The Hawking cascades of gravitons from higher-dimensional Schwarzschild black holes

It has recently been shown that the Hawking evaporation process of $(3+1)$-dimensional Schwarzschild black holes is characterized by the dimensionless ratio $\eta\equiv\tau_{\text{gap}}/\tau_{\text{emission}}\gg1$, where $\tau_{\text{gap}}$ is the characteristic time gap between the emissions of successive Hawking quanta and $\tau_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the Schwarzschild black hole. This strong inequality implies that the Hawking cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is extremely {\it sparse}. In the present paper we explore the semi-classical Hawking evaporation rates of {\it higher}-dimensional Schwarzschild black holes. We find that the dimensionless ratio $\eta(D)\equiv{{\tau_{\text{gap}}}/{\tau_{\text{emission}}}}$, which characterizes the Hawking emission of gravitons from the $(D+1)$-dimensional Schwarzschild black holes, is a {\it decreasing} function of the spacetime dimension. In particular, we show that higher-dimensional Schwarzschild black holes with $D\gtrsim 10$ are characterized by the relation $\eta(D)<1$. Our results thus imply that, contrary to the $(3+1)$-dimensional case, the characteristic Hawking cascades of gravitons from these higher-dimensional black holes have a {\it continuous} character.


I. INTRODUCTION
Analyzing the dynamics of quantum fields in curved black-hole spacetimes, Hawking [1] has reached the intriguing conclusion that black holes have a well defined temperature [2,3]. In particular, Hawking [1] has revealed that semiclassical black holes are characterized by quantum emission spectra which have distinct thermal features [4]. This remarkable prediction is certainly one of the most important outcomes of the interplay between quantum field theory and classical general relativity.
It has recently been shown [5] that the semi-classical Hawking radiation flux out of a (3 + 1)-dimensional Schwarzschild black hole is extremely sparse [see also [6][7][8] for earlier discussions of this characteristic property of the (3 + 1)-dimensional Schwarzschild black hole]. In particular, the Hawking emission of gravitons out of a (3 + 1)-dimensional Schwarzschild black hole is characterized by the dimensionless large ratio [5] η ≡ τ gap τ emission ≫ 1 , where τ gap is the characteristic time gap between the emissions of successive black-hole gravitational quanta [see Eqs. (10) and (11) below], and τ emission is the characteristic timescale required for an individual Hawking quantum to be emitted from the black hole [see Eq. (12) below]. The dimensionless large ratio (1) implies that the semi-classical Hawking evaporation of (3 + 1)-dimensional Schwarzschild black holes is indeed sparse. That is, the characteristic time gap between the emissions of successive gravitational quanta out of an evaporating (3 + 1)-dimensional Schwarzschild black hole is very large on the natural timescale 2π/ω [see Eq. (12) below] set by the characteristic energy (frequency) of the emitted Hawking quanta. The characteristic large ratio (1) therefore suggests a simple physical picture in which an evaporating (3 + 1)-dimensional Schwarzschild black hole [9,10] typically emits Hawking quanta one at a time [5][6][7][8].
One naturally wonders whether the strong inequality (1), which characterizes the Hawking evaporation process of the (3 + 1)-dimensional Schwarzschild black hole, is a generic feature of all (D + 1)-dimensional Schwarzschild black-hole spacetimes? In order to answer this interesting question, we shall explore in this paper the semi-classical Hawking evaporation of higher-dimensional Schwarzschild black holes. Below we shall show that the dimensionless ratio η(D) ≡ τ gap /τ emission , which characterizes the Hawking evaporation process of (D +1)-dimensional Schwarzschild black holes, is a decreasing function of the spacetime dimension. In particular, we shall show that higher-dimensional Schwarzschild black holes with D > ∼ 10 are characterized by the relation η(D) < 1. Thus, our analysis (to be presented below) reveals the fact that the Hawking cascades of gravitons from these higher-dimensional evaporating black holes have a continuous character.

II. THE HAWKING EVAPORATION PROCESS OF (D + 1)-DIMENSIONAL SCHWARZSCHILD BLACK HOLES
We study the semi-classical Hawking emission of gravitational quanta by higher-dimensional Schwarzschild black holes. The characteristic Bekenstein-Hawking temperature of an evaporating (D + 1)-dimensional Schwarzschild black hole is given by [11] where r H is the black-hole horizon radius [12][13][14][15].
The radiation flux (that is, the number of quanta emitted per unit of time) and the radiation power (that is, the energy emitted per unit of time) for one bosonic degree of freedom out of a (D + 1)-dimensional Schwarzschild black hole are given respectively by the integral relations [1,16,17] and where j denotes the angular harmonic indices of the emitted field mode. The frequency-dependent coefficients Γ = Γ(ω; j, D) are the dimensionless greybody factors [17] which quantify the partial scattering of the emitted field modes by the effective curvature potential that surrounds the (D + 1)-dimensional Schwarzschild black hole. These factors are determined by the higher-dimensional version of the Regge-Wheeler equation [18] d 2 where r * is a 'tortoise' radial coordinate which is determined by the relation dr * /dr = [1−(r H /r) D−3 ] −1 . The effective curvature potential in the Schrödinger-like wave equation (5) is given by [18] V (r; where l is the angular harmonic index of the perturbation mode and p = 0, 2 for gravitational tensor perturbations and gravitational vector perturbations, respectively [18,19]. Substituting into (3) and (4) the expression for the D-dimensional volume in frequency-space of the shell (ω, ω + dω), one finds and for the semi-classical Hawking radiation flux and the semi-classical Hawking radiation power which characterize the evaporating (D + 1)-dimensional Schwarzschild black holes.

