Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown

TL;DR

The paper analyzes the motion of a spinning compact body with a spin-induced quadrupole in four-dimensional type-D Einstein spacetimes that admit a Killing-Yano tensor, under the Tulczyjew-Dixon spin supplementary condition. It develops a covariant Hamiltonian formulation on a 10-dimensional reduced phase space, introduces a quadratic Carter-Rüdiger constant Q for black-hole-like deformability (κ=1), and proves Liouville–Arnold integrability at quadratic order in spin by constructing five commuting first integrals {H,Ξ,𝔛,K,Q}. For generic material or exotic secondaries (κ≠1) the quadratic-in-spin invariant Q does not survive, leading to breakdown of integrability at O($S^2$). Kerr is recovered as a special case, and the results generalize Kerr-based integrability to a broader class of type-D Einstein spacetimes with KY symmetry, providing a foundation for symmetry-based, beyond-Kerr EMRI modeling and potential probes of the secondary’s internal structure through orbital dynamics.

Abstract

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.

Figures (1)

• Figure 1: Different sub-manifolds are necessary to lift the degeneracies associated to the existence of Casimir invariants $(\mathcal{M}\rightarrow\mathcal{N})$ and correctly implement the TD SSC $(\mathcal{N}\rightarrow\mathcal{T}\rightarrow\mathcal{P})$. The geodesic Hamiltonian Schm.02HinFle.08 on the 8D phase space $(x^\alpha,\,p_\alpha)\in\mathcal{G}$ is simply obtained by setting $S^{\alpha\beta}=0$ which projects automatically onto the leaves $S_\circ=0$. Figure is reproduced from Ra.Iso.PapII.24.