The quantum emission spectra of rapidly-rotating Kerr black holes: discrete or continuous?

Bekenstein and Mukhanov (BM) have suggested that, in a quantum theory of gravity, black holes may have discrete emission spectra. Using the time-energy uncertainty principle they have also shown that, for a (non-rotating) Schwarzschild black hole, the natural broadening $\delta\omega$ of the black-hole emission lines is expected to be small on the scale set by the characteristic frequency spacing $\Delta\omega$ of the spectral lines: $\zeta^{\text{Sch}}\equiv\delta\omega/\Delta\omega\ll1$. BM have therefore concluded that the expected discrete emission lines of the quantized Schwarzschild black hole are unlikely to overlap. In this paper we calculate the characteristic dimensionless ratio $\zeta(\bar a)\equiv\delta\omega/\Delta\omega$ for the predicted BM emission spectra of rapidly-rotating Kerr black holes (here $\bar a\equiv J/M^2$ is the dimensionless angular momentum of the black hole). It is shown that $\zeta(\bar a)$ is an increasing function of the black-hole angular momentum. In particular, we find that the quantum emission lines of Kerr black holes in the regime $\bar a\gtrsim 0.9$ are characterized by the dimensionless ratio $\zeta(\bar a)\gtrsim1$ and are therefore effectively blended together. Our results thus suggest that, even if the underlying mass (energy) spectrum of these rapidly-rotating Kerr black holes is fundamentally discrete as suggested by Bekenstein and Mukhanov, the natural broadening phenomenon (associated with the time-energy uncertainty principle) is expected to smear the black-hole radiation spectrum into a continuum.


I. INTRODUCTION
Analyzing the quantum properties of fundamental fields in classical black-hole spacetimes, Hawking [1] has revealed that black holes are actually not completely black. In particular, according to Hawking's result, semi-classical black holes are characterized by continuous evaporation spectra in which the emitted field quanta have the familiar blackbody statistical distribution with a well defined temperature [1,2]. This intriguing theoretical prediction is certainly one of the most important outcomes of the interplay between quantum field theory and classical general relativity.
It is important to realize, however, that Hawking's seminal analysis [1] has a semi-classical nature: while the fundamental fields are properly analyzed at the quantum level, the curved black-hole spacetime is treated as a classical background. Although we do not have a self-consistent theory of quantum gravity, it is natural to expect that some modifications to the (semi-classically predicted) Hawking emission spectrum [1] may arise if the black-hole spacetime itself would properly be treated as a dynamical quantum entity [3].
An interesting heuristic quantization of the black-hole area (mass) spectrum was proposed long ago by Bekenstein [3]. In his influential work, Bekenstein has pointed out that the black-hole surface area behaves as a classical adiabatic invariant [3]. Since classical adiabatic invariants are usually related to physical quantities with discrete quantum spectra [4], Bekenstein has suggested that the black-hole surface area has a discrete (quantum) spectrum of the form [3,5] A n = 4γ · n ; n = 1, 2, 3, . . . .
Here γ is a dimensionless constant of order unity. Three possible values of γ are often used in the literature: γ = 2π [3], γ = ln 2 [6,7], and γ = ln 3 [8]. It is worth mentioning that, using different quantization schemes, several authors (see  and references therein) have re-derived the uniformly spaced black-hole area spectrum (1). The discrete black-hole area spectrum (1) also implies a discrete mass (energy) spectrum {M n } for quantum black holes. Thus, as pointed out in [3], a quantized black hole is expected to be characterized by a discrete line emission. In particular, Bekenstein and Mukhanov (BM) [6] have advocated the idea that, within the framework of a quantum theory of gravity, the radiation emitted by a quantized Schwarzschild black hole of mass M n should be at integer multiples of the fundamental frequency [3,6] where is the Bekenstein-Hawking temperature [1,3] of the Schwarzschild black hole. According to the quantization scheme presented in [3,6], the decay of a quantized Schwarzschild black hole of mass M n into the M n−k mass level [28] is accompanied by the emission of a field quantum of frequency ω k = (M n − M n−k )/ = k · ω 0 [see Eq. (1)]. This implies that the discrete line emission {ω 0 , 2ω 0 , 3ω 0 , ...} suggested by BM [3,6] for a quantized Schwarzschild black hole is characterized by the constant frequency spacing between adjacent emission lines.

II. THE NATURAL BROADENING OF THE SCHWARZSCHILD SPECTRAL LINES
The interesting question of the natural broadening of the Schwarzschild black-hole emission lines has been addressed by BM [6] (see also [29,30]). Arguing from the time-energy uncertainty principle, it was suggested in [6] to estimate the natural broadening δω of the Schwarzschild black-hole spectral lines from the relation δω = 1/τ [4], where τ is the characteristic lifetime (as measured by asymptotic observers) of the nth black-hole mass (energy) level [31].