III. THE CHARACTERISTIC TIMESCALES OF THE HAWKING EVAPORATION PROCESS
An important timescale which characterizes the Hawking evaporation process of the (D + 1)-dimensional Schwarzschild black holes is given by the time gap between the emissions of successive Hawking quanta. There are several distinct (though closely related) ways to quantify this fundamental time scale. Here we shall use two natural definitions for this characteristic time gap: (1) One can use the reciprocal of the black-hole radiation flux (8) in order to quantify the characteristic time gap between the emissions of successive Hawking quanta. That is, (2) One can also use the reciprocal of the black-hole radiation power (9) and the characteristic peak frequency ω peak of the semi-classical Hawking radiation spectrum in order to quantify the characteristic time gap between the emissions of successive Hawking quanta. That is, Below we shall show that the characteristic time gaps obtained from these two definitions [Eqs. (10) and (11)] are of the same order of magnitude. A distinct timescale which characterizes the Hawking evaporation process of the (D + 1)-dimensional Schwarzschild black holes is given by the time τ emission required for an individual Hawking quantum to be emitted from the evaporating black hole. This fundamental timescale can be bounded from below by the time-period it takes to the characteristic wave field emitted from the black hole to complete a full oscillation cycle [5,10]. That is, Using the natural timescales (10), (11), and (12), one can define the fundamental dimensionless ratio which provides important information about the Hawking evaporation process of the semi-classical black holes [20]. In particular, physical situations which are characterized by the relation η ≫ 1 describe Hawking evaporation processes which are extremely sparse (that is, the individual Hawking quanta emitted from the black hole are well separated in time), whereas physical situations which are characterized by the relation η ≪ 1 describe Hawking evaporation processes which are effectively continuous.
In the next sections we shall investigate the functional dependence of the dimensionless ratios η (i) (D) ≡ τ (i) gap /τ emission on the spacetime dimension D + 1 of the evaporating black-hole spacetime.

V. HIGHER-DIMENSIONAL BLACK HOLES: INTERMEDIATE D-VALUES
In the previous section we have seen that the (3 + 1)-dimensional Schwarzschild black hole is characterized by large values of the dimensionless ratios η (i) (D = 3) [see Eq. (14)]. In the present section we shall show that the dimensionless ratios η (i) (D), which characterize the Hawking evaporation process of (D + 1)-dimensional Schwarzschild black holes, are decreasing functions of the spacetime dimension.
The semi-classical Hawking emission of gravitons from higher-dimensional Schwarzschild black holes was investigated numerically in [18]. In Table I [21]. One finds from Table I that the characteristic dimensionless ratios τ (i) gap /τ oscillation of the (D + 1)dimensional Schwarzschild black holes are decreasing functions of the spacetime dimension D + 1.
In particular, we find that Schwarzschild black holes in the regime D = O(10) are characterized by the dimensionless ratios τ (i) gap /τ oscillation = O(1). These higher-dimensional black holes therefore mark the boundary between sparse (that is, with η (i) ≫ 1) Hawking cascades of gravitons and continuous (that is, with η (i) ≪ 1) Hawking cascades of gravitons.

VI. HIGHER-DIMENSIONAL BLACK HOLES: THE LARGE-D REGIME
The D-dimensional frequency distribution ω D /(eh ω/TBH − 1) [see Eq. (9)] is characterized by the peak frequencȳ where W (x) is the Lambert function. Using the small argument relation W (x → 0) → 0 and taking cognizance of the Bekenstein-Hawking temperature (2), one finds the characteristic strong inequality [22] in the large D ≫ 1 regime. The strong inequality (16) reflects the fact that, for (D+1)-dimensional Schwarzschild black holes in the large D ≫ 1 regime, the characteristic wavelengths in the semi-classical Hawking emission spectrum are very short as compared to the curvature radius of the corresponding black-hole spacetime. As demonstrated in [22], this fact implies that the semi-classical Hawking evaporation of these higher-dimensional black holes is described extremely well by the eikonal (geometric-optics) approximation [23]. In particular, one finds [22] for the semi-classical Hawking radiation flux, and [22] for the semi-classical Hawking radiation power in the large-D regime [which, as explained above, corresponds to the short wavelengths (geometric-optics) approximation, see Eq. (16)]. Taking the asymptotic large D ≫ 1 limit in Eqs. (17) and (18), one obtains the compact asymptotic expressions and Taking cognizance of Eqs. (10), (11), (12), (19), and (20), one finds the remarkably small dimensionless ratios which characterize the evaporating (D + 1)-dimensional Schwarzschild black holes in the large-D limit [24]. These relations (and, in particular, the strong inequalities τ (i) gap ≪ τ emission ) imply that the characteristic Hawking cascades from these higher-dimensional black holes have a continuous character.
In the present paper we have analyzed the Hawking emission rates of higher-dimensional Schwarzschild black holes. It was shown that the dimensionless ratios η (i) (D) ≡ τ (i) gap /τ emission , which characterize the semi-classical Hawking emission of gravitons from the (D + 1)-dimensional Schwarzschild black holes, are decreasing functions of the spacetime dimension. In particular, we have shown that higher-dimensional Schwarzschild black holes with D > ∼ 10 are characterized by the relation η (i) (D) < 1. This fact implies that the corresponding Hawking cascades from these higher-dimensional black holes have a continuous character with the property τ gap < τ emission .
Moreover, we have shown that the semi-classical Hawking emission spectra of higher-dimensional Schwarzschild black holes in the large-D regime are characterized by the strong inequality [see Eq. (21)] Our results therefore imply that, contrary to the (3 + 1)-dimensional case, the characteristic Hawking cascades from these higher-dimensional black holes [25] have a continuous character.