Quantum-corrected black hole thermodynamics to all
orders in the Planck length
Khireddine Nouicer∗
Laboratory of Theoretical Physics and Department of Physics,
Faculty of Sciences, University of Jijel
Bp 98, Ouled Aissa, 18000 Jijel, Algeria.
Abstract
We investigate the effects to all orders in the Planck length from a generalized un-
certainty principle (GUP) on black holes thermodynamics. We calculate the corrected
Hawking temperature, entropy, and examine in details the Hawking evaporation process.
As a result, the evaporation process is accelerated and the evaporation end-point is a zero
entropy, zero heat capacity and finite non zero temperature black hole remnant (BHR). In
particular we obtain a drastic reduction of the decay time, in comparison with the result
obtained in the Hawking semi classical picture and with the GUP to leading order in the
Planck length.
PACS: 04.60.-m, 05.70.-a
Key Words: Quantum Gravity, Generalized Uncertainty Principle, Thermodynamics
of Black Holes
1 Introduction
Recently a great interest has been devoted to the study of the effects of generalized uncertainty
principles (GUPs) and modified dispersion relations (MDRs) on black holes thermodynamics.
The concepts of GUPs and MDRs originates from several studies in string theory approach to
quantum gravity [1- 4], loop quantum gravity [5], noncommutative space-time algebra [6 - 8]
and black holes gedanken experiments [9 - 10]. All these approaches indicate that the standard
Heisenberg uncertainty principle must be generalized to incorporate additional uncertainties
when quantum gravitational effects are taken into account. Actually it is believed that any
∗Email: khnouicer@mail.univ-jijel.dz / khnouicer@yahoo.fr
http://arxiv.org/abs/0704.1261v1
promising candidate for a quantum theory of gravity must include the GUPs and/or MDRs
as central ingredients.
The main consequence of the GUP is the appearance of a minimal length scale of the order of
the Planck length which cannot be probed, providing a natural UV cut-off, and thus corrections
to black holes thermodynamic parameters are expected at the Planck scale.
The consequences of GUPs and/or MDRs on black holes thermodynamics have been considered
intensively in the recent literature on the subject [11 - 16]. Notably, it has been shown that
GUP prevents black holes from complete evaporation, exactly like the standard Heisenberg
principle prevents the hydrogen atom from total collapse [17]. Then at the final stage of the
Hawking radiation process of a black hole, a inert black hole remnant (BHR) continue to exist
with zero entropy, zero heat capacity and a finite non zero temperature. The inert character of
the BHR, besides gravitational interactions, makes this object a serious candidate to explain
the nature of dark matter [18, 19]. On the other hand, a particular attention has been also
devoted to the computation of the entropy of a black hole and the sub-leading logarithmic
correction [20 - 34].
All the above studies have been performed with a GUP to leading order in the Planck length.
However, recent generalization of the GUP induces quantitative corrections to the entropy and
then influences the evaporation phase of the black hole [35]. Besides this growing interest in
quantum gravity phenomenology, a intense activity is actually devoted to possible production of
black holes at particle colliders [36, 37] and in ultrahigh energy cosmic ray (UHECR) airshowers
[38, 39]. The next generation of particle colliders are planned to reach a c-m energy of the order
of few TeV , a scale at which the complete evaporation of BH is expected to end, leaving up
in a scenario with GUP a inert BHR. Then, it is phenomenologically relevant, to obtain the
corrections to BH thermodynamic parameters in the framework of a GUP beyond the leading
order in the Planck length.
In this paper we discuss the effects, brought by a generalization of the GUP to all orders in the
Planck length, on thermodynamic parameters of the Schwarzschild black hole . Hereafter, we
refer to this version of GUP as GUP∗.
The organization of this work is as follows. In section 2, we introduce a deformed position and
momentum operators algebra leading to GUP∗ and examine its various implications. In section
3, the Hawking temperature and entropy are computed and the departures from the standard
case shown. In section 4, we calculate the deviation from the standard Stefan-Boltzmann law
of the black body radiation spectrum and investigate the Hawking evaporation process of black
holes by a calculation of the evaporation rate, the decay time and the heat capacity. Finally
we compare our results with the ones obtained in the context of the GUP to leading order in
the Planck length commonly used in the literature. Our conclusions are summarized in the last
section.
2 Generalized uncertainty principle
Loop quantum gravity and string theory approach to quantum gravity predict slight deviations
in the laws describing photons propagation in vacuum. It is expected that these effects, leading
to a modified dispersion relation (MDR), could be amplified by cosmological distances and then
become observables [40]. On the other hand, quantum gravity phenomenology has been tackled
within effective models based on MDRs and/or GUPs and containing the minimal length as
a natural UV cut-off. Recently the relation between these approaches has been clarified and
established [41].
The idea of a minimal length can be modelled in terms of a quantized space-time and goes
back to the early days of quantum field theory [42] (see also [40− 43] ). An other approach is
to consider deformations to the standard Heisenberg algebra [7, 8], which lead to generalized
uncertainty principles. In this section we follow the latter approach and exploit results recently
obtained. Indeed, it has been shown in the context of canonical noncommutative field theory
in the coherent states representation [47] and field theory on non-anticommutative superspace
[48, 49], that the Feynman propagator display an exponential UV cut-off of the form exp (−ηp2),
where the parameter η is related to the minimal length. This framework has been further
applied, in series of papers [50], to the black hole evaporation process.
At the quantum mechanical level, the essence of the UV finiteness of the Feynman propagator
can be also captured by a non linear relation, p = f(k), between the momentum and the wave
vector of the particle [41]. This relation must be invertible and has to fulfill the following
requirements:
1. For smaller energies than the cut-off the usual dispersion relation is recovered.
2. For large energies, the wave vector asymptotically reaches the cut-off.
In this case, the usual momentum measure dnp is deformed and becomes dnp
. In the
following, we will restrict ourselves to the isotropic case and work with one space-like dimension.
Following [47, 49] and setting η =
we have
= ~exp
α2L2P l
, (1)
where α is a dimensionless constant of order one.
From Eq.(1) we obtain the dispersion relation
k (p) =
2αLP l
αLP l
, (2)
from which we have the following minimum Compton wavelength
λ0 = 4
παLP l. (3)
Let us show that these results can be obtained from the following representation of the position
and momentum operators
X = i~ exp
α2L2P l
∂p P = p. (4)
The corrections to the standard Heisenberg algebra become effective in the so-called quantum
regime where the momentum and length scales are of the order of the Planck mass MP l and of
the Planck length LP l respectively.
