Relativistic and Newtonian Core-Shell Models: Analytical and Numerical Results

We make a detailed analysis of the exact relativistic core-shell models recently proposed to describe a black hole or neutron star surrounded by an axially 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 their analyticity at the black hole horizon and to the full elimination of axial (conical) singularities. We also consider Newtonian and linearized core-shell models, on their own to account for dust shells and rings around galaxies and supernovae and star remnants around their centers, and also as limiting cases of the corresponding relativistic models to gain physical insight. Second, these models are generic enough to numerically study the role played by the presence/lack of discrete reflection symmetries about planes, i.e., the presence/lack of equatorial planes, in the chaotic behavior of the 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 we change a true central black hole by a Newtonian central mass. Our main numerical findings are as follows: (1) The breakdown of the reflection symmetry about the equatorial plane in both Newtonian and relativistic core-shell models (a) enhances in a significant way the chaotic behavior of orbits in reflection symmetric oblate shell models and (b) inhibits significantly also the occurrence of chaos in reflection symmetric prolate shell models. In particular, in the prolate case the lack 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., with a true central black hole) are significantly larger than in the corresponding Newtonian ones (which have just a -1/r central potential).