III. THE SPECTRAL EMISSION LINES OF ROTATING KERR BLACK HOLES AND THEIR NATURAL BROADENING
In this paper we shall generalize the analyzes of [6, 29, 30] to the regime of rotating Kerr black holes. In particular, our main goal is to study the dependence of the fundamental dimensionless ratio ζ(ā) ≡ δω/∆ω on the black-hole rotation parameterā ≡ J/M 2 [37].
and Ω H = a 2M r + , where are the black-hole horizon radii.
Taking cognizance of the discrete black-hole area spectrum (1) suggested by Bekenstein [3], and using the first law of black-hole mechanics [38] one finds that a quantized rotating Kerr black hole is expected to be characterized by the discrete emission frequencies Here m is the azimuthal harmonic index of the emitted field quanta. Note that the discrete black-hole radiation spectrum (10) is characterized by the frequency spacing [40] ∆ω = ω k+1,m − ω k,m = γT BH (11)

B. The natural broadening of the Kerr spectral lines
Following the Bekenstein-Mukhanov analysis of the Schwarzschild black-hole emission spectrum presented in [6], we shall now use the time-energy uncertainty principle [4] in order to estimate the natural broadening δω = 1/τ of the Kerr black-hole emission lines. In particular, following [6] we shall assume that the characteristic lifetime τ of the Kerr nth mass level (that is, the average time gap between quantum leaps) can be estimated as the reciprocal of the semi-classical [1,[32][33][34][35] black-hole emission rate [see Eq. (15) below] [41].
The semi-classical emission rate of a rotating Kerr black hole (that is, the number of quanta emitted from the black hole per unit of time) is given by the Hawking relation [1,32,42] Here s, m and l ≥ max(s, |m|) are respectively the spin parameter and the harmonic indices (azimuthal and spheroidal) of the emitted field quanta. The energy-dependent grey-body factors (absorption probabilities) Γ = Γ slm (ω;ā) [32] in (12) quantify the interaction of the emitted field quanta with the effective curvature potential that surrounds the emitting black hole. The dependence of the semi-classical emission rateṄ (ā) [see Eq. (12)] on the black-hole rotation parameterā can be computed along the lines of the numerical procedure described in [32]. In particular, one finds that, in the regime of rapidly-rotating Kerr black holes, the semi-classical radiation spectrum of a massless spin-s field is greatly dominated by the [32,43] l = m = s (13) angular mode. In addition, the characteristic thermal (exponential) factor that appears in the expression (12) for the black-hole emission rate implies that, in the regime of rapidly-rotating (near-extremal, T BH → 0) black holes, the emission of high energy quanta with ω > mΩ H is exponentially suppressed. Thus, the semi-classical emission spectra of rapidly-rotating Kerr black holes are dominated by field quanta in the energy interval [44] As discussed above, following [6] we shall assume that the lifetime τ of the meta-stable black-hole state (that is, the average time gap between the emissions of successive black-hole quanta) is given by the reciprocal of the black-hole semi-classical emission rate [33]. Namely, whereṄ is given by Eq. (12). Using the time-energy uncertainty principle [4], one finds (see also [6]) for the natural broadening of the black-hole emission lines.
In Table I we display the dimensionless ratio ζ(ā) ≡ δω/∆ω [45] for the emission of massless gravitons (s = 2) and photons (s = 1) [43,46] by rapidly-rotating Kerr black holes [Here the natural broadening δω of the black-hole spectral lines is given by Eqs. (12) and (16), and the characteristic frequency spacing ∆ω between adjacent emission lines [40] is given by (11)]. One finds that ζ(ā) is an increasing function of the dimensionless black-hole rotation parameterā. In particular, we find that rapidly-rotating Kerr black holes in the regimeā 0.9 are characterized by the relation [47,48] δω ∆ω .
The inequality (17) implies that the emission lines of rapidly-rotating Kerr black holes are effectively blended together.   (12) and (16)] and ∆ω [40,45] is the characteristic frequency spacing between adjacent emission lines [see Eq. (11)]. One finds that ζ(ā) is an increasing function of the black-hole angular momentum.

IV. SUMMARY
Starting with the seminal work of Bekenstein [3], many authors (see  and references therein) have predicted the existence of a uniformly spaced area spectrum [see Eq. (1)] for quantized black holes. This intriguing prediction suggests that, in a quantum theory of gravity [49], black holes may have discrete emission spectra.
Using the time-energy uncertainty principle [4], Bekenstein and Mukhanov [6] have shown that, for (non-rotating) Schwarzschild black holes, the natural broadening δω Sch of the spectral lines is small on the scale set by the characteristic frequency spacing ∆ω Sch of the lines: δω Sch /∆ω Sch ≪ 1. It was therefore concluded in [6] that the discrete emission lines which are expected to characterize a quantized Schwarzschild black hole are unlikely to overlap. In other words, the Schwarzschild line spectrum is expected to be sharp.
Motivated by this important conclusion, in this paper we have analyzed the natural broadening of the emission lines which, according to [3], are expected to characterize quantized rotating Kerr black holes. In particular, we have studied the dependence of the dimensionless ratio ζ(ā) ≡ δω/∆ω on the black-hole rotation parameterā ≡ J/M 2 . It was shown that ζ(ā) is an increasing function of the black-hole angular momentum. In particular, one finds that the quantum emission lines of rapidly-rotating Kerr black holes in the regimeā 0.9 are characterized by the dimensionless ratio ζ(ā) 1 [47,48]. These emission lines are therefore effectively blended together.
Our results thus suggest that, even if the underlying mass (energy) spectrum of these rapidly-rotating Kerr black holes is fundamentally discrete as suggested by Bekenstein and Mukhanov, the natural broadening phenomenon (associated with the time-energy uncertainty principle [4]) is expected to smear the black-hole emission spectrum into a continuum.