A lower bound on the Bekenstein-Hawking temperature of black holes

We present evidence for the existence of a quantum lower bound on the Bekenstein-Hawking temperature of black holes. The suggested bound is supported by a gedanken experiment in which a charged particle is dropped into a Kerr black hole. It is proved that the temperature of the final Kerr-Newman black-hole configuration is bounded from below by the relation $T_{\text{BH}}\times r_{\text{H}}>(\hbar/r_{\text{H}})^2$, where $r_{\text{H}}$ is the horizon radius of the black hole.


I. INTRODUCTION
It is well known [1,2] that, for mundane physical systems of spatial size R, the thermodynamic (continuum) description breaks down in the low-temperature regime T ∼h/R [3]. In particular, these low temperature systems are characterized by thermal fluctuations whose wavelengths λ thermal ∼h/T are of order R, the spatial size of the system, in which case the underlying quantum (discrete) nature of the system can no longer be ignored. Hence, for mundane physical systems of spatial size R, the physical notion of temperature is restricted to the high-temperature thermodynamic regime [1,2] T × R ≫h . (1) Interestingly, black holes are known to have a well-defined notion of temperature in the complementary regime of low temperatures. In particular, the Bekenstein-Hawking temperature of generic Kerr-Newman black holes is given by [4,5] where are the radii of the black-hole (outer and inner) horizons [6]. The relation (2) implies that near-extremal black holes in the regime (r + − r − )/r + ≪ 1 are characterized by the strong inequality [7] T BH × r + ≪h .
It is quite remarkable that black holes have a well defined notion of temperature in the regime (4) of low temperatures, where mundane physical systems are governed by finite-size (quantum) effects and no longer have a selfconsistent thermodynamic description.
One naturally wonders whether black holes can have a physically well-defined notion of temperature all the way down to the extremal (zero-temperature) limit T BH × r + /h → 0? In order to address this intriguing question, we shall analyze in this paper a gedanken experiment which is designed to bring a Kerr-Newman black hole as close as possible to its extremal limit. We shall show below that the results of this gedanken experiment provide compelling evidence that the Bekenstein-Hawking temperature of the black holes is bounded from below by the quantum inequality T BH × r + ≫ (h/r + ) 2 .

II. THE GEDANKEN EXPERIMENT
We consider a spherical body of proper radius R, rest mass µ, and electric charge q which is slowly lowered towards a Kerr black hole of mass M and angular momentum J = M a along the symmetry axis of the black hole [8]. The black-hole spacetime is described by the line element [9-11] where ∆ ≡ r 2 − 2M r + a 2 and ρ 2 ≡ r 2 + a 2 cos 2 θ. The test-particle approximation implies that the parameters of the body are characterized by the strong inequalities These relations imply that the particle which is lowered into the black hole has negligible self-gravity (that is, µ/R ≪ 1) and that it is much smaller than the geometric length-scale set by the black-hole horizon radius. In addition, the weak (positive) energy condition implies that the radius of the charged body is bounded from below by its classical radius [12][13][14] R ≥ R c ≡ q 2 2µ .
This inequality ensures that the energy density inside the spherical charged body is positive [15]. The energy [16] of the charged body in the near-horizon black-hole spacetime is given by [15,17] where r = r 0 is the radial coordinate of the body's center of mass in the black-hole spacetime. The first term on the r.h.s of (8) represents the energy associated with the rest mass µ of the body red-shifted by the black-hole gravitational field [4,18]. The second term on the r.h.s of (8) represents the self-energy of the charged body in the curved black-hole spacetime [15,17,19,20]. The proper height l of the body's center of mass above the black-hole horizon is related by the integral relation [4] to the Boyer-Lindquist radial coordinate r 0 . In the near-horizon l ≪ r + region one finds the relation where α ≡ r 2 + + a 2 . Taking cognizance of Eqs. (8) and (10), one finds for the energy of the body in the near-horizon l ≪ r + region. Suppose now that the charged object is slowly lowered towards the black hole until its center of mass lies a proper height l 0 (with l 0 ≥ R) above the black-hole horizon. The object is then released to fall into the black hole. The assimilation of the charged body by the black hole produces a final Kerr-Newman black-hole configuration whose physical parameters (mass, charge, and angular momentum) are given by The change in the black-hole temperature caused by the assimilation of the charged body can be quantified by the dimensionless physical function whereā ≡ a/M is the dimensionless angular momentum of the black hole [21]. Our goal is to bring the black hole as close as possible to its extremal (zero-temperature) limit. Thus, we would like to minimize the value of the dimensionless physical parameter Θ. In particular, we would like to examine whether Θ(ā), the dimensionless change in the black-hole temperature, can be made negative all the way down to the extremal a → 1 (zero-temperature, T BH → 0) limit.
We shall henceforth consider black holes in the regimē in which case a minimization of the energy delivered to the black hole also corresponds to a minimization of the Bekenstein-Hawking temperature of the final black-hole configuration [22]. The fact that the energy E(l 0 ) of the charged particle in the black-hole spacetime is an increasing function of the dropping height l 0 [see Eq. (11)] implies that, in order to minimize the physical parameter Θ(ā) in the regime (14), one should release the body to fall into the black hole from a point whose proper height above the black-hole horizon is as small as possible. We therefore face the important question: How small can the dropping height l 0 be made? As pointed out by Bekenstein [4], the expression (11) for the energy of our charged spherical object in the black-hole spacetime is only valid in the restricted regime l 0 ≥ R, where every part of the body is still outside the horizon. This fact implies, in particular, that the adiabatic (slow) descent of the charged spherical body towards the black hole must stop when its center of mass lies a proper height l 0 → R + above the horizon. At this point the bottom of the body is almost swallowed by the black hole and the body [having a minimized (red-shifted) energy E(l 0 → R)] should then be released to fall into the black hole [4]. In addition, remembering that the weak (positive) energy condition sets the lower bound (7) on the proper radius of the charged spherical body, one finds the relation [23] for the optimal [24] dropping point of the charged body. Substituting (15) into (11), one finds the remarkably simple (and universal [25]) expression for the minimal energy delivered to the black hole by the charged body. Taking cognizance of Eqs. (2), (12), (13), and (16), one finds the universal expression [26-28] for the smallest possible (most negative) value of the dimensionless physical parameter Θ(ā) which quantifies the change in the black-hole temperature caused by the assimilation of the charged body. Interestingly, one finds from (17) the characteristic inequality which is valid for all valuesā ∈ [0, 1) of the black-hole rotation parameter. The simple inequality (18) implies that, by absorbing charged particles, the black hole can approach arbitrarily close to the extremal (zero-temperature) T BH → 0 limit. It is important to emphasize again that this conclusion is based on the assumption [4] that the charged body can be lowered adiabatically (slowly) until its bottom almost touches the black-hole horizon [29]. In the next section we shall show, however, that Thorne's famous hoop conjecture [30] implies that, for near-extremal black holes, the charged body cannot be lowered adiabatically all the way down to the horizon of the black hole.

