**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

p_{c} to the central energy density ρ_{c}, 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;