The hermiticity condition of the position operator implies the following modified completeness
relation
p2|p〉〈p| = 1 (5)
and modified scalar product
〈p| p′〉 = e
δ (p− p′) . (6)
From Eq.(5) we observe that we have reproduced the Gaussian damping factor in the Feyn-
man propagator [47, 49]. The algebra defined by Eq. (4) leads to the following generalized
commutator and generalized uncertainty principle (GUP∗)
[X,P ] = i~ exp
α2L2P l
, (δX) (δP ) ≥
α2L2P l
. (7)
In order to investigate the quantum implications of this deformed algebra, we consider the
saturate GUP∗ and solve for (δP ). Using the property 〈P 2n〉 ≥ 〈P 2〉n and (δP )2 = 〈P 2〉−〈P 〉2
the saturate GUP∗ is then given by
(δX) (δP ) =
α2L2P l
(δP )
+ 〈P 〉2
. (8)
Taking the square of this expression we obtain
W (u) eW (u) = u, (9)
where we have set W (u) = −2α
(δP )
and u = − α
2(δX)
〈P 〉2
The equation given by Eq.(9) is exactly the definition of the Lambert function [51]. The
LambertW function is a multivalued functions. Its different branches are labelled by the integer
k = 0,±1,±2, · · · . When u is a real number Eq.(9) have two real solutions for 0 ≥ u ≥ −1
denoted by W0(u) and W−1(u), or it can have only one real solution for u ≥ 0, namely W0(u) .
For -∞ < u < −1
, Eq.(9) have no real solutions.
Using Eq.(9) the uncertainty in momentum is then given by
(δP ) =
〈P 〉2
2 (δX)
2L2P le
2α2L2
〈P 〉2
2 (δX)
 . (10)
Then from the argument of the Lambert function in Eq.(10) we have the following condition
α2L2P le
2α2L2
〈P 〉2
2 (δX)
, (11)
which leads to a minimal uncertainty in position given by
αLP le
〈P 〉2
. (12)
The absolutely smallest uncertainty in position or minimal length is obtained for physical
states for which we have 〈P 〉 = 0 and (δP ) = ~/
2αLP l
, and is given by
(δX)0 =
αLP l (13)
In terms of the minimal length the momentum uncertainty becomes
(δP ) =
2 (δX)
(δX)0
. (14)
Here we observe that 1
(δX)0
< 1 is a small parameter, by virtue of the GUP∗, and perturbative
expansions to all orders in the Planck length can be safely performed.
Indeed a series expansion of Eq.(14) gives the corrections to the standard Heisenberg principle
2 (δX)
(δX)0
(δX)0
(δX)0
+ . . .
. (15)
This expression of (δP ) containing only odd powers of (δX) is consistent with a recent analysis
in which string theory and loop quantum gravity, considered as the most serious candidates for
a theory of quantum gravity, put severe constraints on the possible forms of GUPs and MDRs
[20].
Before ending this section and for later use let us recall the form of the GUP to leading order
in the Planck length widely used in the literature on quantum gravity phenomenology. This
GUP is given by
(δX) (δP ) ≥
α2L2P l
(δP )
. (16)
A simple calculation leads to the following minimal length
(δX)0 = αLP l, (17)
which is of order of the Planck length. However, as nicely noted in [41], this form of GUP do
not fulfill the second requirement listed above. In the following sections we use the form of the
GUP given by Eq.(14) and investigate the thermodynamics of the Schwarzschild black hole.
We use units ~ = c = kB = 1 which imply LP l = M
P l = T
P l =
3 Black hole thermodynamics
The metric of a four-dimensional Schwarzschild black hole is given by
ds2 =
1− 2MG
dt2 −
1− 2M
dr2 − r2dΩ2, (18)
where M represents the mass of the black hole. The Schwarzschild horizon radius, located at
rh, is defined by
rh = 2MG. (19)
Near-horizon geometry considerations suggests to set δX ≃ rh, and then Eq.(19) leads to
minimum horizon radius and minimum mass given by
rh = (δX)0 =
αLP l, M0 =
MP l. (20)
Therefore, black holes with mass smaller than M0 do not exist.
In the standard Hawking picture, temperature and entropy of the Schwarzschild black hole of
mass M are [52, 53]
, S = 4πGM2. (21)
Let us then examine the corrections to the above expressions due to the GUP∗. Following the
heuristic argument of Bekenstein we have
. (22)
Using Eq.(14), the GUP∗-corrected Hawking temperature is
8πML2P l
. (23)
On substituting Eq.(20) into Eq.(23) we obtain the following black hole maximum temperature
TmaxH =
. (24)
The corrections to the standard Hawking temperature are obtained by expanding Eq.(23) in
terms of 1
(M0/M). Indeed we obtain
8πML2P l
+ . . .
. (25)
The variation of the Hawking temperature, Eq.(23), with the mass of the black hole is shown
in figure 1.
It is interesting to inverse Eq.(23) and write the mass of the black hole as a function of the
temperature
8πTHL
TmaxH
. (26)
This relation shows that for temperatures larger than TmaxH , the black hole mass increases with
temperature. In our framework, such a behavior is forbidden by the cut-off brought by GUP∗.
However, in the noncommutative approach to radiating black hole, this behavior is allowed
because of a lack of a generalized uncertainty principle [50].
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 1: The temperature versus BH mass. From left to right: the Hawking result (black solid
line), GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α=1 (green), α=1.25 (blue)
respectively.
We turn now to the calculation of the micro canonical entropy of a large black hole. In the stan-
dard situation the entropy is proportional to the black hole horizon-area. Following heuristic
considerations due to Bekenstein, the minimum increase of the area of a black hole absorbing a
classical particle of energy E and size R is given by (∆A)0 ≃ 4L2P l (ln 2)ER. At the quantum
mechanical level the size and the energy of the particle are constrained to verify R ∼ 2δX and
E ∼ δP . Then we have (∆A)0 ≃ 8L2P l (ln 2) δXδP.
Extending this approach to the case with GUP∗ and using near horizon geometry considerations,
we obtain
(∆A)0 ≈ 4L
P l ln 2 exp
, (27)
where A = 4π (δX)
and A0 = 4π (δX)
0 are respectively the horizon area and minimum horizon
area of the black hole. With the aid of the Bekenstein calibration factor for the minimum
increase of entropy (∆S)0 = ln 2 we have
(∆S)0
(∆A)0
4L2P l
. (28)
Before integrating over A we note that the existence of a minimum horizon area enforces us to
set the lower limit of integration as A0. Then the entropy, up to a irrelevant constant, is
4L2P l
dA.. (29)
The relation e
W (x)
x/W (x) allows us to write Eq.(29) as
4eL2P l
∫ − 1
2 [W (y)]
2 dy, (30)
where PV means the Cauchy principal value of the integral. Setting y = −1
and performing
the integration we obtain the GUP∗-corrected black hole entropy
8eL2P l
))− 1
e− Ei
, (31)
where Ei (x) is the exponential function.