III. THE HOOP CONJECTURE AND THE LOWER BOUND ON THE BLACK-HOLE TEMPERATURE
In the previous section we have seen that, by absorbing a charged particle, a black hole can approach arbitrarily close to the extremal (zero-temperature) T BH → 0 limit. As we have emphasized above, this interesting conclusion rests on the assumption that the charged body can be lowered slowly all the way down to the horizon of the black hole [29]. In the present section we shall show, however, that Thorne's famous hoop conjecture [30] sets a lower bound on the minimal proper height l min 0 that the charged body can approach the black-hole horizon without being absorbed, a bound which may be stronger than the previously assumed bound (15).
The Thorne hoop conjecture [30] asserts that a physical system of total mass (energy) M forms a black hole if its circumference radius r c is equal to (or smaller than) the corresponding radius r Sch = 2M of the Schwarzschild black hole. It is worth emphasizing that the validity of this version of the hoop conjecture is supported by several studies [31]. However, it is also important to emphasize the fact that there are known spacetime solutions of the Einstein field equations which provide explicit counterexamples to this version of the hoop conjecture [32,33].
A weaker (and therefore a more robust) version of the hoop conjecture for spacetimes with no angular momentum was suggested in [34,35]. Here we would like to generalize this weaker version of the hoop conjecture to the generic case of spacetimes which possess angular momentum and electric charge. In particular, we conjecture that: A physical system of mass M , angular momentum J, and electric charge Q forms a black hole if its circumference radius r c is equal to (or smaller than) the corresponding Kerr-Newman black-hole radius r KN = M + M 2 − (J/M ) 2 − Q 2 . That is, we conjecture that In the context of our gedanken experiment, this weaker version of the hoop conjecture implies that a new (and larger) horizon is formed if the charged body reaches the radial coordinate r 0 = r hoop , where r hoop (µ, q) is defined by the Kerr-Newman functional relation [see Eq. (3)] Substituting (8) into (20), and assuming r hoop − r + ≪ r + − r − ≪ r + [36], one finds [37] for the radius of the new horizon, where Substituting the radial coordinate (21) into Eq. (10), one finds Taking cognizance of Eqs. (15) and (23) one realizes that, in the regime a new (and larger) horizon is formed [38] before the spherical charged body [39] touches the horizon of the original black hole. Thus, in the regime (24), one should take [40] l min in Eq. (11) in order to minimize the energy delivered to the black hole by the charged body [41]. This implies [37] E min (ā) = 4βµ 2 + q 2 4r + (26) for the smallest possible energy delivered by the charged particle to the black hole in the regime (24). Taking cognizance of Eqs. (2), (12), (13), and (26), one finds the relation in the regime (24). Interestingly, one finds from (27) that the black-hole-charged-body system is characterized by the inequality in the regime (24). Note, in particular, that the inequality (24) is satisfied by near-extremal black holes whose dimensionless temperature τ is characterized by the relation [see Eqs. (22) and (23)] Taking cognizance of Eqs. (28) and (29) one realizes that, in our gedanken experiment, the Bekenstein-Hawking temperature of the black holes cannot be lowered below the critical value [42] T c where µ and q are the proper mass and electric charge of the absorbed particle, respectively.

