Analytic solution of the Mathisson–Papapetrou–Dixon equations

Derive an analytic solution of the coupled, nonlinear Mathisson–Papapetrou–Dixon equations governing the momentum, spin, and velocity of a spinning extended object in curved spacetime.

Background

The paper models spinning extended objects using the Mathisson–Papapetrou–Dixon equations together with the Tulczyjew–Dixon condition and conserved mass and spin. These equations form a highly coupled nonlinear system for the object’s momentum, spin tensor, and four-velocity. The paper states that no analytic solution is known and therefore relies on perturbative and numerical methods to study finite-size effects and gravitational-memory imprints along the worldline.

References

These are highly coupled, non-linear system of equations for which no analytic solution is known.

— On the Emergence of the Hysteresis Effect in Gravitational Systems  (2608.21269 - Moti et al., 21 Aug 2026) in Section III, immediately following equations (MP-Momentum)–(MP-Velocity)

Finally, the incorporation of gravitational radiation reaction into an analytic scheme of this kind remains an open challenge, but it would open the possibility of using piecewise hypergeometric representations for the efficient modeling of EMRIs in spacetimes with a cosmological constant.

— Analytic motions of spinning particles in Schwarzschild-(anti-)de Sitter spacetime  (2609.09755 - Li et al., 9 Sep 2026) in Section 7, Concluding remarks