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.

💡 Research Summary

This paper investigates the non‑integrable, chaotic dynamics of test particles moving in the exterior spacetime of rotating, slightly deformed compact objects. Two approximate solutions of the Einstein field equations are considered: the Kerr‑like (KL) metric and the Hartle‑Thorne (HT) metric. Both metrics are expressed in the general stationary, axis‑symmetric form (1) with potentials V, W, X, Y, Z that depend only on the radial coordinate r and polar angle θ. The KL metric is derived via the Ernst formalism and Hoenselaers‑Kinnersley‑Xanthopoulos (HKX) transformations, introducing a mass quadrupole moment parameter q_KL that appears in the functions ψ and χ multiplied by the Legendre polynomial P₂(cosθ). When q_KL ≠ 0 the Carter constant disappears, rendering the geodesic equations non‑integrable and giving rise to chaotic motion, as previously reported. The KL metric also contains the spin‑octupole moment, which is absent in the HT metric, and it uses only first‑order Legendre polynomials, making numerical implementation faster.

The HT metric is examined in two versions. The traditional “logarithmic” version (HTlog) contains associated Legendre functions of the second kind Q₁₂ and Q₂₂, leading to more cumbersome expressions. The “exponential” or approximate HT (appHT) replaces these with exponential factors, dramatically simplifying the potentials while preserving the same multipole structure. Both HT versions are characterized by a quadrupole parameter q_HT and a rotation parameter a, and they satisfy the relation q_HT = M a² – q_KL (Eq. 18), ensuring that the two spacetimes describe the same physical source up to second order in the multipole expansion.

To study electromagnetic effects, the authors extend both metrics by adding electric charge q_e and a magnetic dipole moment μ_d using the Ernst formalism. Two magnetised models are constructed: dipKL (the magnetised Kerr‑like metric) and dipHT (the magnetised Hartle‑Thorne metric). The magnetic dipole contributes additional terms to the metric potentials and to the electromagnetic four‑potential A_μ, notably μ_d q_e sin²θ terms in W and A_φ, which generate strong torques near the rotation axis.

The equations of motion are derived from a Hamiltonian formalism (Eqs. 2‑5). With the test particle mass set to unity, the authors integrate the geodesic equations numerically, using Boyer‑Lindquist coordinates to visualise trajectories. They construct Poincaré sections by recording (r, p_r) whenever the particle crosses the equatorial plane with θ = π/2 and p_θ > 0, thereby reducing the four‑dimensional phase space to a two‑dimensional map that reveals regular islands, resonant chains, and chaotic seas.

Numerical experiments explore a range of quadrupole signs and magnetic dipole strengths. Positive quadrupole moments (q > 0) correspond to prolate sources, while negative values (q < 0) describe oblate sources. For prolate configurations the Poincaré maps display a large central island of stability surrounded by resonant islands (e.g., 3:2, 2:1) and thin chaotic layers. Oblate configurations show a reduced central island and a broader chaotic region. Increasing μ_d progressively destroys the regular islands, enlarges the chaotic sea, and creates new stochastic layers, especially in the KL spacetime where the effect is more pronounced due to the presence of higher‑order multipoles.

Performance benchmarks indicate that the KL metric requires roughly 30 % less CPU time and 25 % less memory than the HT metric for comparable accuracy, owing to its simpler angular dependence. This makes KL particularly suitable for long‑term integrations needed in gravitational‑wave source modelling or accretion‑disk dynamics.

In conclusion, the study demonstrates that incorporating both a mass quadrupole moment and a magnetic dipole moment yields a rich variety of chaotic phenomena that are absent in the pure Kerr solution. The Kerr‑like metric, with its inclusion of higher‑order multipoles and computational efficiency, provides a more realistic framework for modelling the spacetime of rapidly rotating, deformed compact objects such as millisecond pulsars or magnetised neutron stars. The authors suggest that future work should quantify how these chaotic structures influence observable signatures—gravitational‑wave phase evolution, quasi‑periodic oscillations in X‑ray binaries, and the transport properties of surrounding accretion disks—by extending the analysis to higher multipole orders and fully relativistic magnetohydrodynamic simulations.