Expanding Eq.(31) in the parameter 1
(A0/A) we have
4L2P l
8L2P le
25πα2
192e2
343πα2
2304e3
+ . . .+ C
where the constant is given by
8L2P le
γ − 1− 2 ln (2e)− 2
e− Ei
≃ −4.60
L2P l
and γ is the Euler constant. The dependence on the Planck length is contained in A0 ∼ L2P l.
We observe that we have reproduced, in our framework with GUP∗, the log-area correction
with a negative sign. Other approaches like string theory, loop quantum gravity and effectif
models with GUPs and/or MDRs, lead to the same sub-leading logarithmic correction. Setting
ρ = −πα2
and β = 3π
in Eqs. (25) and (32) we obtain
M2P l
ρ2 + β/4
, (34)
4L2P l
+ ρ ln
L2P l
βL2P l
. (35)
These expressions are exactly the temperature and entropy obtained in loop quantum gravity
and string theory approach quantum gravity.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 2: The entropy versus BH mass. From left to right: the Hawking result (black solid line),
GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α = 1 (green) and α=1.25 (blue)
respectively.
From figures 1 and 2 it follows that the GUP∗-corrected temperature and entropy are respec-
tively higher and smaller than the semi classical results.
4 Black holes evaporation
As a warming to study the Hawking radiation process of the Schwarzschild black hole, we
examine the effects of GUP∗ on the black body radiation spectrum. With the aid of the
squeezed momentum measure given by Eq.(5), which suppress the contribution of unwanted
high momenta, the energy density of a black body at temperature T is defined by
d3pe−α
T − 1
. (36)
Using the variable y = βp (β = 1/T ) and expanding the exponential, equation (36) becomes
Eγ = 8πT 4
(−1)n
(αT/TP l)
y2n+3
ey − 1
. (37)
Now with the help of the following definition of the Riemann zeta function
ey − 1
= Γ (s) ζ (s) , (38)
we obtain
Eγ = 8πT 4
(−1)n
(αT/TP l)
Γ (2n+ 4) ζ (2n+ 4) . (39)
This energy density is defined only for values of temperatures below some characteristic scale.
In fact Eq.(39) is an alternating series which converge if and only if
(αT/TP l)
Γ (2n + 4) ζ (2n+ 4)
= 0. (40)
From this relation it follows that
T < α−1TP l, (41)
as expected from the Gaussian damping factor in Eq.(36). However, we note that we have a
stronger condition on T . Indeed in our framework, the maximum temperature of the black hole
is given by Eq.(24) and it is approximately 0.1TP l for α of order one. Then the condition on
the BH temperature is rewritten as T/TP l < 0.1. For our purpose, the latter constraint allows
us to cut the series in Eq.(39) at n = 1. Using ζ (4) = π
and ζ (6) = π
and Eq.(24) we finally
obtain, from Eq.(39) , the following expression
Eγ (T ) =
1− 15
TmaxH
. (42)
The first term is the standard Stefan-Boltzmann law while the second term is the correction
brought by GUP∗.
We are now ready to study the Hawking evaporation process. The intensity emitted by a black
hole of mass M is defined by
I = AEγ (TH) , (43)
where A is the BH horizon area. Invoking energy conservation, the evaporation rate of the
black hole is
= −AEγ (TH) . (44)
Using Eq. (23) for the corrected Hawking temperature we obtain
= − γ1
M2L4P l
1− 8γ2
with γ1 =
, γ2 =
16128
. The deviations from the standard expression are obtained by applying
a series expansion in 1
(M0/M)
= − γ1
M2L4P l
1− 2γ2
1− 72γ2
25eγ1
+ . . .
The variation of the evaporation rate with the black hole mass is shown in Figure 3. We clearly
observe that the divergence for M → 0 in the standard description of the black hole evaporation
process is now completely regularized by the GUP∗. This regularization is also reflected by the
constraint (41), which suppress the evaporation process beyond the Planck temperature. This
phenomenon is similar to the prevention, by the standard uncertainty principle, of the hydrogen
atom from total collapse. In our picture, the regularization can be considered as a dynamical
effect and not as a consequence of any quantum symmetry in the theory.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 3: The evaporation rate versus BH mass. From left to right: the Hawking result (black solid line) ,
GUP (doted line) and GUP∗ results (solid line) for α = 0.90 (red) , α = 1. (green) and α = 1.25
(blue).
On the other hand, we observe that the evaporation phase ends when the BH mass becomes
equal to M0 with a minimum rate given by
(γ1 − 8γ2)M4P l. (47)
Thus the evaporation process of a black hole with initial mass M > M0 continue until the
horizon radius becomes (δX)0 , leaving a massive relic referred to, in the literature, as a black
hole remnant (BHR). To find the nature of the BHR we calculate the heat capacity defined by
. (48)
Using the expression of temperature given by (23) we easily obtain
C = −8πM2L2P l
. (49)
This expression vanishes when 1 + W
(M0/M)
= 0, whose solution is M = M0. We
conclude that the heat capacity of the black hole vanishes at the end point of the evaporation
process characterized by a BHR with mass M0.Besides the gravitational interaction with the
surrounding, the vanishing of the heat capacity reveals the inert character of the BHRs and
thus make them as potential candidates to explain the origin of dark matter [18, 19]. Finally
we note that, as it is the case with the form of the GUP to leading order in the Planck length,
the BHRs are also a consequence of GUP∗ [17, 55].
We have drawn the variation of the heat capacity with BH mass in figure 4. In it we see that,
the heat capacity vanishes for M0 ≃ 0.50, 0.75 in the case with GUP and M0 ≃ 0.58, 0.87 in
the case with GUP∗.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 4: The heat capacity versus BH mass. From left to right: the standard result (black solid
line), GUP (doted line) and GUP∗ results (solid line) for α = 1 (red) and α = 1.5 (blue) respectively.
Taylor expanding Eq.(49) we have
C = −8πM2L2P l
− . . .
. (50)
The standard expression of the heat capacity C = −8πM2 is reproduced in the limit of black
holes with mass larger than the minimum mass M0. The correction terms to the heat capacity
due to GUP∗ are all positive indicating that the evaporation process is accelerated and leading
to a corrected decay time smaller than the decay time in the standard case.
