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arXiv · 2210.03866

Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry

Paul Ramond·Soichiro Isoyama
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Abstract

We study the Hamiltonian dynamics of a neutral, massive spinning test body at linear order in spin, as governed by the Mathisson--Papapetrou--Tulczyjew--Dixon equations in a 4-dimensional spacetime admitting a non-degenerate Killing--Yano tensor. The Tulczyjew--Dixon spin supplementary condition is imposed as a set of algebraic constraints, and the resulting constraint surface is equipped with a Poisson--Dirac bracket, yielding a non-degenerate, 10-dimensional physical phase space. On this reduced space we identify five functionally independent first integrals in involution: the autonomous Hamiltonian, two constants of motion associated with two commuting Killing vectors, a generalised Carter constant, and the R\"udiger constant; all four descending from the Killing--Yano tensor. All calculations are covariant, and the result is a purely geometric statement: it requires no background field equations and holds for any metric admitting a non-degenerate Killing--Yano tensor, beyond Kerr, beyond vacuum and beyond general relativity. Our results identify Killing--Yano symmetry as the sole geometric source of linear-in-spin integrability. Extensions to quadratic-in-spin dynamics, including the spin-induced quadrupole, and to tidally-induced quadrupolar effects for non-spinning bodies, are treated in companion papers.

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Paul Ramond, Soichiro Isoyama. 2022-10-08. Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry. https://arxiv.org/abs/2210.03866

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