The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length

Recently, the Heisenberg's uncertainty principle has been extended to incorporate the existence of a large (cut-off) length scale in de Sitter or anti-de Sitter space, and the Hawking temperatures of the Schwarzshild-(anti) de Sitter black holes have been reproduced by using the extended uncertainty principle. I generalize the extended uncertainty to the case with an absolute minimum length and compute its modification to the Hawking temperature. I obtain a general trend that the generalized uncertainty principle due to the absolute minimum length ``always'' increases the Hawking temperature, implying ``faster'' decay, which is in conformity with the result in the asymptotically flat space. I also revisit the ``black hole-string'' phase transition, in the context of the generalized uncertainty principle.


I. INTRODUCTION
The Heisenberg's uncertainty principle provides a basic limitation of measuring the classical trajectories in the atomic or sub-atomic scale. But here, there is no absolute minimum or maximum uncertainty in the position and momentum themselves, though there is "conditional" minimum in them when one of them is fixed. So, in this regards, there have been arguments that the Heisenberg's uncertainty principle needs some modifications when the gravitational interaction is considered in quantum mechanics since there is an absolute minimum uncertainty in the position of any gravitating quantum [1,2]. And also, its several interesting implications have been studied in the literatures. Especially, it has been found that the generalized uncertainty principle (GUP) increases the Hawking temperature, resulting in "faster" decay of Schwarzschild black holes in any dimension [3,4].
However, the GUP does not have any limitation on the maximum uncertainty in the position such as it can not be naively applied to the case with the large (cut-off) length scales, like as in de Sitter or anti-de Sitter space. Actually, the Hawking temperature of black holes in (anti) de Sitter space can not be reproduced by the Heisenberg's uncertainty principle or the GUP. Recently, an extended uncertainty principle ( I will call this "EUP", simply) has been introduced to incorporate the existence of the large length scales and it is found that the Hawking temperatures of the Schwarzshild-(anti) de Sitter black holes have been correctly reproduced [5].
In this paper, I generalize the EUP to the case with an absolute minimum uncertainty in the position as well and compute its modification to the Hawking temperature. I obtain a general trend that the generalized uncertainty principle due to the absolute minimum length always increases the Hawking temperature, implying faster decay, which is in conformity with the result of the asymptotically flat space. I also revisit the black hole-string phase transition, in the context of the generalized uncertainty principle.

