Relativistic and Newtonian core-shell models: Analytical and numerical results

We make a detailed analysis of the exact relativistic core-shell models rec ently proposed to describe a black hole or neutron star surrounded by an ax ially symmetric, hollow halo of matter and in a seminal sense also galaxies , since there are massive shell-like structures-as, for example, rings and shells-surrounding many of them and also evidence for many galactic nuclei hiding black holes. We discuss the unicity of the models in relation to the ir analyticity at the black hole horizon and to the full elimination of axi al (conical) singularities. We also consider Newtonian and linearized core- shell models, on their own to account for dust shells and rings around gala xies and supernovae and star remnants around their centers, and also as Lim iting cases of the corresponding relativistic models to gain physical insig ht. Second, these models are generic enough to numerically study the role p layed by the presence/lack of discrete reflection symmetries about planes, i.e., the presence/lack of equatorial planes, in the chaotic behavior of th e orbits. This is to be contrasted with the almost universal acceptance of reflection symmetries as default assumptions in galactic modeling. We also compare the related effects if are change a true central black hole by a Ne wtonian central mass. Our main numerical findings are as follows: (1) The b reakdown of the reflection symmetry about the equatorial plane in both Newt onian and relativistic core-shell models (a) enhances in a significant way the chaotic behavior of orbits in re reflection symmetric oblate shell mode ls and (b) inhibits significantly also the occurrence of chaos in reflectio n symmetric prolate shell models. In particular, in the prolate case the la ck of the reflection symmetry provides the phase space with a robust family of regular orbits that is otherwise not found at higher energies. (2) The relative extents of the chaotic regions in the relativistic cases (i.e., wi th a true central black hole) are significantly larger than in the correspo nding Newtonian ones (which have just a -1/r. central potential).