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

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.

• The paper interprets chaotic dynamics in Kerr-like and Hartle-Thorne metrics across a range of mass quadrupole moments.
• The analysis shows that the Hartle-Thorne exponential variant (appHT) matches Kerr-like dynamics within the quadrupole approximation
• The combined effect of strong magnetic fields and high quadrupole moments in the Kerr-like metric stabilizes chaotic orbits, suggesting stable astrophysical orbits around deformed, magnetized stars, specifically pulsar planets

Overview

This paper compares the geodesic dynamics of a unit-mass test particle in three approximate stationary, axisymmetric spacetimes: the Kerr-like (KL) metric with mass quadrupole moment (2602.00565), the logarithmic Hartle–Thorne metric (HTlog), and an exponential ("approximate") Hartle–Thorne metric (appHT). The authors extend the comparison to their magnetized counterparts — the dipolar Kerr-like metric (KLdip) and the dipolar Hartle–Thorne metric (HTdip) — which include the magnetic dipole moment $\mu_d$ of the source. The central diagnostic is the Poincaré section of the equatorial phase space, with the mass quadrupole $q_{KL}$ serving as the control parameter for both prolate ($q_{KL} > 0$) and oblate ($q_{KL} < 0$) sources.

The motivation rests on two facts. First, the Kerr metric describes only a perfectly spherical rotating source and admits no interior extension, making it unsuitable for neutron stars, white dwarfs, or planets. Second, any nonzero mass quadrupole destroys the Carter constant, rendering the Hamiltonian system non-integrable and opening chaotic regions near the strong-field zone — behavior previously established for Manko–Novikov, Quevedo–Mashhoon, Zipoy–Voorhees, and Hartle–Thorne spacetimes.

Metrics and their relation

All metrics considered share the general stationary form with potentials $V$, $W$, $X$, $Y$, $Z$ depending only on $(r,\theta)$, so that $q_{KL}$0 and $q_{KL}$1 are conserved but no third isolating integral exists once $q_{KL}$2. The KL metric, constructed via the Ernst formalism and Hoenselaers–Kinnersley–Xanthopoulos transformations, contains Legendre polynomials of the first kind only, whereas HTlog involves associated Legendre functions of the second kind with logarithmic terms — a computational disadvantage the authors emphasize.

A key methodological element is the matching of the Geroch–Hansen multipole moments via the Fodor–Hoenselaers–Perjés algorithm. Since $q_{KL}$3 for HT and $q_{KL}$4 for KL, the correspondence

$q_{KL}$5

allows both metrics to describe the same physical source. Under this identification, Taylor-expanded forms of both metrics agree to second order, and the KL metric additionally carries the spin octupole moment absent in HT. The appHT variant replaces the logarithmic terms of HTlog by exponential factors while preserving the same quadrupole relation. The authors explicitly note that KL and HT are not isometric as parametrized, and that the exact Quevedo–Mashhoon metric was excluded because it is not isometric to either and is computationally expensive.

For the charged/magnetized extensions, test-particle motion is treated via minimal coupling and the super-Hamiltonian formalism, $q_{KL}$6, with full canonical equations given. Throughout the dynamical study, both the source charge $q_{KL}$7 and test-particle charge $q_{KL}$8 are set to zero, retaining only $q_{KL}$9.

Dynamics without electromagnetic fields

Simulations use Runge–Kutta–Fehlberg integration with fixed parameters $q_{KL} > 0$0, $q_{KL} > 0$1, $q_{KL} > 0$2, $q_{KL} > 0$3, $q_{KL} > 0$4, scanning $q_{KL} > 0$5. The principal findings are:

• Unperturbed limit: For $q_{KL} > 0$6, the KL metric reduces exactly to Kerr and shows only the main island of stability. Both HT metrics, however, reduce only to Lense–Thirring and carry a residual $q_{KL} > 0$7, producing narrow resonances (near $q_{KL} > 0$8 and $q_{KL} > 0$9) and incipient chaos even at nominally zero quadrupole. This is a direct consequence of the imperfect Kerr reduction of HT and propagates into all subsequent comparisons.
• Prolate case ($q_{KL} < 0$0, $q_{KL} < 0$1): KL exhibits rich chaotic structure around $q_{KL} < 0$2 that HTlog lacks, while appHT matches KL closely — attributed to its Taylor expansion agreeing more tightly with KL's.
• Strong prolate case ($q_{KL} < 0$3): KL and appHT remain qualitatively similar (main islands near $q_{KL} < 0$4), differing mainly in hyperbolic-point location ($q_{KL} < 0$5 vs. $q_{KL} < 0$6) and satellite-island structure. HTlog diverges markedly, placing its main island near $q_{KL} < 0$7.
• Oblate cases ($q_{KL} < 0$8 and $q_{KL} < 0$9): the pattern inverts — KL and appHT sections are more stable than HTlog, which shows extra higher-order islands, hyperbolic points, and at $V$0 a wholesale destruction of tori with few remaining structures.

The overall conclusion of this section is that KL and appHT agree well even at large $V$1, whereas HTlog diverges from both, consistent with the claim that appHT has a wider validity range in quadrupole than HTlog, and that KL's higher-order multipoles make it more accurate.

Effects of the magnetic dipole

With $V$2 added:

• For a spherical source ($V$3), both KLdip and HTdip display resonances and chaos from the magnetic perturbation alone, with broadly similar main islands near $V$4; HTdip shows an additional resonance at $V$5.
• Combining $V$6 with $V$7 produces a striking result: the secondary structures and chaotic regions vanish entirely, leaving sections that appear integrable. The authors are careful not to claim integrability — rather, the combined perturbation destroys the less stable geodesics. They report that additional unshown simulations reproduce this behavior.

The authors draw a physical implication from this suppression: spinning, deformed compact objects with strong magnetic fields may admit stable satellite orbits, offering a partial explanation for pulsar planets. They immediately qualify this: the chosen parameters do not correspond to a realistic neutron star, since large $V$8 implies extreme deformation, and parameters were selected to maximize visible phase-space structure.

Limitations and open questions

Several caveats are stated plainly in the paper. The comparison rests on the assumption that matching $V$9 suffices to equate the sources, even though KL carries higher multipoles (notably spin octupole) that HT lacks — so residual dynamical differences between KL and appHT cannot be attributed solely to numerical accuracy. The event horizons of these approximate metrics are mathematical constructs without physical meaning for material stars. The apparent "integrabilization" under combined $W$0–$W$1 perturbation is demonstrated only for one parameter set plus unspecified additional cases, without a systematic scan or an analytical explanation of the destructive interference mechanism. Whether the stabilizing effect persists for realistic magnetar parameters ($W$2 ratios, moderate $W$3) remains open, as does quantitative chaos characterization beyond visual Poincaré inspection (e.g., Lyapunov exponents or rotation numbers).

Conclusion

Using matched Geroch–Hansen quadrupole moments, this work establishes that the exponential-form Hartle–Thorne metric tracks the Kerr-like metric's phase-space dynamics across prolate and oblate deformations, while the traditional logarithmic Hartle–Thorne metric diverges increasingly with $W$4. The KL family's higher multipole content and computational simplicity favor it for numerical relativity tests. The most consequential finding is the suppression of chaotic structures when magnetic dipole and quadrupole perturbations act jointly, suggesting a dynamical route to long-term orbital stability around strongly magnetized, deformed compact objects — a result whose astrophysical relevance awaits testing with realistic stellar parameters.