Let us consider a black hole starting the evaporation process with a mass M and ending the
process with the minimum mass M0. Using (45) and the variable y = −1e (M0/M)
, the decay
time is given by
t = (−1)7/2
∫ − 1
(M0/M)
W−5/2 (y) e−
W (y)dy
+ (−1)5/2
2γ21e
2L2P l
∫ − 1
(M0/M)
W−3/2 (y) e−
W (y)dy. (51)
Performing the integration we obtain
4 (1− ǫ)
W (y)
−W (y)
W 2 (y)
+ C, (52)
where the constant C is the value for y = −1/e and ǫ = 3γ2
∼ 10−6 for α of order
one. Ignoring ǫ and performing a series expansion in y we have
M3L4P l
+ . . .
Then to first order in 1
(M0/M) the relative correction to the decay time is
, (54)
where t0 =
is the decay time without GUP∗. From Eq.(54) , it follows that black holes
with GUP∗ are hotter and decay faster than in the standard case.
Let us now turn to a comparison of the corrected BH thermodynamics with GUP∗ with the
corrections brought by the GUP to leading order in the Planck length. Since our comparison
is quantitative we use the Planck units. Repeating the same calculations as above with the
GUP given by Eq.(16), the temperature, the entropy and the heat capacity of the black hole
are respectively given by
TGUP =
, (55)
SGUP = 2πM
CGUP = πα
1− α2
1− α2
. (57)
The minimum black hole mass and maximum temperature allowed by GUP areM0 = (δX)0 /2 =
and Tmax = 1/2πα. In figures 1, 2, 3 and 4 we have plotted, besides the results obtained with
GUP∗, the variation of temperature, entropy, evaporation rate and heat capacity with GUP as
functions of the black hole mass for different values of the parameter α. Figure 2 shows, that in
the scenario with GUP∗, the BH entropy decreases compared to the entropy in the standard
case and the scenario with GUP. This reveals the deeper quantum nature of the black hole
in the scenario with GUP∗. Thus quantum effects become manifest at an earlier stage of the
evaporation phase than was predicted by the semi classical Hawking analysis [54] and the GUP
analysis [55].
The calculation of the evaporation rate in the framework with GUP requires a careful analysis.
In all the calculations done until now, the validity of the Stefan-Boltzmann law is assumed,
ignoring the UV cut-off implemented by GUP. However, it was pointed in [16] that the effect
of the GUP should be also reflected in a modification of the de Broglie wave length relation
1 + α2p2
. (58)
This relation must be translated into a modification of the momentum measure such that the
contributions of high momenta are suppressed. As shown in [7], the GUP to leading order in
the Planck length leads to a squeezing of the momentum measure by a factor 1
(1+α2L2Plp2)
. Then
following the same calculation leading to Eq.(39) , the energy density of a black body with GUP
(1 + α2L2P lp
T − 1
. (59)
Performing the integral and using the same argument as before, we obtain the expression given
by Eq. (42) . We note, that in a recent calculation of the Stefan-Boltzmann law with GUP
[24], the sign of the correction term is positive, in contradiction with the role of the UV cut-off
implemented by the GUP.
The correct evaporation rate with GUP is then given by
128π2M6
1024π2M8
63α10
. (60)
In figure 3 we observe that the evaporation process with GUP is retarded compared to the
process with GUP∗ and that the process ends at a mass M0 = α/2 with a minimum rate given
min,GUP
= − 32π
1260M20
, (61)
which is greater (in absolute value) than the one obtained with GUP∗.
In table 1 we show the GUP and GUP∗-corrected thermodynamics of two black holes with
initial mass equal to 2MP l and 5MP l for α = 1. The first row gives the semi classical Hawking
results. The second row gives the GUP-corrected results and the third row the GUP∗−corrected
ones. It is interesting to note that, in the scenario with GUP∗, the final stage of the evaporation
phase is a remnant with a mass larger than the one obtained with GUP and that the decay time
is drastically reduced. In a scenario with extra dimensions, these results may have important
consequences on possible black holes production at particle colliders and in ultrahigh energy
cosmic ray (UHECR) air-showers.
Finally, let us notice that the corrections to the black hole thermodynamics become indistin-
guishable in the two version of GUP in the limit of large mass and small values of α. However, for
growing values of the parameter α, corresponding to strong gravitational effects, the predictions
of the two GUPs concerning the entropy become different even for massive black holes.
Table 1. GUP and GUP∗-corrected thermodynamics for two BHs with mass M = 2 and M = 5 (in
Planck units). The deviations from the Hawking results are also given.
M = 2
α Minimum mass Initial temp Final temp Decay time Entropy
0 - 0.019 ∞ 129.69 50.27
1.0 0.5 0.020 (+3%) 0.16 111.92 (−14%) 44.66 (−11%)
1.0 0.58 (+16%) 0.020 (+3%) 0.11 (−31%) 3.33 (−97%) 43.73 (−13%)
M = 5
α Minimum mass Initial temp Final temp Decay time Entropy
0 - 0.008 ∞ 2026.42 314.16
1.0 0.5 0.008 0.16 1976.60 (−2.5%) 307.10 (−2%)
1.0 0.58 (+16%) 0.008 0.11 (−31%) 22.17 (−99%) 306.18 (−2.2%)
5 Conclusion
In this paper we have studied how black holes thermodynamic parameters are affected by a
GUP to all orders in the Planck length. We have obtained exact analytic expressions for the
Hawking temperature and entropy. Particularly we found that a black hole with a mass smaller
than a minimum mass do not exist. The existence of a energy scale which is one order below the
Planck scale allowed us to calculate, to leading order, the deviations from the standard Stefan-
Boltzmann law. Then we investigated the Hawking radiation process of the Schwarzschild
black hole and shown that at the end of the evaporation phase a inert massive relic continue
to exist as a black hole remnant (BHR) with zero entropy, zero heat capacity and non zero
finite temperature. For completeness, we have also compared our results with the semi classical
results and the predictions of the GUP to leading order in the Planck length. In particular,
we have shown that the entropy in our framework is smaller than the entropy in the standard
case and with GUP. We have also made the correct calculation of the evaporation rate with
GUP. Finally, we have shown that black holes with the form of GUP used in this paper are
hotter, shorter-lived and tend to evaporate less than black holes in the semi classical and the
GUP to leading order pictures. On the other hand, we have found that the predictions of the
GUP to all orders in the Planck length and the GUP to leading order in the Planck length,
concerning the entropy, become different for strong gravitational effects and large black holes
mass, suggesting a further investigation of the early universe thermodynamics in the framework
with the GUP to all orders in the Planck length. In a future work we will examine the effects of
the GUP to all orders in the Planck length on black holes thermodynamics in a scenario with
extra dimensions.
