The quantum emission spectra of rapidly-rotating Kerr black holes: discrete or continuous?. (arXiv:1909.04057v1 [gr-qc])
<a href="http://arxiv.org/find/gr-qc/1/au:+Hod_S/0/1/0/all/0/1">Shahar Hod</a>
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 $deltaomega$ of the
black-hole emission lines is expected to be small on the scale set by the
characteristic frequency spacing $Deltaomega$ of the spectral lines:
$zeta^{text{Sch}}equivdeltaomega/Deltaomegall1$. 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 $zeta(bar
a)equivdeltaomega/Deltaomega$ for the predicted BM emission spectra of
rapidly-rotating Kerr black holes (here $bar aequiv J/M^2$ 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 $bar agtrsim 0.9$ 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.
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 $deltaomega$ of the
black-hole emission lines is expected to be small on the scale set by the
characteristic frequency spacing $Deltaomega$ of the spectral lines:
$zeta^{text{Sch}}equivdeltaomega/Deltaomegall1$. 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 $zeta(bar
a)equivdeltaomega/Deltaomega$ for the predicted BM emission spectra of
rapidly-rotating Kerr black holes (here $bar aequiv J/M^2$ 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 $bar agtrsim 0.9$ 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.
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