The Aharonov-Bohm effect demonstrates that quantum particles accumulate phase from electromagnetic potentials even in regions where the field vanishes. The potential, not the field, is the physical quantity. In general relativity, frame-dragging around rotating masses creates an analogous structure: the off-diagonal metric component acts as a gravitational vector potential.
Abrunhosa (arXiv:2602.20337) calculates the Aharonov-Bohm phase shift for Cooper pairs in Kerr spacetime. The frame-dragging couples to the macroscopic phase of the superconducting condensate, producing a gravitomagnetic phase shift. Near Sagittarius A, the predicted shift is approximately 10^24 radians. Near M87, approximately 10^27.
The numbers are absurd and unmeasurable. But the calculation reveals something structural: the coupling between gravity and quantum coherence does not require exotic physics. Cooper pairs — ordinary superconducting quasiparticles — respond to gravitational frame-dragging through exactly the same mechanism by which they respond to electromagnetic potentials. The off-diagonal metric component enters the Ginzburg-Landau equation in the same position as the vector potential.
Tidal forces, which would destroy superconductivity by disrupting Cooper pairs, are negligible at safe distances from the event horizon. The phase shift accumulates without the pairs being destroyed. The information about the spacetime geometry is encoded in the phase, not in the amplitude. The pairs survive; their coherence records the curvature.
The general observation: when a quantum phase couples to a classical field, the coupling can produce enormous phase shifts that are mathematically well-defined even when experimentally inaccessible. The calculation's value is not in predicting a measurement but in establishing that the coupling exists — that spacetime geometry and quantum coherence speak the same mathematical language.