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The superradiant instability regime of the spinning Kerr black hole · Around

2016

The superradiant instability regime of the spinning Kerr black hole

Hod, Shahar

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Spinning Kerr black holes are known to be superradiantly unstable to massive scalar perturbations.

We here prove that the instability regime of the composed Kerr-black-hole-massive-scalar-field system is bounded from above by the dimensionless inequality $M\mu < m \cdot \sqrt{{{2(1+\gamma) (1-\sqrt{1-\gamma^2}) - \gamma^2} \over {4\gamma^2}}}$, where $\{\mu,m\}$ are respectively the proper mass and azimuthal harmonic index of the scalar field and $\gamma\equiv r_-/r_+$ is the dimensionless ratio between the horizon radii of the black hole.
It is further shown that this {\it analytically} derived upper bound on the superradiant instability regime of the spinning Kerr black hole agrees with recent {\it numerical} computations of the instability resonance spectrum.

Built on

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Similar

A. V. Vilenkin, Phys. Lett. B 78
1978
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S. Chandrasekhar, The Mathematical Theory of Black Holes
1983
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T. Hartman, W. Song, and A. Strominger, JHEP 1003:118 (2010)
2010
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P. P. Fiziev, e-print arXiv:0902.1277; R. S. Borissov and P. P. Fiziev, e-print arXiv:0903.3617; P. P. Fiziev, Phys. Rev. D 80
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2010
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T. Damour, N. Deruelle and R. Ruffini, Lett. Nuovo Cimento 15
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2011
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For former analytical bounds on the instability regime of the composed Kerr-black-hole-massive-scalar-field system, see H. R. Beyer, Commun. Math. Phys. 221

Then

H. Yang, D. A. Nichols, F. Zhang, A. Zimmerman, Z. Zhang and Y. Chen, Phys. Rev. D 86
2012
Later among the works it cites.
S. Hod, Phys. Rev. Lett. 100
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S. Hod, Phys. Rev. D 86
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C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. 112
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S. Hod, Phys. Lett. B 746
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V. Cardoso, O. J. C. Dias, J. P. S. Lemos, and S. Yoshida, Phys. Rev. D 70

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2011

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S. Hod, Phys. Lett. B 708

2012

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We use natural units in which G = c = ℏ = 1 G=c=\hbar=1

Cited in the paper.

We shall assume a > 0 a>0 and m > 0 m>0 without loss of generality

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Note that the characteristic mass parameter μ \mu of the scalar field stands for μ / ℏ \mu/\hbar . Hence, this physical parameter has the dimensions of ( ( length OPEN ) − 1 )^{-1}

Cited in the paper.

As shown in [ 2 , 13 ] , an artificial confinement mechanism can be provided by a reflecting mirror which surrounds the black hole and prevents the amplified scalar field modes from radiating their energy to infinity. This composed setup is known as the Kerr-black-hole-scalar-field-mirror bomb

Cited in the paper.

It is worth emphasizing again that the onset of the superradiant instabilities in the composed Kerr-black-hole-massive-scalar-field system is marked by the presence of the stationary (marginally-stable) bound-state field configurations ( 4

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