Gravitational instability of polytropic spheres containing region of trapped null geodesics: a possible explanation of central supermassive black holes in galactic halos
2017
We study behaviour of gravitational waves in the recently
introduced general relativistic polytropic spheres containing a
region of trapped null geodesics extended around radius of the
stable null circular geodesic that can exist for the polytropic
index N > 2.138 and the relativistic parameter, giving ratio of
the central pressure p_c to the central energy density rho_c,
higher than sigma = 0.677. In the trapping zones of such
polytropes, the effective potential of the axial gravitational
wave perturbations resembles those related to the ultracompact
uniform density objects, giving thus similar long-lived axial
gravitational modes. These long-lived linear perturbations are
related to the stable circular null geodesic and due to
additional non-linear phenomena could lead to conversion of the
trapping zone to a black hole. We give in the eikonal limit
examples of the long-lived gravitational modes, their
oscillatory frequencies and slow damping rates, for the
trapping zones of the polytropes with N is an element of
(2.138, 4). However, in the trapping polytropes the long-lived
damped modes exist only for very large values of the multipole
number l > 50, while for smaller values of l the numerical
calculations indicate existence of fast growing unstable axial
gravitational modes. We demonstrate that for polytropes with N
>= 3.78, the trapping region is by many orders smaller than
extension of the polytrope, and the mass contained in the
trapping zone is about 10^(-3) of the total mass of the
polytrope. Therefore, the gravitational instability of such
trapping zones could serve as a model explaining creation of
central supermassive black holes in galactic halos or galaxy
clusters.
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