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 $\delta \omega$ 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: ${\zeta }^{\text{Sch}}\equiv \delta \omega /\Delta \omega \ll 1$. 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 $\zeta (\bar{a})\equiv \delta \omega /\Delta \omega$ 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 $\zeta (\bar{a})\gtrsim 1$ 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.

6 pages

References in corpus (2)

• The physical interpretation of the spectrum of black hole quasinormal modes
• Analytic Study of Rotating Black-Hole Quasinormal Modes

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