Gravitational Length Stretching: Curvature-Induced Modulation of Quantum Probability Densities

This paper introduces Gravitational Length Stretching (GLS), a previously unexplored quantum-gravitational phenomenon where spacetime curvature modulates the spatial probability density of particles through amplitude distortions in their wavefunctions. Departing from conventional studies of gravitational phase effects, we focus on the amplitude function $ B_k(t, \vec{x}) $—governed by the covariant continuity equation $ \nabla_\mu (B_k^2 k^\mu) = 0 $—as a key mediator of curvature-induced quantum distortions. For ultracold neutrons (UCNs) in a weak-field Schwarzschild spacetime, we predict a 20\% spatial variation in probability density across a $ 1\,\text{mm} $ vertical potential well, alongside curvature-driven energy level shifts of $ \sim\!10^{-13}\,\text{eV} $, resolvable via gravitational resonance spectroscopy. The GLS mechanism, distinct from special relativistic length contraction or tidal forces, arises from the suppression of $ B_k^2 $ in stronger gravitational fields, stretching quantum spatial profiles.