A Polarised Torsion Balance Test of Spin-Sourced Torsion: The Contact Result and the Reach for Propagating Extensions

A torsion balance carrying spin-polarised test bodies is often proposed as a direct laboratory test of Einstein-Cartan gravity. This paper works out what such an instrument can and cannot see. In Einstein-Cartan theory the torsion tensor is fixed algebraically by the spin density where the spin density is; it does not propagate into the vacuum between two separated bodies. The predicted torque between a polarised source and a polarised pendulum held apart by any vacuum gap is therefore exactly zero, and a null result at any sensitivity confirms the theory while a nonzero result would refute its minimal form. The balance is nevertheless a sharp instrument against extensions in which an axial torsion mode propagates with a Yukawa range. For a pendulum with 9.4e22 polarised electron spins, a 0.05 m separation, and a torque sensitivity of 1e-15 N m, the per-spin energy sensitivity is 6.64e-20 eV, and the bound on a long-range axial spin-spin coupling is g squared over 4 pi below 1.79e-37, about eight orders of magnitude above the gravitational-strength coupling of the electron, 1.75e-45. At a range of 1 mm the bound weakens to 9.28e-16 because of Yukawa attenuation across the gap. Computed from stated assumptions, not measured.