The Projection Hypothesis: Emergent 3D+1 Spacetime from a 4D Quantum Potential

Reconciling quantum mechanics with gravity remains a pivotal challenge in theoretical physics. We propose that 3+1-dimensional spacetime emerges from projecting a 4-dimensional isometric space along an observer’s four-velocity. Extending the de Broglie--Bohm interpretation via a modified Klein--Gordon equation, we derive a 4D quantum potential, \( Q(x) = \frac{\hbar^2}{2m} \frac{\Delta R(x)}{R(x)} \), and a refractive four-force, \( F^\mu = m (\delta^{\mu \nu} - u^\mu u^\nu) \partial_\nu Q(x) \), which guides particle trajectories like a wave bending in a medium, simultaneously inducing an emergent Lorentzian spacetime. Our unified variational principle naturally yields classical gravitational effects---such as Mercury’s perihelion shift---while predicting neutron interferometry phase shifts and unique, falsifiable higher-order corrections. This framework offers novel insights into quantum gravity, applicable to extreme regimes from black holes to the early universe.