Spin-Curvature Coupling Limits on Test Mass Stability in Space-Borne Gravitational Wave Detectors

Space-borne gravitational wave detectors, such as LISA, Taiji, and TianQin, rely on the drag-free motion of test masses (TMs) to serve as geodesic reference points. While ideally treated as point particles, real TMs are extended bodies with finite residual rotation. According to the Mathisson-Papapetrou-Dixon (MPD) equations in General Relativity, a spinning body in a curved spacetime deviates from geodesic motion due to the coupling between its spin tensor $S^{\mu\nu}$ and the ambient Riemann curvature tensor $R^\mu{}_{\nu\rho\sigma}$. In this work, we rigorously derive the acceleration noise induced by this spin-curvature coupling in the context of the Solar System's gravitational field. We establish an analytical upper bound on the permissible residual angular velocity of the TMs to ensure the acceleration noise remains within the LISA budget ($3 \times 10^{-15} \, \text{m/s}^2/\sqrt{\text{Hz}}$). The derivation of the MPD deviation force is formally verified using the Lean 4 theorem prover, ensuring the tensorial consistency of the noise model. Our results indicate that while currently negligible for LISA, MPD effects may become a limiting noise source for next-generation detectors (e.g., DECIGO) targeting the deci-hertz band with sensitivity goals of $10^{-18} \, \text{m/s}^2/\sqrt{\text{Hz}}$.