Chaotic Dynamics due Prolate and Oblate Sources in Kerr-like and Hartle-Thorne Spacetimes with and without Magnetic Field

TL;DR

The paper investigates chaotic dynamics of test particles in Kerr-like and Hartle--Thorne spacetimes that model deformed rotating bodies via their mass quadrupole moments, focusing on how $q_{\mathrm{KL}}$ and $q_{\mathrm{HT}}$ break integrability by eliminating the Carter constant. It compares the Kerr-like (KL) metric, HTlog, and exponentially improved appHT variants, including their magnetic-dipole extensions (dipKL, dipHT), using Hamiltonian dynamics and Poincaré sections to reveal phase-space structures and chaos, and it relates the quadrupole moments through $q_{\mathrm{HT}} = Ma^2 - q_{\mathrm{KL}}$. Key findings show that nonzero quadrupoles induce chaotic islands and resonances, with KL and appHT generally yielding similar qualitative behavior and HTlog exhibiting stronger deviations due to logarithmic terms; introducing a magnetic dipole can either enhance or suppress chaos depending on parameter choice, sometimes producing an effectively integrable appearance. The results have implications for understanding orbital dynamics around magnetized, deformed compact objects and for testing GR multipole structure in astrophysical environments, such as around pulsars. Overall, the work highlights the value of KL and KLdip models for robust numerical studies of non-Kerr spacetimes and the nuanced role of magnetic fields in shaping geodesic chaos.

Abstract

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.

Figures (22)

• Figure 1: Representation of mass distribution according its mass quadrupole moment $q_{KL}$, for Erez--Rozen metric and Kerr-like metric. The sign is reversed for Hartle--Thorne metric.
• Figure 2: Poincaré section using KL metric for $\theta = \pi/2$ and $p_\theta \geq 0$ and parameters $M = 1.0$, $a = 0.1$, $E = 0.95$, $L_z = 3.0$ and $q_{KL} = 0$, there is no structure beside the main island of stability.
• Figure 3: Poincaré section using HTlog metric for $\theta = \pi/2$ and $p_\theta \geq 0$ and parameters $M = 1.0$, $a = 0.1$, $E = 0.95$, $L_z = 3.0$ and $q_{KL} = 0$, there is a small structure on the tip nearest to the compact object (left) and another resonance near $r = 4.595$.
• Figure 4: Poincaré section using appHT metric for $\theta = \pi/2$ and $p_\theta \geq 0$ and parameters $M = 1.0$, $a = 0.1$, $E = 0.95$, $L_z = 3.0$ and $q_{KL} = 0$. Again, there is a resonance on the tip nearest to the compact object surrounded by second order islands and chaos (left) and another resonance with center near $r = 4.606$.
• Figure 5: Poincaré section using KL metric for $\theta = \pi/2$ and $p_\theta \geq 0$ and parameters $M = 1.0$, $a = 0.1$, $E = 0.95$, $L_z = 3.0$ and $q_{KL} = 0.1$.