Physics Letters B 635 (2006) 232–234

www.elsevier.com/locate/physletb

Black hole remnants due to GUP or quantum gravity?

Michael Maziashvili

Department of Physics, Tbilisi State University, 3 Chavchavadze Ave., Tbilisi 0128, GeorgiaInstitute of High Energy Physics and Informatization, 9 University Str., Tbilisi 0186, Georgia

Received 5 December 2005; received in revised form 20 February 2006; accepted 8 March 2006

Available online 13 March 2006

Editor: N. Glover

Abstract

Based on the micro-black hole gedanken experiment as well as on general considerations of quantum mechanics and gravity the generalizeduncertainty principle (GUP) is analyzed by using the running Newton constant. The result is used to decide between the GUP and quantumgravitational effects as a possible mechanism leading to the black hole remnants of about Planck mass.© 2006 Elsevier B.V. All rights reserved.

PACS: 03.65.-w; 04.60.-m; 04.70.-s; 04.70.Dy

The cogent argument for the black hole to evaporate entirelyis that there are no evident symmetry or quantum number pre-venting it. Nevertheless, the heuristic derivation of the Hawkingtemperature with the use of GUP prevents a black hole fromcomplete evaporation, just like the prevention of hydrogen atomfrom collapse by the uncertainty principle [1]. The general-ized uncertainty relation takes into account the gravitationalinteraction of the photon and the particle being observed. Thisconsideration relies on classical gravitational theory [2,3]. Thequantum corrected Schwarzschild space–time obtained with theuse of running Newton constant also indicates that the blackhole evaporation stops when its mass approaches the criticalvalue of the order of Planck mass [6]. On the other hand thequantum corrected gravity modifies this GUP as well. It is fairto ask whether the halt of black hole radiation is provided byGUP or it is due to quantum gravitational effects. Let us give acritical view of this problem.

Let us briefly discuss the modification of Heisenberg un-certainty principle due to gravitational interaction. The mainconceptual point concerning GUP is that there is an additionaluncertainty in quantum measurement due to gravitational in-teraction. We focus on consideration of this problem presented

E-mail addresses: maziashvili@ictsu.tsu.edu.ge,mishamazia@hotmail.com (M. Maziashvili).

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.03.009

in [2], (h̄ = c = 1 is assumed in what follows). The approachproposed in this Letter relying on classical gravity is to cal-culate the displacement of electron caused by the gravitationalinteraction with the photon and add it to the position uncer-tainty. The photon due to gravitational interaction imparts toelectron the acceleration given by a = G0�E/r2 (G0 is exper-imentally observed value of Newton’s constant for macroscopicvalues of distances). Assuming r0 is the size of the interactionregion the variation of the velocity of the electron is given by�v ∼ G0�E/r0 and correspondingly �xg ∼ G0�E. There-fore the total uncertainty in the position is given by

(1)�x � 1

2�E+ αG0�E,

where α is the factor of order unity in respect with the stringyinduced GUP [4]. Eq. (1) exhibits the minimal observable dis-tance. However, the minimal observable distance is determinedrather due to collapse of �E than merely by Eq. (1), because itputs simply the bound on the measurement procedure. In thisway one gets that for α � 2 the �xmin = √

2αG0 while forα < 2 the minimal observable distance is given by �xmin =√

8G0/(4 − α). In the framework of this discussion one canobtain the GUP in higher-dimensional case as well as on thebrane [5].

The heuristic derivation of black hole evaporation proceedsas follows [1]. (In paper [1] α = 1 is assumed.) The black hole is

M. Maziashvili / Physics Letters B 635 (2006) 232–234 233

modeled as an object with linear size equal to the two times thegravitational radius 2rg and the minimum uncertainty conditionis assumed for the radiation. Then the lower value of �E

(2)�E = rg − (r2g − G0)

1/2

2G0,

that comes from uncertainty relation is identified to the charac-teristic temperature of the black hole emission with the constantof proportionality 1/2π . As one sees from Eq. (2) the GUPamended Hawking temperature becomes complex if the massof the black hole is less than 1/2G

1/20 , leading thereby to the

nonzero minimal black hole mass [1].As it is evident the GUP assumes two �E values for a given

�x. But the choice of the lower value is well motivated physi-cally because for relatively small values of �E the gravitationaluncertainty becomes negligible in comparison with the stan-dard term and therefore in the limit �x � √

G0 one has torecover the standard uncertainty relation. On the other handsuch a choice is motivated by the correct asymptotic depen-dance of Hawking temperature on the black hole mass in theframework of this heuristic approach.

