8,158 research outputs found
Quasi-bound states of massive scalar fields in the Kerr black-hole spacetime: Beyond the hydrogenic approximation
Rotating black holes can support quasi-stationary (unstable) bound-state
resonances of massive scalar fields in their exterior regions. These spatially
regular scalar configurations are characterized by instability timescales which
are much longer than the timescale set by the geometric size (mass) of the
central black hole. It is well-known that, in the small-mass limit
(here is the mass of the scalar field), these
quasi-stationary scalar resonances are characterized by the familiar hydrogenic
oscillation spectrum: , where
the integer is the principal quantum number of
the bound-state resonance (here the integers and
are the spheroidal harmonic index and the resonance parameter of the field
mode, respectively). As it depends only on the principal resonance parameter
, this small-mass () hydrogenic spectrum is obviously
degenerate. In this paper we go beyond the small-mass approximation and analyze
the quasi-stationary bound-state resonances of massive scalar fields in
rapidly-spinning Kerr black-hole spacetimes in the regime . In
particular, we derive the non-hydrogenic (and, in general, non-degenerate)
resonance oscillation spectrum
, where is the generalized
principal quantum number of the quasi-stationary resonances. This analytically
derived formula for the characteristic oscillation frequencies of the composed
black-hole-massive-scalar-field system is shown to agree with direct numerical
computations of the quasi-stationary bound-state resonances.Comment: 7 page
Marginally stable resonant modes of the polytropic hydrodynamic vortex
The polytropic hydrodynamic vortex describes an effective -dimensional
acoustic spacetime with an inner reflecting boundary at . This
physical system, like the spinning Kerr black hole, possesses an ergoregion of
radius and an inner non-pointlike curvature singularity of
radius . Interestingly, the fundamental ratio
which characterizes the effective geometry is
determined solely by the dimensionless polytropic index of the
circulating fluid. It has recently been proved that, in the
case, the effective acoustic spacetime is characterized by an {\it infinite}
countable set of reflecting surface radii,
, that can support static
(marginally-stable) sound modes. In the present paper we use {\it analytical}
techniques in order to explore the physical properties of the polytropic
hydrodynamic vortex in the regime. In particular, we prove
that in this physical regime, the effective acoustic spacetime is characterized
by a {\it finite} discrete set of reflecting surface radii,
, that can support
the marginally-stable static sound modes (here is the azimuthal harmonic
index of the acoustic perturbation field). Interestingly, it is proved
analytically that the dimensionless outermost supporting radius
, which marks the onset of superradiant
instabilities in the polytropic hydrodynamic vortex, increases monotonically
with increasing values of the integer harmonic index and decreasing values
of the dimensionless polytropic index .Comment: 13 page
Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis
The spheroidal harmonics have attracted the attention of
both physicists and mathematicians over the years. These special functions play
a central role in the mathematical description of diverse physical phenomena,
including black-hole perturbation theory and wave scattering by nonspherical
objects. The asymptotic eigenvalues of these functions have
been determined by many authors. However, it should be emphasized that all
previous asymptotic analyzes were restricted either to the regime
with a fixed value of , or to the complementary regime with a
fixed value of . A fuller understanding of the asymptotic behavior of the
eigenvalue spectrum requires an analysis which is asymptotically uniform in
both and . In this paper we analyze the asymptotic eigenvalue spectrum
of these important functions in the double limit and
with a fixed ratio.Comment: 5 page
On the number of light rings in curved spacetimes of ultra-compact objects
In a very interesting paper, Cunha, Berti, and Herdeiro have recently claimed
that ultra-compact objects, self-gravitating horizonless solutions of the
Einstein field equations which have a light ring, must possess at least {\it
two} (and, in general, an even number of) light rings, of which the inner one
is {\it stable}. In the present compact paper we explicitly prove that, while
this intriguing theorem is generally true, there is an important exception in
the presence of degenerate light rings which, in the spherically symmetric
static case, are characterized by the simple dimensionless relation [here is the radius of the
light ring and are respectively the energy density and
tangential pressure of the matter fields]. Ultra-compact objects which belong
to this unique family can have an {\it odd} number of light rings. As a
concrete example, we show that spherically symmetric constant density stars
with dimensionless compactness possess only {\it one} light ring
which, interestingly, is shown to be {\it unstable}.Comment: 5 page
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