Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown
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| Format: | Preprint |
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2026
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| author | Ramond, Paul Isoyama, Soichiro Druart, Adrien |
| author_facet | Ramond, Paul Isoyama, Soichiro Druart, Adrien |
| contents | We develop a covariant Hamiltonian formulation of the Mathisson-Papapetrou-Tulczyjew-Dixon dynamics at quadratic order in spin under the Tulczyjew-Dixon spin supplementary condition (TD SSC). In four-dimensional, type-D Einstein (vacuum/$Λ$-vacuum) spacetimes admitting a non-degenerate Killing-Yano (KY) tensor, we reduce via a Dirac bracket to the 10-dimensional physical phase space and model the quadratic sector with a spin-induced quadrupole characterized by a deformability $κ$ ($κ=1$ for black-hole--like; $κ\neq 1$ for material or exotic compact objects). For $κ=1$, we construct five independent first integrals -- an autonomous Hamiltonian, two KY-generated Killing invariants, a linear Rüdiger constant, and a quadratic Carter-Rüdiger constant -- establishing Liouville-Arnold integrability at quadratic order in spin. For $κ\neq 1$, the symmetry-generated invariants are not conserved in general and integrability does not persist at this order. The proof proceeds via covariant Poisson-bracket computations using a null bivector decomposition; Kerr is recovered as a special case. These results show that integrability can persist beyond Kerr and beyond the linear-in-spin regime, laying groundwork for symmetry-based, beyond-Kerr modelling of asymmetric-mass, spinning compact binaries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_06416 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown Ramond, Paul Isoyama, Soichiro Druart, Adrien General Relativity and Quantum Cosmology Exactly Solvable and Integrable Systems We develop a covariant Hamiltonian formulation of the Mathisson-Papapetrou-Tulczyjew-Dixon dynamics at quadratic order in spin under the Tulczyjew-Dixon spin supplementary condition (TD SSC). In four-dimensional, type-D Einstein (vacuum/$Λ$-vacuum) spacetimes admitting a non-degenerate Killing-Yano (KY) tensor, we reduce via a Dirac bracket to the 10-dimensional physical phase space and model the quadratic sector with a spin-induced quadrupole characterized by a deformability $κ$ ($κ=1$ for black-hole--like; $κ\neq 1$ for material or exotic compact objects). For $κ=1$, we construct five independent first integrals -- an autonomous Hamiltonian, two KY-generated Killing invariants, a linear Rüdiger constant, and a quadratic Carter-Rüdiger constant -- establishing Liouville-Arnold integrability at quadratic order in spin. For $κ\neq 1$, the symmetry-generated invariants are not conserved in general and integrability does not persist at this order. The proof proceeds via covariant Poisson-bracket computations using a null bivector decomposition; Kerr is recovered as a special case. These results show that integrability can persist beyond Kerr and beyond the linear-in-spin regime, laying groundwork for symmetry-based, beyond-Kerr modelling of asymmetric-mass, spinning compact binaries. |
| title | Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown |
| topic | General Relativity and Quantum Cosmology Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2601.06416 |