Acknowledgments
The author thanks the referees for their comments and valuable remarks.
References
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	Introduction
	Generalized uncertainty principle
	Black hole thermodynamics
	Black holes evaporation
	Conclusion

  We investigate the effects to all orders in the Planck length from a
generalized un- certainty principle (GUP) on black holes thermodynamics. We
calculate the corrected Hawking temperature, entropy, and examine in details
the Hawking evaporation process. As a result, the evaporation process is
accelerated and the evaporation end-point is a zero entropy, zero heat capacity
and finite non zero temperature black hole remnant (BHR). In particular we
obtain a drastic reduction of the decay time, in comparison with the results
obtained in the Hawking semi classical picture and with the GUP to leading
order in the Planck length.

Introduction
Recently a great interest has been devoted to the study of the effects of generalized uncertainty
principles (GUPs) and modified dispersion relations (MDRs) on black holes thermodynamics.
The concepts of GUPs and MDRs originates from several studies in string theory approach to
quantum gravity [1- 4], loop quantum gravity [5], noncommutative space-time algebra [6 - 8]
and black holes gedanken experiments [9 - 10]. All these approaches indicate that the standard
Heisenberg uncertainty principle must be generalized to incorporate additional uncertainties
when quantum gravitational effects are taken into account. Actually it is believed that any
∗Email: khnouicer@mail.univ-jijel.dz / khnouicer@yahoo.fr
http://arxiv.org/abs/0704.1261v1
promising candidate for a quantum theory of gravity must include the GUPs and/or MDRs
as central ingredients.
The main consequence of the GUP is the appearance of a minimal length scale of the order of
the Planck length which cannot be probed, providing a natural UV cut-off, and thus corrections
to black holes thermodynamic parameters are expected at the Planck scale.
The consequences of GUPs and/or MDRs on black holes thermodynamics have been considered
intensively in the recent literature on the subject [11 - 16]. Notably, it has been shown that
GUP prevents black holes from complete evaporation, exactly like the standard Heisenberg
principle prevents the hydrogen atom from total collapse [17]. Then at the final stage of the
Hawking radiation process of a black hole, a inert black hole remnant (BHR) continue to exist
with zero entropy, zero heat capacity and a finite non zero temperature. The inert character of
the BHR, besides gravitational interactions, makes this object a serious candidate to explain
the nature of dark matter [18, 19]. On the other hand, a particular attention has been also
devoted to the computation of the entropy of a black hole and the sub-leading logarithmic
correction [20 - 34].
All the above studies have been performed with a GUP to leading order in the Planck length.
However, recent generalization of the GUP induces quantitative corrections to the entropy and
then influences the evaporation phase of the black hole [35]. Besides this growing interest in
quantum gravity phenomenology, a intense activity is actually devoted to possible production of
black holes at particle colliders [36, 37] and in ultrahigh energy cosmic ray (UHECR) airshowers
[38, 39]. The next generation of particle colliders are planned to reach a c-m energy of the order
of few TeV , a scale at which the complete evaporation of BH is expected to end, leaving up
in a scenario with GUP a inert BHR. Then, it is phenomenologically relevant, to obtain the
corrections to BH thermodynamic parameters in the framework of a GUP beyond the leading
order in the Planck length.
In this paper we discuss the effects, brought by a generalization of the GUP to all orders in the
Planck length, on thermodynamic parameters of the Schwarzschild black hole . Hereafter, we
refer to this version of GUP as GUP∗.
The organization of this work is as follows. In section 2, we introduce a deformed position and
momentum operators algebra leading to GUP∗ and examine its various implications. In section
3, the Hawking temperature and entropy are computed and the departures from the standard
case shown. In section 4, we calculate the deviation from the standard Stefan-Boltzmann law
of the black body radiation spectrum and investigate the Hawking evaporation process of black
holes by a calculation of the evaporation rate, the decay time and the heat capacity. Finally
we compare our results with the ones obtained in the context of the GUP to leading order in
the Planck length commonly used in the literature. Our conclusions are summarized in the last
section.
2 Generalized uncertainty principle
Loop quantum gravity and string theory approach to quantum gravity predict slight deviations
in the laws describing photons propagation in vacuum. It is expected that these effects, leading
to a modified dispersion relation (MDR), could be amplified by cosmological distances and then
become observables [40]. On the other hand, quantum gravity phenomenology has been tackled
within effective models based on MDRs and/or GUPs and containing the minimal length as
a natural UV cut-off. Recently the relation between these approaches has been clarified and
established [41].
The idea of a minimal length can be modelled in terms of a quantized space-time and goes
back to the early days of quantum field theory [42] (see also [40− 43] ). An other approach is
to consider deformations to the standard Heisenberg algebra [7, 8], which lead to generalized
uncertainty principles. In this section we follow the latter approach and exploit results recently
obtained. Indeed, it has been shown in the context of canonical noncommutative field theory
in the coherent states representation [47] and field theory on non-anticommutative superspace
[48, 49], that the Feynman propagator display an exponential UV cut-off of the form exp (−ηp2),
where the parameter η is related to the minimal length. This framework has been further
applied, in series of papers [50], to the black hole evaporation process.
At the quantum mechanical level, the essence of the UV finiteness of the Feynman propagator
can be also captured by a non linear relation, p = f(k), between the momentum and the wave
vector of the particle [41]. This relation must be invertible and has to fulfill the following
requirements:
1. For smaller energies than the cut-off the usual dispersion relation is recovered.
2. For large energies, the wave vector asymptotically reaches the cut-off.
In this case, the usual momentum measure dnp is deformed and becomes dnp
. In the
following, we will restrict ourselves to the isotropic case and work with one space-like dimension.
Following [47, 49] and setting η =
we have
= ~exp
α2L2P l
, (1)
where α is a dimensionless constant of order one.
From Eq.(1) we obtain the dispersion relation
k (p) =
2αLP l
αLP l
, (2)
from which we have the following minimum Compton wavelength
λ0 = 4
παLP l. (3)
Let us show that these results can be obtained from the following representation of the position
and momentum operators
X = i~ exp
α2L2P l
∂p P = p. (4)
The corrections to the standard Heisenberg algebra become effective in the so-called quantum
regime where the momentum and length scales are of the order of the Planck mass MP l and of
the Planck length LP l respectively.