II. THE GUP AND HAWKING TEMPERATURE IN ASYMPTOTICALLY FLAT SPACE
In this section, I review, with some new interpretations and remarks, the GUP and the derivation of Hawking temperature from the uncertainty principle in the asymptotically flat space [3,4].
The GUP is given by where x i and p j (i, j = 1, · · · , d − 1) are the spatial coordinates and momenta, respectively; l P = (hG) 1/(d−2) is the Planck length and α is a dimensionless real constant of order one [1]. In the absence of the second term in the right hand side, this reduces to the usual Heisenberg's uncertainty principle without any "absolute" bound of ∆x i nor ∆p j themselves. But, in the presence of the second term, there exists an absolute minimum in the position uncertainty and the uncertainty in the momentum is given bȳ The left inequality in (3) provides some small corrections to the Heisenberg's uncertainty principle for ∆x i ≫ αl P (i.e., semi-classical regime), On the other hand, the right inequality implies that ∆p i can not be arbitrarily large in order that the correction in (1) makes sense. Of course, this upper bound can be higher with the higher order terms in the right hand side of the GUP (1), but the absolute minimum in ∆x i can be also lowered or even disappeared, depending on the parameters [6]. Another more interesting interpretation would be that the upper bound corresponds to the limit where the quantum gravity effects are very strong such as a black hole-string phase transition can occur [7]. Actually, the inequality can be written also as which can be directly derived also from the high momentum uncertainty ∆p j limit in (1), and it is saturated by the linear relation ∆p i =h∆x i /α 2 l 2 P , which coincides with that of strings at the high energy limit, by identifying the string scale l S ≈ αl P [2,5]. Now, let me derive the Hawking temperature from the uncertainty principle and general properties of black holes. To this end, let me first consider a d−dimensional Schwarzshild black hole with a metric given by where and Ω d−2 is the area of the unit sphere S n−2 [8]. By modeling a black hole as a black box with linear size r + , the uncertainty in the position of an emitted particle by the Hawking effect is with the radius of the event horizon r + . In the absence of the GUP effect, the horizon radius is given by from the metric (6). On the other hand, in the presence of the GUP effect, the precise form of the horizon radius r + = r + (M, αl P ) is not known unless the GUP corrected metric is known, which is beyond the scope of this paper. However, I note that the relation (8) would be generally valid even with the GUP effect, In the absence of the GUP, there is no absolute minimum radius for the black hole evaporation (thin line). With the GUP, the Hawking temperature becomes hotter, implying faster decay, and also there is a minimum radius r + = 2αl P where the curve ends, implying that the black hole evaporation stops (thick line). Here, I have plotted the cases with h = l P = 1, α = 0.5 and the GUP curve stops at r + = 1.
with understanding r + as the GUP corrected horizon already. Then, the uncertainty in the energy of the emitted particle is ( by neglecting the mass of the emitted particle ) 1 By assuming that ∆E, which can be identified as the characteristic temperature of the Hawking radiation, saturates the left inequality 2 , one can obtain the Hawking temperature Here, the "calibration" factor '(d − 3)/4π' has been introduced in order to have agreements with the usual Hawking temperature of the Schwarzschild black hole in the leading term, for a large black hole , i.e., r + ≫ αl P [8,9]: Before finishing this section, I remark first that the formula (8), as a result (10), is still valid even for the small black holes up to the absolute minimum, which is order of Planck length l P , though the series formula (11) is valid only for a large r + . The black hole evaporation stops at r + = 2αl P , where the curve ends, and this would correspond to a "melting" of 1 There might exist high energy modifications in the dispersion relation (9), generally [10]. But, I will not consider this possibility in this paper. 2 This assumption would correspond to the Bekenstein bound of the entropy of an arbitrary bounded system BH dM BH ) whose upper bound is saturated by that of black holes, S BH , for a given mass M = M BH [11].
the black hole which is followed by the string phase, according to the new interpretation [7]. Second, the effect of the GUP with an absolute minimum length increases the Hawking temperature always and this implies that it decays faster than the usual Schwarzschild black hole without the GUP (Fig.1).

III. THE EUP AND HAWKING TEMPERATURE IN (A)DS SPACE
The GUP can not be naively applied to the space with the large length scales like as in (A)dS space 3 . In this section, I consider an extension of the uncertainty principle in order to incorporate the large-length scales and derivation of Hawking temperature from the uncertainty principle.
The extended uncertainty principle (EUP) is given by 4 where l is the characteristic, large length scale and β is a dimensionless real constant of order one [5] 5 . (For some gedanken experiments' derivation, even without considering black holes, see also Ref. [16].) Then, it is easy to see that there is an absolute minimum in the momentum uncertainty Here, I note that the first inequality is an "exact" relation drawn from (12), without considering any limit as in (4). Now, using the approach in Sec. II, it is straightforward to see that the Hawking temperature of the Schwarzshild-AdS black holes from the EUP (13) 6 . To this end, let me first consider a d-dimensional Schwarzshild-AdS black hole with the metric function in the metric (6) and a cosmological constant Λ = −(d −1)(d −2)/2l 2 AdS [13]. Then, with the same identifications (8) and (9) for the Hawking-emitted particles, which do not depends on the large scale behaviors but only on the local structure near the horizon, one can obtain the Hawking temperature T EU P ≈ ∆p i , 3 This has been noted earlier by Konishi et al. also [2]. See also Ref. [12] for another related work. 4 The parameter β in Ref. [5] is related to here's by β T here = (l P /l)β Here . 5 This has been considered earlier by Kempf et. al. also [15], but its physical consequences have not been studied. 6 For an alternative derivation from the laws of classical physics and Heisenberg's uncertainty principle, see Ref. [17]. But, there is no room for the GUP in that derivation.
with the same calibration factor '(d − 3)/4π' as in the asymptotically flat case, implying its universality, and β = (d − 1)/(d − 3), l = l AdS ; r + is the radius of the event horizon which solves N 2 (r) = 0. Here, the existence of the absolute minimum in ∆p i and so in T EU P (AdS) is a general consequence of the EUP of (12) (see Fig. 2 (thin line)).
So far, I have shown that the EUP in (12) applies to the AdS space. Now, the EUP for the dS space can be easily constructed by considering l 2 → −l 2 in (12): Then, in contrast to (12), there is an absolute maximum in ∆x i as in order that ∆p i is not negative 7 , Note that the absolute maximum in ∆x i does not haveh such as this is a purely classical result. The Hawking temperature of the Schwarzshild-dS black hole with a cosmological constant [14] is similarly computed as, by considering l 2 AdS → −l 2 dS in (15), Here, the maximum bound reads ( l = l dS , β = (d − 1)/(d − 3) ) which is the Nariai bound where the black hole horizon and the cosmological horizon meet [19]. So, the condition (17) reflects the fact that the uncertainty in the position can not exceed the cosmological horizon, which is the size of the casually connected world in a dS space (see Fig.3 (thin line)).

