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

TL;DR

This paper investigates whether the quantum emission spectra of rapidly-rotating Kerr black holes are discrete or continuous, finding that natural broadening likely causes the spectra to appear continuous despite an underlying discrete energy spectrum.

Contribution

It extends Bekenstein and Mukhanov's analysis to Kerr black holes, showing that high angular momentum leads to spectral blending due to broadening effects.

Findings

For Kerr black holes with high spin, emission lines are effectively blended.

The ratio of broadening to spectral line spacing increases with angular momentum.

Rapidly-rotating black holes likely have smeared, continuum-like emission spectra.

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: $\zeta^{\text{Sch}}\equiv\delta\omega/\Delta\omega\ll1$. 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.