Emergent Spacetime and Entanglement. Quantum Mechanical Derivation of the Schwarzschild Radius

Abstract

We develop a relational extension of special relativity in which temporal structure is represented by a two-dimensional phase variable and the kinematics of free particles correspond to motion of fixed magnitude in this time phase plane. Spatial and temporal notions arise only through comparisons between physical systems, and the resulting “completed” relativistic framework reproduces standard Lorentz kinematics in inertial regimes while reducing to Newtonian dynamics in the non-relativistic limit. Gravitational phenomena are introduced through local phase shifts in the temporal and spatial components of particle trajectories, without invoking curvature of an underlying spacetime manifold. Within this formulation, entangled configurations correspond to superpositions occupying the same spacetime phase point from the observer’s perspective, thereby preserving locality. Applying the phase evolution rules to particles propagating parallel to a gravitational field yields a critical radius identical to the Schwarzschild boundary, derived here from quantum and geometric resolvability conditions rather than from Einstein’s equations. Information is confined to a finite shell outside the critical radius. No central singularity appears. Zero-frequency gravitons remain untapped by the horizon. An optical test is proposed using intersecting monochromatic laser beams. The model predicts enhanced photon–photon coupling when the temporal phases of the beams are aligned.