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 flat 4D Euclidean space along an observer’s four-velocity, dynamically inducing a Lorentzian signature. 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 governs particle trajectories. This framework recovers gravitational attraction, post-Newtonian corrections, and the perihelion shift of Mercury from first principles, without assuming time or curvature a priori. Using an effective field theory expansion of \( Q(x) \), we derive a falsifiable higher-order correction to the COW neutron interferometry phase shift, providing an Earth-based test of the model. Finally, we show that the emergent metric arises as a real slice of a Hermitian form in twistor space, interpreting spacetime curvature as refractive deformation of incidence relations. This predictive framework derives general relativity from quantum mechanics and supports Penrose’s vision of gravity as a quantum-geometric phenomenon.