The hermiticity condition of the position operator implies the following modified completeness
relation
p2|p〉〈p| = 1 (5)
and modified scalar product
〈p| p′〉 = e
δ (p− p′) . (6)
From Eq.(5) we observe that we have reproduced the Gaussian damping factor in the Feyn-
man propagator [47, 49]. The algebra defined by Eq. (4) leads to the following generalized
commutator and generalized uncertainty principle (GUP∗)
[X,P ] = i~ exp
α2L2P l
, (δX) (δP ) ≥
α2L2P l
. (7)
In order to investigate the quantum implications of this deformed algebra, we consider the
saturate GUP∗ and solve for (δP ). Using the property 〈P 2n〉 ≥ 〈P 2〉n and (δP )2 = 〈P 2〉−〈P 〉2
the saturate GUP∗ is then given by
(δX) (δP ) =
α2L2P l
(δP )
+ 〈P 〉2
. (8)
Taking the square of this expression we obtain
W (u) eW (u) = u, (9)
where we have set W (u) = −2α
(δP )
and u = − α
2(δX)
〈P 〉2
The equation given by Eq.(9) is exactly the definition of the Lambert function [51]. The
LambertW function is a multivalued functions. Its different branches are labelled by the integer
k = 0,±1,±2, · · · . When u is a real number Eq.(9) have two real solutions for 0 ≥ u ≥ −1
denoted by W0(u) and W−1(u), or it can have only one real solution for u ≥ 0, namely W0(u) .
For -∞ < u < −1
, Eq.(9) have no real solutions.
Using Eq.(9) the uncertainty in momentum is then given by
(δP ) =
〈P 〉2
2 (δX)
2L2P le
2α2L2
〈P 〉2
2 (δX)
 . (10)
Then from the argument of the Lambert function in Eq.(10) we have the following condition
α2L2P le
2α2L2
〈P 〉2
2 (δX)
, (11)
which leads to a minimal uncertainty in position given by
αLP le
〈P 〉2
. (12)
The absolutely smallest uncertainty in position or minimal length is obtained for physical
states for which we have 〈P 〉 = 0 and (δP ) = ~/
2αLP l
, and is given by
(δX)0 =
αLP l (13)
In terms of the minimal length the momentum uncertainty becomes
(δP ) =
2 (δX)
(δX)0
. (14)
Here we observe that 1
(δX)0
< 1 is a small parameter, by virtue of the GUP∗, and perturbative
expansions to all orders in the Planck length can be safely performed.
Indeed a series expansion of Eq.(14) gives the corrections to the standard Heisenberg principle
2 (δX)
(δX)0
(δX)0
(δX)0
+ . . .
. (15)
This expression of (δP ) containing only odd powers of (δX) is consistent with a recent analysis
in which string theory and loop quantum gravity, considered as the most serious candidates for
a theory of quantum gravity, put severe constraints on the possible forms of GUPs and MDRs
[20].
Before ending this section and for later use let us recall the form of the GUP to leading order
in the Planck length widely used in the literature on quantum gravity phenomenology. This
GUP is given by
(δX) (δP ) ≥
α2L2P l
(δP )
. (16)
A simple calculation leads to the following minimal length
(δX)0 = αLP l, (17)
which is of order of the Planck length. However, as nicely noted in [41], this form of GUP do
not fulfill the second requirement listed above. In the following sections we use the form of the
GUP given by Eq.(14) and investigate the thermodynamics of the Schwarzschild black hole.
We use units ~ = c = kB = 1 which imply LP l = M
P l = T
P l =
3 Black hole thermodynamics
The metric of a four-dimensional Schwarzschild black hole is given by
ds2 =
1− 2MG
dt2 −
1− 2M
dr2 − r2dΩ2, (18)
where M represents the mass of the black hole. The Schwarzschild horizon radius, located at
rh, is defined by
rh = 2MG. (19)
Near-horizon geometry considerations suggests to set δX ≃ rh, and then Eq.(19) leads to
minimum horizon radius and minimum mass given by
rh = (δX)0 =
αLP l, M0 =
MP l. (20)
Therefore, black holes with mass smaller than M0 do not exist.
In the standard Hawking picture, temperature and entropy of the Schwarzschild black hole of
mass M are [52, 53]
, S = 4πGM2. (21)
Let us then examine the corrections to the above expressions due to the GUP∗. Following the
heuristic argument of Bekenstein we have
. (22)
Using Eq.(14), the GUP∗-corrected Hawking temperature is
8πML2P l
. (23)
On substituting Eq.(20) into Eq.(23) we obtain the following black hole maximum temperature
TmaxH =
. (24)
The corrections to the standard Hawking temperature are obtained by expanding Eq.(23) in
terms of 1
(M0/M). Indeed we obtain
8πML2P l
+ . . .
. (25)
The variation of the Hawking temperature, Eq.(23), with the mass of the black hole is shown
in figure 1.
It is interesting to inverse Eq.(23) and write the mass of the black hole as a function of the
temperature
8πTHL
TmaxH
. (26)
This relation shows that for temperatures larger than TmaxH , the black hole mass increases with
temperature. In our framework, such a behavior is forbidden by the cut-off brought by GUP∗.
However, in the noncommutative approach to radiating black hole, this behavior is allowed
because of a lack of a generalized uncertainty principle [50].
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 1: The temperature versus BH mass. From left to right: the Hawking result (black solid
line), GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α=1 (green), α=1.25 (blue)
respectively.
We turn now to the calculation of the micro canonical entropy of a large black hole. In the stan-
dard situation the entropy is proportional to the black hole horizon-area. Following heuristic
considerations due to Bekenstein, the minimum increase of the area of a black hole absorbing a
classical particle of energy E and size R is given by (∆A)0 ≃ 4L2P l (ln 2)ER. At the quantum
mechanical level the size and the energy of the particle are constrained to verify R ∼ 2δX and
E ∼ δP . Then we have (∆A)0 ≃ 8L2P l (ln 2) δXδP.
Extending this approach to the case with GUP∗ and using near horizon geometry considerations,
we obtain
(∆A)0 ≈ 4L
P l ln 2 exp
, (27)
where A = 4π (δX)
and A0 = 4π (δX)
0 are respectively the horizon area and minimum horizon
area of the black hole. With the aid of the Bekenstein calibration factor for the minimum
increase of entropy (∆S)0 = ln 2 we have
(∆S)0
(∆A)0
4L2P l
. (28)
Before integrating over A we note that the existence of a minimum horizon area enforces us to
set the lower limit of integration as A0. Then the entropy, up to a irrelevant constant, is
4L2P l
dA.. (29)
The relation e
W (x)
x/W (x) allows us to write Eq.(29) as
4eL2P l
∫ − 1
2 [W (y)]
2 dy, (30)
where PV means the Cauchy principal value of the integral. Setting y = −1
and performing
the integration we obtain the GUP∗-corrected black hole entropy
8eL2P l
))− 1
e− Ei
, (31)
where Ei (x) is the exponential function.