IV. THE GENERALIZED EUP (GEUP)
In the EUP (12), there is an absolute minimum in the uncertainty of the momentum. In this section, I generalize the EUP to have a minimum length scale as well, by combining the GUP and the EUP, and study the effect of the minimum length to the Hawking temperature from the EUP, i.e., the Hawking temperature of Schwarzshild-(A)dS black holes.

The generalized EUP (GEUP) is given by
where I have considered the case of the AdS space, first. Then, by inverting (21), one has the inequalities, Here, one finds that there are, now, both the absolute minimum in ∆x i and ∆p i from the reality of ∆p (±) i and ∆x (±) i , respectively, with the condition The left inequality in (22), as in (3) of the GUP, provides some small corrections to the Heisenberg's uncertainty principle, due to the minimum length and momentum, for αl P ≪ ∆x i ≪ l/β, By repeating the same arguments as in the GUP and the EUP cases (with understanding that r + as the GUP corrected horizon), one can obtain the Hawking temperature T GEU P ≈ ∆p  Here, the third term is purely the GUP correction and the second term in the first bracket { } is the GEUP effect, and these correction terms are all positive. This shows that the Hawking temperature of the AdS black hole is increased also by the minimum uncertainty in the position, with the GUP.
The analysis for the dS case is also straightforward. From the GEUP with l 2 → −l 2 , one has with the minimum uncertainty in ∆x i (but none in ∆p i ) Moreover, in order that ∆p is not negative one obtains the same condition as (17) which is unchanged by the GUP effect (i.e., no α dependence), in contrast to the lower bound in (31). Then, one finds the Hawking temperature which gives for semiclassical dS black holes with αl P ≪ r + ≪ (d − 3)/(d − 1)l dS . Here, note that the maximum bound of the black hole horizon (20) is not changed by the existence of the minimal length but the temperature is always increasing: The second term in the first bracket { } gives a negative correction but this is dominated by the third term, which is always positive. Now, one finds a quite general trend that the GUP due to a minimal length increases always the Hawking temperature (Fig.2, 3), regardless of being asymptotically flat or (A)dS space. This can be traced back to the universal appearance of the term "+α 2 l P /r 3 + " in the temperature formula, which makes the decay to be faster. This seems to be also true in other forms of the deformation of the uncertainty principle [20].
Finally, two remarks are in order. First, one might consider the first law of thermodynamics to compute the GUP corrected black hole entropy from the same ADM mass formula as that of the case without the GUP [3]. But, it still unclear how to fix uniquely the GUP corrected mass formula from the GUP corrected Hawking temperature, without knowing the precise form the GUP corrected gravity and its black hole solutions.
Second, I note that, in the d = 3 limit of the AdS black holes (i.e., the BTZ black hole limit), one has the Hawking temperature from the series formula (29), though it needs a scale tuning l AdS → ∞, d → 3, with ' √ d − 3l AdS = a fixed large number'. This shows also an increase of the temperature, implying faster decay from the GUP effect, compared to that of the usual BTZ black hole, T BT Z =hr + /(2πl 2 AdS ). But, remarkably, there is a minimum temperature at r + = √ 2αl P and growing temperature for smaller black holes, in contrast to the monotonically decreasing temperature as r + becomes smaller in the BTZ black hole without the GUP. If this were true, the Hawking-Page transition [13] would occur even in three-dimensional AdS space, due to the GUP effect. But, this does not seem to occur from (24), which implies r + ≥ 2αl P for consistency of the exact formula (22), such as the evaporation stops before reaching the absolute minimum of the temperature at r + = √ 2αl P . This needs more rigorous analysis which can be well-defined in the three dimension, from the start [20].