Investigating Phase Field Cosmology: Observer-induced Lorentzian Geometry from a Scalar Field on Euclidean E4

This is the authors' abstract. We don't add key points for this paper.

Starting from a single complex scalar field \(Ψ(X^A)\) defined on \(E^4\), we connect to GR sufficiently to derive the Einstein-Infeld-Hoffmann (EIH) equations—describing the post-Newtonian dynamics of gravitating bodies—entirely from the internal structure of a single complex scalar field \(Ψ(X^A)\) defined on a flat four-dimensional Euclidean space \(E^4\). No prior spacetime geometry, time direction, or gravitational dynamics are assumed. Instead, all such structures emerge from the phase and amplitude of \(Ψ\) alone. This model requires no free parameters and adheres rigorously to first principles derivations. A probability current \(J^A = R^2 ∂^A S\) defines an intrinsic observer field \(U^A\), which induces a 3+1 foliation with \( U^A ∂_A E = 0 \) adherence, an effective Lorentzian signature, and extrinsic curvature \(KAB\) associated with gravitational behavior. A geometric action built from \(KAB\) leads to field equations consistent with general relativity in the weak-field limit. Applying the Madelung decomposition and identifying \(ρ = m R^2\) as effective mass density, we recover classical gravitational dynamics, culminating in a post-Newtonian expansion that matches the standard EIH equations through \(O(v^4/c^4)\). The emergent force law reveals that gravitational curvature arises directly from misalignment in internal phase flow, not background geometry, allowing us to simulate Mercury's precession and cosmic expansion entirely from flat-space dynamics. This demonstrates that gravitational motion, including relativistic corrections, can emerge from flat-space scalar field dynamics without invoking curved spacetime or metric postulates.