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

TL;DR

This work derives a complete analytic solution of the Mathisson--Papapetrou--Dixon equations at linear order in spin for a general plane gravitational wave with polarization profiles $h_{+}(u)$ and $h_{\times}(u)$. By combining a parallel-transported orthonormal tetrad with the plane-wave Killing symmetries, six conserved quantities (three from Killing vectors and three from spin components) are obtained to fully determine the spin, momentum, and worldline at $ ext{O}(s)$. The transverse motion decouples and is expressible as single integrals over the retarded time $u$, providing a model-independent, closed-form description of spin--curvature–induced deviations suitable for gravitational memory studies and high-precision GW detectors. This framework offers a concrete tool for assessing spin-dependent effects in plane-wave spacetimes and lays groundwork for connecting classical spin dynamics to spin–graviton interactions in semi-classical contexts.

Abstract

The interaction between spin and gravitational waves causes spinning bodies to deviate from their geodesics. In this work, we obtain the complete analytic solution of the Mathisson--Papapetrou--Dixon equations at linear order in the spin for a general plane gravitational wave with arbitrary polarization profiles. 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 for realistic or theoretical gravitational-wave signals. This analytic solution offers a versatile tool for studying spin-dependent effects in gravitational memory, Penrose-limit geometries, and future high-precision space-based detectors.

Figures (2)

• Figure 1: Evolution of the coordinate components $s^y(u)$ and $s^z(u)$ for a particle initially at rest, with transverse spin magnitude $s_\perp = 0.1\,m$ and $s_\perp = 0.05\,m$. The amplitude of the gravitational wave is $h=10^{-1}$. Although the tetrad components of the spin remain constant, the coordinate components undergo a precession driven by the plane wave geometry. The arrows indicate the initial spin direction.
• Figure 2: Evolution of the transverse momentum components $p^y(u)$ and $p^z(u)$ for a particle initially at rest with transverse spin magnitude $s_\perp=0.1\,m$. The amplitude of the gravitational wave is $h=10^{-10}$. Different colors correspond to different initial orientations in the $(e_2,e_3)$ plane. The oscillatory structure reflects the spin--curvature coupling, whereas a geodesic particle would maintain $p^y=p^z=0$.