However for the black hole with a radius not too far abovethe Planck length the quantum fluctuations of the metric playan important role. The effective Newton constant obtained bymeans of the Wilson-type effective action has the form [6]

(3)G(r) = G0r3

r3 + 2.504G0(r + 4.5G0M),

where M is the mass of the source. The effective Newton con-stant Eq. (3) depends on the mass of source and correspond-ingly the mass of the test particle is implied to be negligiblysmall in comparison with the source mass. Since the gravi-tational uncertainty becomes appreciable when the energy ofphoton approaches the Planck scale, which in turn exceeds verymuch the mass of the standard model particles, one can safelyuse the Eq. (3). An interesting observation made in [6] is thatfor M less than the critical value Mcr = 3.503G

−1/20 the horizon

disappears. So that the black hole evaporation process comes toa complete halt when the mass reaches to the critical value. Bytaking the quantum corrected equation for gravitational radiusas [6]

rg = 2G(rg)�E,

one finds the following expression for outer horizon

rg/G0�E

= 0.667

+0.265(16 − 139.5t + 10.392

√t (t − 0.204)(176.392 + t)

)1/3

(4)

− 0.41(3t − 4)

(16 − 139.5t + 10.392√

t (t − 0.204)(176.392 + t) )1/3,

where t ≡ 2.504m2p/�E2 and mp ≡ G

−1/20 . The critical value

of mass below which the horizon disappears is given by tcr =0.204, �Ecr = 3.503G

−1/2. Following the paper [3] one can

0

combine this gravitational radius and standard position uncer-tainty as

(5)�x �{

1/2�E if �E � 3.503G−1/20 ,

rg(�E) if �E > 3.503G−1/20

where rg(�E) is given by Eq. (4). From Eq. (5) one gets the

following minimal length �xmin = 0.143G1/20 . The linear com-

bination

(6)�x � 1

2�E+ rg(�E),

does not make any sense for �E < �Ecr since for these val-ues of energy there is no horizon at all. (Let us comment thatthe idea of paper [3] to determine the total uncertainty for�E > �Ecr by Eq. (6) is not quite clear because the collapseof �E puts simply the limitation on the measurement.) Onehas to define the gravitational disturbance of the electron posi-tion directly. Taking the quantum corrected potential around thephoton to be −�EG(r)/r then the acceleration imparted to theelectron is

a =∣∣∣∣ �E2G0r

r3 + 2.504G0(r + 4.5G0�E)

(7)− �EG0r2(3r2 + 2.504G0)

[r3 + 2.504G0(r + 4.5G0�E)]2

∣∣∣∣.The characteristic time and length scale for the interaction whenone uses the energy �E for the measurement is given by �E−1

[7]. Thus for the GUP when �E � �Ecr one gets

(8)

�x = 1

2�E+ α

|22.536G30�E5 + 2.504G2

0�E3 − G0�E|[1 + 2.504G0(�E2 + 4.5G0�E4)]2

.

Assuming α = 1 it is easy to check that the minimal distancethat comes from Eq. (8) �xmin = �x(�Ecr) = 0.147G

1/20 <

rg(�Ecr) = 4.484G1/20 . But the object with the size �xmin can-

not be black hole at all for the quantum corrected Schwarzschildspace–time does not admit the black hole with the size less thanrg(�Ecr) [6]. In general the parameter α is of order unity, butthis numerical factor cannot change the result because it shouldbe about 1000 the �xmin to be comparable to the rg(�Ecr).So, one concludes that black hole evaporation is halt due toquantum gravitational effects. To be strict the disappearanceof the horizon beneath the mass scale Mcr results in the subtlequestion what actually is the object left behind the evaporation,whether it is unambiguously black hole or the classical remnantis also allowed in general. But in the real situation this questiondoes not arise because when the mass approaches Mcr the emis-sion temperature becomes zero [6] and due to absorption of thebackground radiation this limit is simply unattainable.

Another very interesting issue that comes from effectiveaverage action and its associated exact renormalization groupequation is the fractal structure of spacetime on sub-Planckiandistances with effective dimensionality 2 [8]. It is of interest toknow if presence of minimum uncertainty in the position con-sidered above (�xmin = 0.147G

1/20 ) allows one to observe the

“ripples” of spacetime at sub-Planckian distances.

234 M. Maziashvili / Physics Letters B 635 (2006) 232–234

Acknowledgements

The author is greatly indebted to Z. Berezhiani, M. Makhvi-ladze and R. Percacci for useful conversations. The work wassupported by the grant FEL. REG. 980767.

References

[1] R. Adler, P. Chen, D. Santiago, Gen. Relativ. Gravit. 33 (2001) 2101, gr-qc/0106080.

[2] R. Adler, D. Santiago, Mod. Phys. Lett. A 14 (1999) 1371, gr-qc/9904026.[3] F. Scardigli, Phys. Lett. B 452 (1999) 39, hep-th/9904025.[4] G. Veneziano, Europhys. Lett. 2 (1986) 199;

D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 197 (1987) 81;D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 216 (1989) 41;D. Gross, P. Mende, Phys. Lett. B 197 (1987) 129;D. Gross, P. Mende, Nucl. Phys. B 303 (1988) 407;K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B 234 (1990) 276.

[5] M. Makhviladze, M. Maziashvili, D. Nozadze, gr-qc/0512044.[6] A. Bonanno, M. Reuter, Phys. Rev. D 62 (2000) 043008, hep-th/0002196.[7] L. Landau, E. Lifshitz, Quantum Mechanics, Nauka, Moscow, 1989.[8] O. Lauscher, M. Reuter, Phys. Rev. D 65 (2002) 025013, hep-th/0108040;

O. Lauscher, M. Reuter, Phys. Rev. D 66 (2002) 0250026, hep-th/0205062;O. Lauscher, M. Reuter, JHEP 0510 (2005) 050, hep-th/0508202.