Tensor Spin Polarization Induced by Curved Freeze-Out Hypersurface

We investigate how the curvature of the freeze-out hypersurface polarizes massive vector bosons in relativistic heavy-ion collisions. Starting from the Proca Lagrangian and using the Wigner function formalism, we perform a systematic gradient expansion to obtain a covariant spin-polarization tensor expressed in terms of hydrodynamic fields and the curvature tensor of the freeze-out hypersurface. Analytic results for Bjorken and Gubser flows show that curvature anisotropy generates a non-zero tensor polarization. For $\phi$ meson, we estimate the curvature contribution to its spin alignment as $\delta {\mathrm{\Theta }}_{yy}\sim -{10}^{-4}$ to $-{10}^{-3}$. We also find that the curvature contribution grows as the system size decreases, suggesting that spin polarization measurements in small systems may provide a clean probe of this geometric effect.