Publication date: Jun 2017
Abstract:
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
pc to the central energy density ρc, higher than σ
= 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 in
(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.
Authors:
Stuchlík, Zdeněk; Schee, Jan; Toshmatov, Bobir; Hladík, Jan; Novotný, Jan;