Analytic Solution for the Motion of Spinning Particles in Plane Gravitational Wave Spacetime

Abstract

The interaction between spin and gravitational waves causes spinning bodies to deviate from their geodesics. In this work, we obtain the analytic solution of the Mathisson--Papapetrou--Dixon equations at linear order in the spin for plane gravitational wave spacetimes. Our approach combines a parallel-transported tetrad with the translational Killing symmetries of plane wave spacetimes, yielding six conserved quantities that fully determine the momentum, spin evolution, and worldline. The resulting transverse and longitudinal motions are expressed in closed form as single integrals of the retarded time, providing a unified and model-independent framework for computing spin--curvature-induced deviations. This analytic solution offers a versatile tool for studying spin-dependent effects in gravitational memory, Penrose-limit geometries, and high-energy scattering regimes.