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

Abstract

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 δω of the black-hole emission lines is expected to be small on the scale set by the characteristic frequency spacing Δω of the spectral lines: ζSch≡δω/Δω≪1. 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 ζ(a¯)≡δω/Δω for the predicted BM emission spectra of rapidly-rotating Kerr black holes (here a¯≡J/M2 is the dimensionless angular momentum of the black hole). It is shown that ζ(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 a¯≳0.9 are characterized by the dimensionless ratio ζ(a¯)≳1 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.