PPN Structure and Substrate Back‑Reaction in FRCFD

Substrate Self‑Glow and PPN Refinement in FRCFD

Derek Flegg

Southern Ontario, March 2026

Table of Contents

1. Self‑Sourced Substrate Field Equation

Beyond the test‑particle limit, the substrate field S carries its own energy density and therefore contributes to its own sourcing. In a static, spherically symmetric vacuum outside a central mass, the field equation becomes: 

d²S/dr² + (2/r)dS/dr - beta*S³ = -g * rho_substrate

where:
rho_substrate = 1/2 (dS/dr)² + (beta/4) * S⁴

This nonlinear feedback means the substrate “feels” its own intensity. In the far field, the nonlinear terms decay faster than the Laplacian, preserving the Newtonian limit. Closer to the source, the substrate’s self‑energy adds a small but finite contribution to the effective mass profile—an effect we call the Self‑Glow.

2. Transition to Saturation

Numerical solutions of the substrate equation reveal three distinct regimes:

  • Newtonian Tail: At large radii, S ∝ 1/r and rho_substrate is negligible.
  • Inertial Thickening: In the mid‑range, the substrate’s self‑energy increases the local impedance, slightly steepening the potential.
  • Saturation Plateau: As S → S_max, the response function f(S) → 0. The field “freezes,” flattening dS/dr and preventing divergence.

This plateau produces a finite‑radius, high‑impedance core—replacing the geometric singularity of GR with a physically saturated interior.

3. Second‑Order Metric Expansion

To compare FRCFD with precision gravitational tests, we expand the effective metric using the response function: 

f(S) = exp(-S / S_max)

In the weak‑field regime, where S ≈ U = GM/r, the metric components expand to: 

g_00 = exp(-2S/S_max)
     ≈ 1 - 2(S/S_max) + 2(S/S_max)²

g_rr = -exp(2S/S_max)
     ≈ -(1 + 2(S/S_max) + 2(S/S_max)²)

These corrected expansions ensure consistency with the PPN formalism.

4. Derivation of the PPN β Parameter

The standard PPN metric for a static, spherically symmetric mass is: 

g_00 = 1 - 2U + 2 beta U²
g_rr = -(1 + 2 gamma U)

Identifying the Newtonian potential with the substrate field, U = S, we compare coefficients:

  • From g_rr: gamma = 1 / S_max
  • From g_00: the quadratic term gives beta = (1 / S_max)²

beta = (1 / S_max)²

Thus, when S_max = 1, FRCFD reproduces the Einsteinian values gamma = 1 and beta = 1.

The substrate’s self‑energy rho_substrate introduces additional O(U²) corrections, meaning β may deviate slightly from (1/S_max)² for compact objects. This deviation is a testable prediction.

5. Physical Consistency and GR Limit

As S_max → 1, both PPN parameters converge to their GR values. This demonstrates that FRCFD is not merely an alternative to General Relativity, but a one‑parameter extension that contains GR as a special capacity state.

The theory’s rigidity is now explicit: a single parameter S_max simultaneously controls:

  • linear light‑bending (γ)
  • nonlinear perihelion precession (β)
  • strong‑field saturation behavior
  • the size of the high‑impedance core

This unified structure makes FRCFD directly falsifiable across weak‑field, strong‑field, and imaging regimes.

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