A1 The Metric Gradient Force—A Flagship Test of Geometric Unification: From Legacy Stern-Gerlach Data to a BEC Exclusive Crucial Experiment

Abstract

Physical interactions can be classified by their time-reversal properties into two categories: reversible geometric forces and irreversible dissipative forces. General relativity describes the former—particles follow geodesics in curved spacetime, with acceleration arising from the geometric curvature of space. Newtonian mechanics and statistical physics describe the latter—a particle in a viscous fluid experiences a damping force that depends on the direction of velocity. Yet how to experimentally distinguish these two types of forces remains an open question. Starting from the order-parameter spacetime theory, this paper proposes an experimental protocol that can be executed on existing cold-atom BEC platforms to answer this question. The core idea of the protocol is: the metric gradient force is an even function of velocity (acceleration proportional to the square of velocity, sign independent of velocity direction), whereas all known dissipative forces are odd functions of velocity (they change sign when velocity reverses). This symmetry difference provides a universal criterion for distinguishing geometric forces from dissipative forces. The argument proceeds at two levels. Level one (pilot verification): a reanalysis of the 87Rb Bose-Einstein condensate Stern-Gerlach experiment data published by the MPQ group in 2003. By extracting the center-of-mass position data of the molecular cloud at 16 time-slices from Figure 4 of the experiment, we find that the molecular cloud undergoes harmonic oscillation in a 100 G/cm magnetic field gradient, with an oscillation frequency of 56 Hz and an amplitude of 0.68 mm. Comparing the experimental acceleration with the theoretically predicted metric gradient force, we verify the a ∝ v² scaling relation and deduce the logarithmic gradient of the order-parameter space metric ∂ ln g/∂x ≈ 3.4 × 10³ m⁻¹. The left and right extreme positions of the molecular cloud oscillation are approximately symmetric about the equilibrium position, consistent with the prediction of the metric gradient force as an even function of velocity. This reanalysis of legacy data provides a pilot signal for the existence of the metric gradient force. Level two (exclusive crucial experiment): based on the above pilot signal, we design a crucial experimental protocol using 87Rb BEC and Feshbach resonance. The protocol creates a spatially non-uniform effective metric via a spatially inhomogeneous magnetic field, and measures the mean displacement of the atomic cloud center-of-mass for two opposite initial velocity directions. The dual self-calibration feature of the protocol—the mean of forward and backward displacements cancels dissipative forces, while the control group cancels conservative forces—eliminates all known interferences, leaving a clean geometric force signal. We systematically provide five signal identification criteria and three independent verification schemes for scaling relations. Under baseline experimental parameters, the expected displacement is about 1.47 microns; by optimizing the detuning, the expected displacement can be enhanced to about 4.1 microns, with a combined signal-to-noise ratio of about 8 standard deviations. The reanalysis results of the Stern-Gerlach legacy data provide empirical support for this prospective experiment. A positive result would demonstrate that the effective metric in condensed matter systems and the spacetime metric obey the same geometric dynamical rules—they are projections of the same geometric structure in different systems, rather than a mere analogy. A negative result would directly falsify the geometric core of Postulate One of the order-parameter spacetime theory.