# Gravitational instability of polytropic spheres containing region of trapped null geodesics: a possible explanation of central supermassive black holes in galactic halos

## TL;DR

This paper investigates gravitational wave behavior in relativistic polytropic spheres with trapped null geodesics, suggesting such zones could lead to black hole formation and explain supermassive black holes in galactic centers.

## Contribution

It introduces a model of polytropic spheres with trapping zones that exhibit long-lived gravitational modes, potentially leading to black hole formation, a novel explanation for supermassive black holes.

## Key findings

- Long-lived gravitational modes exist for high multipole numbers in trapping zones.
- Unstable modes dominate for lower multipole numbers, indicating possible collapse.
- Mass in trapping zones is a small fraction of total mass, supporting black hole formation hypothesis.

## 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 $p_\mathrm{c}$ to the central energy density $\rho_\mathrm{c}$, 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 $N \in (2.138,4)$. However, in the trapping polytropes the long-lived damped modes exist only for very large values of the multipole number $\ell>50$, while for smaller values of $\ell$ the numerical calculations indicate existence of fast growing unstable axial gravitational modes. We demonstrate that for polytropes with $N \geq 3.78$, 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.