Reciprocity Theorem from Null Geodesics in FRCFD

Einstein showed that light stretches when space expands. But what if light stretches even when space doesn’t?

The Null-Geodesic Derivation of the Reciprocity Theorem in FRCFD

Derek Flegg

Southern Ontario, March 21, 2026

1. Introduction: From Analogy to Derivation

In standard FLRW cosmology, the relationship between luminosity distance (d_L) and angular diameter distance (d_A) — the Etherington Reciprocity Theorem — is a direct consequence of metric expansion. Any static-background alternative must reproduce the same scaling: 

d_L = d_A * (1 + z)^2

In Finite-Response Coupled Field Dynamics (FRCFD), this relation is not assumed. It emerges naturally from the substrate response function f(S) and the structure of the effective propagation metric.

2. The Cosmological Substrate Metric

We model the cosmological background as a cumulative substrate stress field: 

S(chi) = gamma * chi

The effective FRCFD metric is: 

ds^2 = f(S)^2 dt^2 - f(S)^(-2) dchi^2 - Sigma(chi)^2 dOmega^2

with the response function: 

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

For null geodesics (ds^2 = 0), the radial propagation satisfies: 

dchi / dt = f(S)^2

3. Step-by-Step Derivation

3.1 Redshift (1 + z)

The observed frequency is determined by the time-component of the metric: 

1 + z = f(S_emit)^(-1) = exp( gamma * chi / S_max )

3.2 Angular Diameter Distance (d_A)

A physical transverse size Y at emission corresponds to an observed angle theta. The transverse geometry is scaled by f(S) at the emission point: 

d_A = Sigma(chi) * f(S)
d_A = Sigma(chi) / (1 + z)

3.3 Luminosity Distance (d_L)

Flux is reduced by:

  • Energy redshift: (1 + z)^(-1)
  • Arrival-rate stretching: (1 + z)^(-1)

Combined with geometric dilution over the wavefront area: 

d_L = Sigma(chi) * (1 + z)

4. Comparison of Observational Scalings

Parameter Standard FLRW FRCFD (Emergent)
Redshift Mechanism Metric Expansion Substrate Impedance
d_L / d_A Ratio (1 + z)^2 (1 + z)^2
Surface Brightness (1 + z)^(-4) (1 + z)^(-4)

5. Conclusion: A Unified Cosmology

By solving the null geodesics of the f(S)-weighted metric, the Etherington Reciprocity Theorem emerges naturally within FRCFD. The geometric factor (1 + z)^2 is not imposed — it follows directly from the symmetry between the temporal f(S)^2 and spatial f(S)^(-2) components of the substrate.

FRCFD therefore provides a self-consistent, non-expanding cosmology that preserves all major observational scalings while replacing geometric expansion with finite-response substrate dynamics.

PPN Structure and Substrate Back‑Reaction in FRCFD

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