Werner M. Vieira

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).

Chaos in black holes surrounded by gravitational waves

The occurrence of chaos for test particles moving around a Schwarzschild black hole perturbed by a special class of gravitational waves is studied in the context of the Melnikov method. The explicit integration of the equations of motion for the homoclinic orbit is used to reduce the application of this method to the study of simple graphs.

On the integrability of halo dipoles in gravity

Abstract We stress that halo dipole components are nontrivial in core-halo systems in both Newtons gravity and general relativity. To this end, we extend a recent exact relativistic model to include also a halo dipole component. Next, we consider orbits evolving in the inner vacuum between a monopolar core and a pure halo dipole and find that, while the Newtonian dynamics is integrable, its relativistic counterpart is chaotic. This shows that chaoticity due only to halo dipoles is an intrinsic relativistic gravitational effect.

Study of chaos in Hamiltonian systems via convergent normal forms

Abstract We use Mosers normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space, rather than just the Poincare section. The formalism was applied to the Henon-Heiles hamiltonian, producing strong evidence that the region of convergence of these normal forms extends over that orginally established by Moser.

Chaos and Taub-NUT related spacetimes

Abstract The occurrence of chaos for test particles moving in a Taub-NUT spacetime with a dipolar halo perturbation is studied using Poincare sections. We find that the NUT parameter (magnetic mass) attenuates the presence of chaos.

Chaos in periodically perturbed monopole + quadrupole-like potentials

Abstract The motion of a particle subjected to simple inner (outer) periodic perturbations when it evolves around a center of attraction modeled by an inverse square law plus a quadrupole-like term is studied. The equations of motion are used to reduce the Melnikov method to the study of simple graphics.

Extended convergence of normal forms around unstable equilibria

Abstract There is strong numerical evidence that the convergence of normal forms around saddle points of Hamiltonian systems should extend beyond the region originally established by Moser. We show that these normal forms do converge along a neighbourhood of the stable and unstable manifolds emanating from Mosers region if the Hamiltonian is analytical. A possible further extension will allow the calculation of homoclinic orbits as intersections of the analytical images of the stable and the unstable subspaces for the normal form.

Addendum: Chaos around a H\'enon-Heiles-Inspired Exact Perturbation of a Black Hole

except that it contains much more points, this time suucient to exhibit (three small) chaotic regions (one around each vertex of the triangular region).4