Dynamics of Spinning Test Body in quadratic Einstein-Cartan Theory and its Free-fall Test

We study the dynamics of the non-relativistic spinning test body (STB) in the framework of Einstein-Cartan theory(ECT), in which the weak equivalence principle is violated by the spin-gravitational interaction. We derive the general equation of geodesic in terms of comoving tetrads. More concretely, we consider the case of the quadratic form of the lagrangian, within the environment of weak and static spherically symmetric space-time. We find that the trajectories of STB deviate from the traditional Mathisson\textendash Papapetrou equation, which is due to the coupling of the spin of the test particle to the torsion field of the environment. This allows us to test the theory with free-fall experiment in the laboratory, such as atom interferometer. By using the previous data, we find the upper bound of the possible torsion field on Earth is given by up to $2.0\times 10^{1} \mathrm{~m^{-1}}$ and torsion gradient up to $3.1 \times 10^{-6}\mathrm{~m^{-2}}$. This result may enable us to provide a theoretical foundation for future precision measurements of the existence of the fifth force.