Expanding Eq.(31) in the parameter 1
(A0/A) we have
4L2P l
8L2P le
25πα2
192e2
343πα2
2304e3
+ . . .+ C
where the constant is given by
8L2P le
γ − 1− 2 ln (2e)− 2
e− Ei
≃ −4.60
L2P l
and γ is the Euler constant. The dependence on the Planck length is contained in A0 ∼ L2P l.
We observe that we have reproduced, in our framework with GUP∗, the log-area correction
with a negative sign. Other approaches like string theory, loop quantum gravity and effectif
models with GUPs and/or MDRs, lead to the same sub-leading logarithmic correction. Setting
ρ = −πα2
and β = 3π
in Eqs. (25) and (32) we obtain
M2P l
ρ2 + β/4
, (34)
4L2P l
+ ρ ln
L2P l
βL2P l
. (35)
These expressions are exactly the temperature and entropy obtained in loop quantum gravity
and string theory approach quantum gravity.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 2: The entropy versus BH mass. From left to right: the Hawking result (black solid line),
GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α = 1 (green) and α=1.25 (blue)
respectively.
From figures 1 and 2 it follows that the GUP∗-corrected temperature and entropy are respec-
tively higher and smaller than the semi classical results.
4 Black holes evaporation
As a warming to study the Hawking radiation process of the Schwarzschild black hole, we
examine the effects of GUP∗ on the black body radiation spectrum. With the aid of the
squeezed momentum measure given by Eq.(5), which suppress the contribution of unwanted
high momenta, the energy density of a black body at temperature T is defined by
d3pe−α
T − 1
. (36)
Using the variable y = βp (β = 1/T ) and expanding the exponential, equation (36) becomes
Eγ = 8πT 4
(−1)n
(αT/TP l)
y2n+3
ey − 1
. (37)
Now with the help of the following definition of the Riemann zeta function
ey − 1
= Γ (s) ζ (s) , (38)
we obtain
Eγ = 8πT 4
(−1)n
(αT/TP l)
Γ (2n+ 4) ζ (2n+ 4) . (39)
This energy density is defined only for values of temperatures below some characteristic scale.
In fact Eq.(39) is an alternating series which converge if and only if
(αT/TP l)
Γ (2n + 4) ζ (2n+ 4)
= 0. (40)
From this relation it follows that
T < α−1TP l, (41)
as expected from the Gaussian damping factor in Eq.(36). However, we note that we have a
stronger condition on T . Indeed in our framework, the maximum temperature of the black hole
is given by Eq.(24) and it is approximately 0.1TP l for α of order one. Then the condition on
the BH temperature is rewritten as T/TP l < 0.1. For our purpose, the latter constraint allows
us to cut the series in Eq.(39) at n = 1. Using ζ (4) = π
and ζ (6) = π
and Eq.(24) we finally
obtain, from Eq.(39) , the following expression
Eγ (T ) =
1− 15
TmaxH
. (42)
The first term is the standard Stefan-Boltzmann law while the second term is the correction
brought by GUP∗.
We are now ready to study the Hawking evaporation process. The intensity emitted by a black
hole of mass M is defined by
I = AEγ (TH) , (43)
where A is the BH horizon area. Invoking energy conservation, the evaporation rate of the
black hole is
= −AEγ (TH) . (44)
Using Eq. (23) for the corrected Hawking temperature we obtain
= − γ1
M2L4P l
1− 8γ2
with γ1 =
, γ2 =
16128
. The deviations from the standard expression are obtained by applying
a series expansion in 1
(M0/M)
= − γ1
M2L4P l
1− 2γ2
1− 72γ2
25eγ1
+ . . .
The variation of the evaporation rate with the black hole mass is shown in Figure 3. We clearly
observe that the divergence for M → 0 in the standard description of the black hole evaporation
process is now completely regularized by the GUP∗. This regularization is also reflected by the
constraint (41), which suppress the evaporation process beyond the Planck temperature. This
phenomenon is similar to the prevention, by the standard uncertainty principle, of the hydrogen
atom from total collapse. In our picture, the regularization can be considered as a dynamical
effect and not as a consequence of any quantum symmetry in the theory.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 3: The evaporation rate versus BH mass. From left to right: the Hawking result (black solid line) ,
GUP (doted line) and GUP∗ results (solid line) for α = 0.90 (red) , α = 1. (green) and α = 1.25
(blue).
On the other hand, we observe that the evaporation phase ends when the BH mass becomes
equal to M0 with a minimum rate given by
(γ1 − 8γ2)M4P l. (47)
Thus the evaporation process of a black hole with initial mass M > M0 continue until the
horizon radius becomes (δX)0 , leaving a massive relic referred to, in the literature, as a black
hole remnant (BHR). To find the nature of the BHR we calculate the heat capacity defined by
. (48)
Using the expression of temperature given by (23) we easily obtain
C = −8πM2L2P l
. (49)
This expression vanishes when 1 + W
(M0/M)
= 0, whose solution is M = M0. We
conclude that the heat capacity of the black hole vanishes at the end point of the evaporation
process characterized by a BHR with mass M0.Besides the gravitational interaction with the
surrounding, the vanishing of the heat capacity reveals the inert character of the BHRs and
thus make them as potential candidates to explain the origin of dark matter [18, 19]. Finally
we note that, as it is the case with the form of the GUP to leading order in the Planck length,
the BHRs are also a consequence of GUP∗ [17, 55].
We have drawn the variation of the heat capacity with BH mass in figure 4. In it we see that,
the heat capacity vanishes for M0 ≃ 0.50, 0.75 in the case with GUP and M0 ≃ 0.58, 0.87 in
the case with GUP∗.
0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 4: The heat capacity versus BH mass. From left to right: the standard result (black solid
line), GUP (doted line) and GUP∗ results (solid line) for α = 1 (red) and α = 1.5 (blue) respectively.
Taylor expanding Eq.(49) we have
C = −8πM2L2P l
− . . .
. (50)
The standard expression of the heat capacity C = −8πM2 is reproduced in the limit of black
holes with mass larger than the minimum mass M0. The correction terms to the heat capacity
due to GUP∗ are all positive indicating that the evaporation process is accelerated and leading
to a corrected decay time smaller than the decay time in the standard case.
