Natural broadening in the quantum emission spectra of higher-dimensional Schwarzschild black holes

Abstract

Following an intriguing heuristic argument of Bekenstein, many researches have suggested during the last four decades that quantized black holes may be characterized by discrete radiation spectra. Bekenstein and Mukhanov (BM) have further argued that the emission spectra of quantized $(3+1)$-dimensional Schwarzschild black holes are expected to be sharp in the sense that the characteristic natural broadening $\delta\omega$ of the black-hole radiation lines, as deduced from the quantum time-energy uncertainty principle, is expected to be much smaller than the characteristic frequency spacing $\Delta\omega=O(T_{\text{BH}}/\hbar)$ between adjacent black-hole quantum emission lines. It is of considerable physical interest to test the general validity of the interesting conclusion reached by BM regarding the sharpness of the Schwarzschild black-hole quantum radiation spectra. To this end, in the present paper we explore the physical properties of the expected radiation spectra of quantized $(D+1)$-dimensional Schwarzschild black holes. In particular, we analyze the functional dependence of the characteristic dimensionless ratio $\zeta(D)\equiv\delta\omega/\Delta\omega$ on the number $D+1$ of spacetime dimensions. Interestingly, it is proved that the dimensionless physical parameter $\zeta(D)$, which characterizes the sharpness of the black-hole quantum emission spectra, is an increasing function of $D$. In particular, we prove that the quantum emission lines of $(D+1)$-dimensional Schwarzschild black holes in the regime $D\gtrsim 10$ are characterized by the dimensionless ratio $\zeta(D)\gtrsim1$ and are therefore effectively blended together.