Parametrized Post-Newtonian Analysis of Density Field Dynamics in the Weak-Field, Slow-Motion Limit

Abstract (Theoretical Research; Preprint)

We present a complete mapping of Density Field Dynamics (DFD) to the ten standard Parametrized Post-Newtonian (PPN) coefficients {γ, β, ξ, α₁,₂,₃, ζ₁,₂,₃,₄} in the weak-field, slow-motion (1PN) regime. Starting from the optical-metric ansatz g₀₀ = −eᵠ, gᵢⱼ = e⁻ᵠδᵢⱼ with ψ = −2U/c² + O(c⁻⁴), we derive γ = β = 1 from the scalar sector. We then solve the vector sector via a transverse (Helmholtz) projection of the mass current to obtain g₀ᵢ = (1/c³)(−7/2 Vᵢ − 1/2 Wᵢ), which implies α₁,₂,₃ = ξ = ζ₁ = 0 at 1PN. Completing g₀₀ at O(c⁻⁴) shows no Whitehead term (ξ = 0), and diffeomorphism invariance with minimal coupling gives local conservation, ζ₁,₂,₃,₄ = 0. Thus DFD reproduces all ten GR PPN values at 1PN. We provide explicit derivations and audit checks, validate against classic observables (deflection, Shapiro, perihelion, frame-dragging), and summarize experimental implications.