Nonlocal Frame Response and Field-Theoretic PPN Extraction

Nonlocal Frame Response and Field-Theoretic PPN Extraction

Derek Flegg

Southern Ontario, March 2026

Table of Contents

1. Nonlocal Frame Response (omega(r))

In FRCFD, rotational effects are not treated as the geometric "dragging" of a manifold, but as a nonlocal coupling between the angular momentum density J_sub and the substrate field. The effective angular velocity omega(r) is mediated by a kernel that accounts for the finite coherence length (ell) of the substrate: 

omega(r) = (kappa/r^2) * integral[ J_sub(r') * exp(-|r-r'|/ell) * f(S(r)) * f(S(r')) dr' ]

This integral representation ensures that frame-dragging is suppressed in high-impedance regions where f(S) -> 0, preventing the unphysical infinite rotational velocities associated with standard Kerr singularities.

2. PPN Parameter Extraction (gamma, beta)

To compare FRCFD with Solar System constraints, we expand the effective metric in the Parametrized Post-Newtonian (PPN) limit. Given f(S) = exp(-S/S_max) and the weak-field approximation S approx GM/r:

2.1 Parameter gamma

The spatial curvature relative to the time-component defines gamma. In the FRCFD metric ds^2 = f(S)^2 dt^2 - f(S)^-2 dr^2: 

f(S)^2 approx 1 - 2(GM/r)(1/S_max) + 2(GM/r)^2(1/S_max)^2
f(S)^-2 approx 1 + 2(GM/r)(1/S_max) + 3(GM/r)^2(1/S_max)^2

Matching the coefficient of (GM/r) to the standard PPN form (1 + 2 gamma GM/r) yields: 

gamma = 1 / S_max

Observationally, since gamma approx 1, this forces S_max approx 1. Deviations at the 10^-5 level provide a direct test of the substrate capacity.

3. Full Geodesic Derivation

We derive the equations of motion using the Lagrangian L_g = 1/2 g_mu_nu (dx^mu/d_tau)(dx^nu/d_tau). For the effective metric, the radial geodesic equation for a massive particle (h=0) is: 

(dr/d_tau)^2 = E^2 - f(S)^2 * [1 + L^2/r^2]

Unlike the heuristic approach, this form accounts for the nonlinear suppression of the effective potential as f(S) vanishes near the saturation limit. The term f(S)^2 acts as a universal barrier, enforcing finite r_ISCO and preventing the "plunge" into a singularity.

4. Energy Conditions and Stability

The stability of the substrate field S is analyzed via the Hamiltonian density. The potential term beta/4 S^4 ensures that for beta > 0, the vacuum is stable.

  • Weak Energy Condition: Validated as long as the matter coupling g S |Psi|^2 does not overwhelm the gradient energy (grad S)^2.
  • Saturation Stability: As S -> S_max, the non-polynomial nature of the response function f(S) provides a "stiffening" effect, preventing the collapse of the matter-substrate system.

5. Numerical Solutions for the S Equation

For static, spherically symmetric sources, the substrate equation reduces to a nonlinear ODE: 

d²S/dr² + (2/r)dS/dr - beta * S^3 = -g * |Psi|^2

Numerical integration using a Runge-Kutta 4th order scheme reveals that for high mass-densities, the S-field exhibits a "plateau" behavior near the core, where dS/dr -> 0 as S -> S_max. This saturation confirms the removal of the 1/r singularity at the origin.

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