Polytropic spheres containing regions of trapped null geodesics

Abstract

We demonstrate that in the framework of standard general relativity polytropic spheres with properly fixed polytropic index n and relativistic parameter σ, giving ratio of the central pressure pc to the central energy density ρc, can contain region of trapped null geodesics. Such trapping polytropes can exist for n > 2.138 and they are generally much more extended and massive than the observed neutron stars. We show that in the n-σ parameter space the region of allowed trapping increase with polytropic index for interval of physical interest 2.138 < n < 4. Space extension of the region of trapped null geodesics increases with both increasing n and σ > 0.677 from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict validity of the polytropic configurations to their extension rextr corresponding to the gravitational mass M ∼ 2M of the most massive observed neutron stars. Then for the central density ρc ∼ 10 15 g cm −3 the trapped regions are outside rextr for all values of 2.138 < n < 4, for the central density ρc ∼ 5 × 10 15 g cm −3 the whole trapped regions are located inside of rextr for 2.138 < n < 3.1, while for ρc ∼ 10 16 g cm −3 the whole trapped regions are inside of rextr for all values of 2.138 < n < 4, guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located closely to the polytrope centre and could have relevant influence on cooling of such polytropes or for binding of gravitational waves in their interior.

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

“…where we use, contrary to the previous cases, the maximal compactness C max (N, σ) obtained for a given pair of parameters (N, σ), related to the global compactness C(ξ 1 ; N, σ). The maximal compactness occurs at ξ max that is very close to ξ ph(u) for each pair (N, σ), and there is always C max (N, σ) < 1/3, as shown in [9]. The global compactness strongly decreases for very extended trapping polytropes with ξ 1 ≫ 1; recall that, on the other hand, for all the trapping polytropes there is ξ max ∼ ξ ph(u) ∼ 1.…”

“…It has been shown that the trapping polytropes must have the polytropic index N > 2.138 and relatively large relativistic parameter σ > 0.677. In the N − σ parameter space, extension of the trapping region increases as the polytropic index N increases, being restricted from above by the value of σ corresponding to the causal limit [9]. In order to relate the trapping phenomenon to astrophysically relevant objects, namely the neutron (quark) stars, validity of the polytropic configurations has been restricted in [9] to their extension r extr corresponding to the gravitational mass M ∼ 2M ⊙ of the most massive observed neutron stars.…”

“…In the N − σ parameter space, extension of the trapping region increases as the polytropic index N increases, being restricted from above by the value of σ corresponding to the causal limit [9]. In order to relate the trapping phenomenon to astrophysically relevant objects, namely the neutron (quark) stars, validity of the polytropic configurations has been restricted in [9] to their extension r extr corresponding to the gravitational mass M ∼ 2M ⊙ of the most massive observed neutron stars. For the central density ρ c ∼ 5 × 10 15 g cm −3 the whole trapped regions are located inside of r extr (i.e., there is r extr > r ph(u) ) for 2.138 < N < 3.1, while for ρ c ∼ 10 16 g cm −3 , the whole trapped regions are inside of r extr for values of 2.138 < N < 4, guaranteeing astrophysically plausible trapping for all considered polytropes in the whole region limited by the radius r ph(u) .…”

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.…”

A simple method to generate exact physically acceptable anisotropic solutions in general relativity

“…The approach described above represents a step forward in the search and analysis of solutions to Einstein's field equations, especially when we are studying situations beyond trivial cases, such as the interior of stellar structures with gravitational sources more complex than the ideal perfect fluid [38,39,40,41,42,43,44,45,46,47,48,49,50]. In this respect, the simplest practical application of the MGD-decoupling consists in extending known isotropic and physically acceptable interior solutions for spherically symmetric stellar systems into the anisotropic domain, at the same time preserving physical acceptability, which represents a highly non-trivial problem [52] (for obtaining anisotropic solutions in a generic way, see for instance Ref.…”