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Marginally stable resonant modes of the polytropic hydrodynamic vortex · Around

2017

Marginally stable resonant modes of the polytropic hydrodynamic vortex

Hod, Shahar

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The polytropic hydrodynamic vortex describes an effective $(2+1)$-dimensional acoustic spacetime with an inner reflecting boundary at $r=r_{\text{c}}$.

This physical system, like the spinning Kerr black hole, possesses an ergoregion of radius $r_{\text{e}}$ and an inner non-pointlike curvature singularity of radius $r_{\text{s}}$.
Interestingly, the fundamental ratio $r_{\text{e}}/r_{\text{s}}$ which characterizes the effective geometry is determined solely by the dimensionless polytropic index $N_{\text{p}}$ of the circulating fluid.
It has recently been proved that, in the $N_{\text{p}}=0$ case, the effective acoustic spacetime is characterized by an {\it infinite} countable set of reflecting surface radii, $\{r_{\text{c}}(N_{\text{p}};n)\}^{n=\infty}_{n=1}$, that can support static (marginally-stable) sound modes.

Built on

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W. H. Press and S. A. Teukolsky, Nature 238
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W. H. Press and S. A. Teukolsky, Astrophys. J. 185
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Ya. B. Zel’dovich, Pis‘ma Zh. Eksp. Teor. Fiz. 14
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J. L. Friedman, Commun. Math. Phys. 63
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W. G. Unruh, Phys. Rev. Lett. 46
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Then

S. R. Dolan, L. A. Oliveira, and L. C. B. Crispino, Phys. Rev. D 82
2012
Later among the works it cites.
V. Cardoso, O. J. C. Dias, J. P. S. Lemos and S. Yoshida, Phys. Rev. D 70
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2013
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V. Cardoso and O. J. C. Dias, Phys. Rev. D 70
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2013
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L. A. Oliveira, V. Cardoso, and L. C. B. Crispino, Phys. Rev. D 89
2014
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S. Hod, Phys. Rev. D 90
Original
2014
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L. A. Oliveira, V. Cardoso, and L. C. B. Crispino, Phys. Rev. D 92

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The static surface of a spinning black hole is composed of all spacetime points at which it is necessary to move at the speed of light in order to appear stationary to asymptotically far inertial observers [ 1 ]

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It is worth noting that, in order to trigger superradiant instabilities by bosonic fields in spinning black-hole spacetimes, one must confine the extracted rotational energy to a finite spacetime region in the vicinity of the outer horizon. There are several known physical mechanisms that can provide the confinement mechanism which is required in order to support the exponentially growing superradiant bosonic fields: (1) An artificial mirror placed inside the rotating black-hole ergoregion [ 5 , 10 ] , (2) A massive field whose bosonic mass term acts as an effective confining mirror [ 11 ] , and (3) A spinning black-hole spacetime with a non-asymptotically flat reflecting boundary [ 12 ]

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R. Brito, V. Cardoso and P. Pani, Superradiance, arXiv:1501.06570

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It is important to stress the fact that, as originally proved in [ 19 ] , the wave equation which characterizes the dynamics of acoustic sound modes in fluids systems [see Eq. ( 4

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As explicitly proved in [ 16 ] , the location of the inner radius r s r_{\text{s}} depends on the circulation of the fluid, its asymptotic mass density, and on the physical parameters k p k_{\text{p}} and N p N_{\text{p}} [see Eq. ( 1

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The outer boundary r = r e r=r_{\text{e}} of the acoustic ergoregion is determined by the circle at which the circular velocity of the fluid equals the propagation speed c s c_{\text{s}} of acoustic sound modes [ 16 ]

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2015

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