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

Abstract

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 longlived 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 ∈ (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.

Anisotropic Tolman VII solution by gravitational decoupling

“…(28) and (32) for the MGD solution have to be satisfied simultaneously with the general conservation law given by Eq. (9). We conclude that both systems, perfect fluid and other gravitational source, conserve independently, i.e., these two systems cannot exchange energy-momentum, and their interaction is solely gravitational [32].…”

Anisotropic solutions by gravitational decoupling

“…Indeed, this simple and systematic method could be conveniently exploited in a large number of relevant cases, such as the Einstein-Maxwell [20] and Einstein-Klein-Gordon system [21][22][23][24], for higher derivative gravity [25][26][27], f (R)-theories of gravity [28][29][30][31][32][33][34], Hořava-aether gravity [35,36], polytropic spheres [37][38][39], among many others. In this respect, the simplest practical application of the MGD-decoupling consists in extending known isotropic and physically acceptable interior solutions for spherically symmetric self-gravitating systems into the anisotropic domain, at the same time preserving physical acceptability, which represents a highly non-trivial problem [40] (for obtaining anisotropic solutions in a generic way, see for instance Refs.…”

Using MGD Gravitational Decoupling to Extend the Isotropic Solutions of Einstein Equations to the Anisotropical Domain

“…In consequence we are able to find analytical internal solutions of Einstein equations that considers an energy‐momentum tensor of the form where α is a coupling constant and is a gravitational source. In fact, with this approach we are able to study different systems as polytropic spheres, Horava‐aether gravity, Einstein‐Maxwell, Einstein Klein‐Gordon, and many others (see for example []).…”

“…where α is a coupling constant and θ μν is a gravitational source. In fact, with this approach we are able to study different systems as polytropic spheres, [27][28][29] Horava-aether gravity, [30,31] Einstein-Maxwell, [32] Einstein Klein-Gordon, [33][34][35][36] and many others (see for example [37][38][39][40][41][42][43][44][45][46] This works for as many gravitational sources as we want…”