IV. THE QUANTUM BUOYANCY EFFECT AND THE LOWER BOUND ON THE BLACK-HOLE TEMPERATURE
Thus far, we have analyzed the gedanken experiment at the classical level. It is important to emphasize, however, that the well known quantum buoyancy effect [43] in the black-hole spacetime should also be taken into account in the present gedanken experiment. This quantum buoyancy effect stems from the fact that the slowly lowered object interacts with the quantum thermal atmosphere of the black-hole spacetime [43,44].
In particular, as shown by Bekenstein [44], the quantum buoyancy effect shifts the optimal dropping point [24] of the object from l min 0 = R [see Eq. (15)] to a slightly higher point whose proper radial distance from the black-hole horizon is given by [44] l min where the dimensionless factor ǫ is given by [44,45] ǫ ≡ N 720π ·h µR (32) and N is the effective number of quantum radiation species [44]. The quantum shift (increase) ǫR [see Eq. (31)] in the radial proper distance of the optimal dropping point results in a quantum increase ǫ · (r + − r − )µR/α [44] in the energy delivered to the black hole. Taking into account this quantum buoyancy increase in the energy delivered to the black hole, one finds that the classical expression (17) for the dimensionless function Θ(ā) acquires a positive quantum correction term. In particular, for near-extremal black holes the quantum-mechanically corrected expression for Θ(ā) is given by [37,39] Interestingly, one finds from (33) that the black-hole-charged-body system is characterized by the inequality in the regime The relations (34) and (35) imply that, due to the quantum buoyancy effect [43,44], the Bekenstein-Hawking temperature of the black holes cannot be lowered below the critical value [42] T c BH × r + = ǫ ·h π .

V. SUMMARY AND DISCUSSION
We have analyzed a gedanken experiment in which a spherical charged particle is lowered into a Kerr black hole. It was shown that if the charged particle can be lowered slowly all the way down to the horizon of the black hole, then the Bekenstein-Hawking temperature of the final black-hole configuration can approach arbitrarily close to the extremal (zero-temperature) T BH → 0 limit. However, we have shown that Thorne's famous hoop conjecture [30] [and also its weaker (and more robust) generalization (19)] implies that, for near-extremal black holes in the regime (29), a new (and larger) horizon is already formed before the charged particle touches the horizon of the original black hole. The hoop conjecture therefore implies that, in our gedanken experiment, the temperature of the final [46] black-hole configuration cannot approach arbitrarily close to zero [47]. In particular, we have proved that the Bekenstein-Hawking temperature of the black holes is an irreducible quantity in the near-extremal regime T BH < T c BH determined by the critical temperature (30). It is worth emphasizing that we have provided in this paper only one specific example, not a general proof, to the fact that the black-hole temperature cannot approach arbitrarily close to zero. Nevertheless, this intriguing conclusion of our gedanken experiment [48] makes it tempting to conjecture that the Bekenstein-Hawking temperature of black holes is bounded from below by the simple universal relation [see Eq. (30)] [49][50][51] We believe that it would be highly important to test the general validity of the conjectured lower bound (37) on the Bekenstein-Hawking temperature of the black holes. 0 ≪ r+ [see Eq. (10)] corresponds to charged particles in the regime q 2 ≪ µr+.
[24] That is, the dropping point for which the energy delivered to the black hole, and thus also the physical parameter Θ(ā), are minimized.
[25] The expression (16) for the minimal energy delivered to the black hole by the charged body is universal in the sense that it is independent of the black-hole rotation parameterā.
[26] The expression (17) for the dimensionless physical quantity Θ min (ā) is universal in the sense that it is independent of the black-hole rotation parameterā.
[28] Note that the relation q 2 = 2µR ≪ r 2 + for our charged spherical object [see Eqs. (6) and (7)] implies |∆T min BH | ≪ TBH. Here ∆T min BH denotes the most negative value which is physically allowed for the change ∆TBH in the black-hole temperature in our gedanken experiment.