciNetGoogle Scholar - 62.P. Chen, Y.C. Ong, Dh Yeom, JHEP
**1412**, 021 (2014)CrossRefADSGoogle Scholar

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