Black-hole evaporation, cosmic censorship, and a quantum lower bound on the Bekenstein-Hawking temperature

The semi-classical Hawking evaporation process of Reissner-Nordstr\"om black holes is analyzed. It is shown that this quantum mechanism may turn a near-extremal black-hole spacetime with $T_{\text{BH}}\lesssim \hbar^2/G^2M^3$ into an horizonless naked singularity, thus violating the Penrose cosmic censorship conjecture. It is therefore conjectured that, within the framework of a self-consistent quantum theory of gravity, the Bekenstein-Hawking temperature should be bounded from below by the simple relation $T_{\text{BH}}\gtrsim \hbar^2/G^2M^3$.


I. INTRODUCTION
Following Bekenstein's proposal that black holes have a well defined entropy [1], Hawking, using a semi-classical analysis [2], has revealed the intriguing fact that these fundamental objects of general relativity are characterized by thermally distributed (filtered black-body) radiation spectra. In particular, the emission spectra of the canonical family of charged and rotating Kerr-Newman black holes are characterized by the well defined Bekenstein-Hawking temperature [3] . (1) Here are the (inner and outer) horizon radii which characterize the black-hole spacetime, where {M, Q, a} are respectively the asymptotically measured black-hole mass, electric charge, and angular momentum per unit mass.
Inspection of the functional expression (1) for T BH = T BH (M, Q, a) immediately reveals the intriguing fact that the semi-classical Bekenstein-Hawking temperature approaches zero in the M 2 − Q 2 − a 2 → 0 limit of near-extremal black holes.
In the present paper we shall address the following physically interesting question: Can the Bekenstein-Hawking temperature of a given mass black hole be made arbitrarily small?
According to Page [4,5], the answer to this question is 'yes'. Page has based his argument for the non-existence of a lower bound on the Bekenstein-Hawking temperature on the fact that, for near-extremal Reissner-Nordström (a = 0) black holes in the large-mass regime the emission of charged massive fields is exponentially suppressed as compared to the emission of neutral massless (electromagnetic and gravitational) fields [4,5]. It was therefore argued by Page [4,5] that, by emitting neutral fields which reduce the black-hole mass (without reducing its electric charge), a charged Reissner-Nordström black hole can approach arbitrarily close to the zero-temperature (extremal) limit T BH → 0 [7][8][9].
The main goal of the present compact paper is to reveal the fact that near-extremal Reissner-Nordström black holes in the regime T BH < ∼h 2 /G 2 M 3 , if they exist, may violate, through the Hawking emission process, the black-hole condition Q ≤ M which is imposed by the Penrose cosmic censorship conjecture [10,11]. This fact, to be proved below, suggests that, in order to guarantee the validity of the fundamental Penrose cosmic censorship conjecture, the Bekenstein-Hawking temperature of a given mass black hole should be bounded from below by the simple relation

II. THE SEMI-CLASSICAL HAWKING EVAPORATION PROCESS OF NEAR-EXTREMAL REISSNER-NORDSTRÖM BLACK HOLES
In the present section we shall analyze the physical and mathematical properties which characterize the Hawking radiation spectra of near-extremal Reissner-Nordström black holes in the regime [12] 0 Here ∆ is the excess energy of the charged Reissner-Nordström spacetime above the minimal mass (extremal) black-hole configuration with M = M min (Q) = Q. As explicitly shown in [4], the semi-classical decay of near-extremal Reissner-Nordström black holes in the largemass regime (3) is dominated by the emission of massless neutral photons with unit angular momentum [13].
The black-hole radiation power for one bosonic degree of freedom is given by the semiclassical Hawking integral relation [14] P =h G 2π l,m where {l, m} are the angular (spheroidal and axial) harmonic indices which characterize the emitted field mode, and the frequency-dependent dimensionless parameters Γ = Γ lm (ω) are the greybody factors which characterize the linearized interaction (scattering) of the field mode with the curved black-hole spacetime [14].
It is interesting to point out that the characteristic thermal factor ω/(eh ω/T BH − 1) that appears in the semi-classical expression (5) for the Hawking radiation power implies that the black-hole-field emission spectra peak at the characteristic dimensionless frequencȳ ity [15] GMT BH h ≪ 1 , one deduces from (6) that, for near-extremal black holes in the regime (7), the characteristic field frequencies which constitute the Hawking black-hole emission spectra are characterized by the strong dimensionless inequality Interestingly, and most importantly for our analysis, it has been demonstrated in [4] that, in the low-frequency regime (8) which dominates the emission spectra of the near-extremal black holes, the dimensionless greybody factors Γ lm (ω) that appear in the integral relation (5) can be expressed in a closed analytic form. In particular, one finds the simple expression for the characteristic greybody factor of unit (l = 1) angular momentum photons [16], where here we have used the dimensionless physical quantities Substituting Eq. (9) into Eq. (5), and using the relations [see Eqs.
From Eq. (12) one finds that the function F (ν), which determines the energy distribution of the radiated field modes, has a maximum at implying that the characteristic photons in the emission spectra of the near-extremal black holes have an energy which is of the order of [see Eqs. (11) and (13)] The Hawking emission of these neutral field modes would reduce the black-hole mass (without reducing its charge) by ∆M = −E. Thus, the mass of the resulting black-hole configuration (after the emission of a characteristic Hawking photon) is given by [see Eqs. (4) and (14)] Taking cognizance of the Penrose cosmic censorship conjecture [10,11], one immediately realizes that, in order for the Hawking evaporation process to respect the black-hole condition (4), the characteristic energy of the emitted photons must be bounded from above. In particular, using the constraint on the physical parameters which characterize the black-hole configuration after the emission of a characteristic Hawking photon, one finds the lower bound [see Eqs. (14), (15), and (16)] on the excess energy of a given mass Reissner-Nordström black hole.
Interestingly, using the relation [see Eqs.
(1), (10), and (11)] one finds from (17)  Our analysis therefore suggests that one of the following physical scenarios should be valid in a self-consistent quantum theory of gravity: (1) The Penrose cosmic censorship conjecture can be violated within the framework of a quantum theory of gravity.
(2) The Hawking semi-classical treatment of the black-hole evaporation process breaks down in the regime T BH < ∼h 2 /G 2 M 3 of near-extremal black holes in such a way that the emission of charged massive fields (which reduce the black-hole charge and thus increase the black-hole temperature) dominates over the emission of neutral massless fields.
(3) Near-extremal black holes with ∆ < ∼h 2 /G 2 M 3 do not exist within the framework of a self-consistent quantum theory of gravity. If true, this last scenario implies that the Bekenstein-Hawking temperature of quantized black holes is bounded from below by the simple relation [17,18] T BH > ∼h