Let us consider a black hole starting the evaporation process with a mass M and ending the
process with the minimum mass M0. Using (45) and the variable y = −1e (M0/M)
, the decay
time is given by
t = (−1)7/2
∫ − 1
(M0/M)
W−5/2 (y) e−
W (y)dy
+ (−1)5/2
2γ21e
2L2P l
∫ − 1
(M0/M)
W−3/2 (y) e−
W (y)dy. (51)
Performing the integration we obtain
4 (1− ǫ)
W (y)
−W (y)
W 2 (y)
+ C, (52)
where the constant C is the value for y = −1/e and ǫ = 3γ2
∼ 10−6 for α of order
one. Ignoring ǫ and performing a series expansion in y we have
M3L4P l
+ . . .
Then to first order in 1
(M0/M) the relative correction to the decay time is
, (54)
where t0 =
is the decay time without GUP∗. From Eq.(54) , it follows that black holes
with GUP∗ are hotter and decay faster than in the standard case.
Let us now turn to a comparison of the corrected BH thermodynamics with GUP∗ with the
corrections brought by the GUP to leading order in the Planck length. Since our comparison
is quantitative we use the Planck units. Repeating the same calculations as above with the
GUP given by Eq.(16), the temperature, the entropy and the heat capacity of the black hole
are respectively given by
TGUP =
, (55)
SGUP = 2πM
CGUP = πα
1− α2
1− α2
. (57)
The minimum black hole mass and maximum temperature allowed by GUP areM0 = (δX)0 /2 =
and Tmax = 1/2πα. In figures 1, 2, 3 and 4 we have plotted, besides the results obtained with
GUP∗, the variation of temperature, entropy, evaporation rate and heat capacity with GUP as
functions of the black hole mass for different values of the parameter α. Figure 2 shows, that in
the scenario with GUP∗, the BH entropy decreases compared to the entropy in the standard
case and the scenario with GUP. This reveals the deeper quantum nature of the black hole
in the scenario with GUP∗. Thus quantum effects become manifest at an earlier stage of the
evaporation phase than was predicted by the semi classical Hawking analysis [54] and the GUP
analysis [55].
The calculation of the evaporation rate in the framework with GUP requires a careful analysis.
In all the calculations done until now, the validity of the Stefan-Boltzmann law is assumed,
ignoring the UV cut-off implemented by GUP. However, it was pointed in [16] that the effect
of the GUP should be also reflected in a modification of the de Broglie wave length relation
1 + α2p2
. (58)
This relation must be translated into a modification of the momentum measure such that the
contributions of high momenta are suppressed. As shown in [7], the GUP to leading order in
the Planck length leads to a squeezing of the momentum measure by a factor 1
(1+α2L2Plp2)
. Then
following the same calculation leading to Eq.(39) , the energy density of a black body with GUP
(1 + α2L2P lp
T − 1
. (59)
Performing the integral and using the same argument as before, we obtain the expression given
by Eq. (42) . We note, that in a recent calculation of the Stefan-Boltzmann law with GUP
[24], the sign of the correction term is positive, in contradiction with the role of the UV cut-off
implemented by the GUP.
The correct evaporation rate with GUP is then given by
128π2M6
1024π2M8
63α10
. (60)
In figure 3 we observe that the evaporation process with GUP is retarded compared to the
process with GUP∗ and that the process ends at a mass M0 = α/2 with a minimum rate given
min,GUP
= − 32π
1260M20
, (61)
which is greater (in absolute value) than the one obtained with GUP∗.
In table 1 we show the GUP and GUP∗-corrected thermodynamics of two black holes with
initial mass equal to 2MP l and 5MP l for α = 1. The first row gives the semi classical Hawking
results. The second row gives the GUP-corrected results and the third row the GUP∗−corrected
ones. It is interesting to note that, in the scenario with GUP∗, the final stage of the evaporation
phase is a remnant with a mass larger than the one obtained with GUP and that the decay time
is drastically reduced. In a scenario with extra dimensions, these results may have important
consequences on possible black holes production at particle colliders and in ultrahigh energy
cosmic ray (UHECR) air-showers.
Finally, let us notice that the corrections to the black hole thermodynamics become indistin-
guishable in the two version of GUP in the limit of large mass and small values of α. However, for
growing values of the parameter α, corresponding to strong gravitational effects, the predictions
of the two GUPs concerning the entropy become different even for massive black holes.
Table 1. GUP and GUP∗-corrected thermodynamics for two BHs with mass M = 2 and M = 5 (in
Planck units). The deviations from the Hawking results are also given.
M = 2
α Minimum mass Initial temp Final temp Decay time Entropy
0 - 0.019 ∞ 129.69 50.27
1.0 0.5 0.020 (+3%) 0.16 111.92 (−14%) 44.66 (−11%)
1.0 0.58 (+16%) 0.020 (+3%) 0.11 (−31%) 3.33 (−97%) 43.73 (−13%)
M = 5
α Minimum mass Initial temp Final temp Decay time Entropy
0 - 0.008 ∞ 2026.42 314.16
1.0 0.5 0.008 0.16 1976.60 (−2.5%) 307.10 (−2%)
1.0 0.58 (+16%) 0.008 0.11 (−31%) 22.17 (−99%) 306.18 (−2.2%)
5 Conclusion
In this paper we have studied how black holes thermodynamic parameters are affected by a
GUP to all orders in the Planck length. We have obtained exact analytic expressions for the
Hawking temperature and entropy. Particularly we found that a black hole with a mass smaller
than a minimum mass do not exist. The existence of a energy scale which is one order below the
Planck scale allowed us to calculate, to leading order, the deviations from the standard Stefan-
Boltzmann law. Then we investigated the Hawking radiation process of the Schwarzschild
black hole and shown that at the end of the evaporation phase a inert massive relic continue
to exist as a black hole remnant (BHR) with zero entropy, zero heat capacity and non zero
finite temperature. For completeness, we have also compared our results with the semi classical
results and the predictions of the GUP to leading order in the Planck length. In particular,
we have shown that the entropy in our framework is smaller than the entropy in the standard
case and with GUP. We have also made the correct calculation of the evaporation rate with
GUP. Finally, we have shown that black holes with the form of GUP used in this paper are
hotter, shorter-lived and tend to evaporate less than black holes in the semi classical and the
GUP to leading order pictures. On the other hand, we have found that the predictions of the
GUP to all orders in the Planck length and the GUP to leading order in the Planck length,
concerning the entropy, become different for strong gravitational effects and large black holes
mass, suggesting a further investigation of the early universe thermodynamics in the framework
with the GUP to all orders in the Planck length. In a future work we will examine the effects of
the GUP to all orders in the Planck length on black holes thermodynamics in a scenario with
extra dimensions.
Acknowledgments
The author thanks the referees for their comments and valuable remarks.
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	Introduction
	Generalized uncertainty principle
	Black hole thermodynamics
	Black holes evaporation
	Conclusion
