Finite-Response Coupled Field Dynamics (FRCFD): From Strong-Field Astrophysics to Cosmological Redshift

Derek Flegg

Southern Ontario, March 2026

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Abstract

Finite-Response Coupled Field Dynamics (FRCFD) is a non-geometric framework in which gravitational, inertial, and cosmological phenomena emerge from a finite-capacity substrate field S(x,t) coupled to matter fields Ψ. The theory replaces geometric curvature with a bounded response function f(S) = exp(-S/Smax), ensuring finite behavior in strong-field regimes while recovering leading-order predictions of General Relativity in the weak-field limit. Singularities are replaced by saturation states, producing finite photon spheres, innermost stable circular orbits (ISCO), and shadow radii. Rotation induces a nonlocal, saturation-limited frame-dragging response rather than geometric twisting. Cosmological redshift is reinterpreted as cumulative impedance-driven frequency decay that preserves spectral coherence. The framework is explicitly falsifiable through cross-domain measurements of the substrate capacity Smax, strong-field imaging, pulsar timing, and precision gravitational tests.

1. Introduction

General Relativity describes gravity as curvature of spacetime, successfully explaining a wide range of phenomena but predicting singularities and relying on geometric ontology. FRCFD proposes an alternative: gravity and time are emergent consequences of a finite-response substrate.

• Recover General Relativity in weak fields
• Eliminate singularities through saturation
• Provide testable deviations in strong-field and cosmological regimes
• Maintain a minimal ontology: one substrate field S, one matter field Ψ

2. Core Framework

2.1 Lagrangian

L = 1/2(∂S)² - β/4 S⁴ + (∂Ψ)² - m²|Ψ|² - g S |Ψ|²

2.2 Coupled Field Equations

∂²S/∂t² - c² ∇²S + β S³ = g |Ψ|²

∂²Ψ/∂t² - v² ∇²Ψ + (m² + g S)Ψ = 0

Matter sources the substrate, while the substrate feeds back into matter propagation—forming a self-regulating nonlinear system.

2.3 Response Function (Update-Rate)

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

This function governs time dilation, signal propagation speed, and effective gravitational behavior.

3. Emergent Metric (Effective Description)

ds² = f(S)² dt² - f(S)⁻² dr² - r² dΩ²

This metric is not fundamental, but an emergent tool describing how matter and light propagate through the substrate.

4. Weak-Field Limit

For small S:

• S ≈ GM/r
• f(S)² ≈ 1 - 2GM/r

Recovering: gravitational redshift, light bending, and perihelion precession.

5. Strong-Field Saturation (No Singularities)

As S → Smax, response saturates, propagation slows but remains finite, and curvature divergences are avoided. Result: Singularities are replaced by high-impedance cores.

6. Photon Sphere and Shadow

r_ph = GM/Smax, R_shadow = e · GM/Smax

These arise from response gradients, not geometric trapping surfaces.

7. ISCO Constraint

r_ISCO = 2GM/Smax

Matching the GR value 6GM gives: Smax ≈ 1/3.

8. Time Dilation as Substrate Loading

Time is identified as the local update-rate of the substrate.

τ ∝ f(S_total), S_total = S_gravity + S_kinetic

This unifies gravitational and relativistic (velocity) time dilation.

9. Rotating Systems (Kerr-Like Behavior)

9.1 Nonlocal Frame-Dragging

ω(r) = κ/r² ∫ J_sub(r') exp(-|r-r'|/ℓ) f(S(r)) f(S(r')) dr'

Rotation produces nonlocal substrate slip, not geometric twisting.

9.2 Effective Rotating Metric

ds² = f(S)² dt² - f(S)⁻² dr² - r² dθ² - r² sin²θ(dϕ - ω dt)²

10. Strong-Field Lensing

Δϕ = 2 ∫ [dr / (r² sqrt(1/b² - f(r)²/r²))] - π

11. Cosmological Redshift as Impedance

11.1 Unified Redshift Law

ln(1 + z) = ∫ (S(x)/Smax) dx

Redshift arises from cumulative update-rate suppression, not geometric expansion. Unlike "Tired Light," FRCFD preserves supernova time dilation and spectral integrity.

12. Cosmic Background Radiation

E ∝ exp(-∫ (S/Smax) dx)

Preserves blackbody spectrum via frequency-independent scaling.

13. Principle of Physical Admissibility

Following Stephen Hawking: infinities indicate breakdown of theory, not reality.

Domain	Standard Artifact	FRCFD Interpretation
Gravity	Singularities	Saturation (Smax)
Quantum	Infinite branching	Coherence limits
Thermodynamics	Entropy → time	Time = update-rate

14. Dimensional Efficiency (Unification)

FRCFD replaces extra dimensions (e.g., String Theory) with nonlinear feedback and finite capacity. All forces emerge as substrate response modes.

15. Observational Tests (Falsification Suite)

• Universal Constant: Smax(EHT) must equal Smax(pulsars).
• Frame-Dragging Limit: vmax = c · f(Smax).
• Precision: Deviation from γ=1 in solar system must be small but non-zero.

17. Core Insight

FRCFD replaces: Geometry → Response, Curvature → Impedance, and Singularity → Saturation. Spacetime becomes an emergent description of a finite-capacity system.

18. Conclusion

Finite-Response Coupled Field Dynamics provides a minimal, testable alternative to geometric gravity. Its strength lies in its rigidity: a small set of parameters yields precise, cross-domain predictions. The theory stands or falls on observation.

— Derek Flegg