Chaotic Dynamics due Prolate and Oblate Sources in Kerr-like and Hartle-Thorne Spacetimes with and without Magnetic Field
As demonstrated by observations, every stellar-mass object rotates around some axis; some objects spin faster than others due to different mechanisms. Furthermore, these spinning objects are slightly deformed and are no longer perfect spheres because of hydrostatic equilibrium. The well-known Kerr solution of the Einstein Field Equations (EFE) represents the spacetime surrounding a rotating spherical gravitational source. However, real objects deviate from a perfect sphere and may be prolate or oblate. There are several solutions of the EFE that represent the spacetime of deformed objects. The Kerr--like (KL) metric represents the spacetime surrounding this kind of object, where the deformation is characterized by the mass quadrupole moment parameter $q_{\mathrm{KL}}$. When $q_{\mathrm{KL}} \neq 0$, the Carter constant no longer exists and the equations of motion (EOM) are no longer integrable; therefore, the system exhibits chaotic orbits. Another widely used solution is the Hartle--Thorne (HT) metric, which has similar characteristics and represents a slightly deformed, slowly rotating star. The HT metric has several versions, and two of them were selected to test their validity. The traditional HT version, which contains logarithmic terms, is less accurate than the version with exponential terms. Moreover, both the KL and HT metrics may be extended to include contributions due to the magnetic dipole moment of the source. The equations of motion (EOM) were computed, and these new dynamical systems display several interesting features, which are shown in their Poincar'e sections.