Introduction

This document presents the full mathematical derivation of the Ives–Stilwell result from first principles of the Lilborn Framework. Our objective is to show that the observed spectral symmetry and second-order Doppler shift can be explained entirely through spatial coherence geometry, without invoking time dilation.

Definitions and Framework Setup

• Let v be the velocity of the moving emitter relative to the observer.
  
• Let ℓ be the structural coherence constant (analogous in value to c but not a speed).
  
• Let k₀ be the intrinsic spatial frequency (coherent cycles per unit spatial length).
  
• Let θ be the angle between the direction of motion and the observer’s line of sight.
  
• Let k be the spatial frequency registered by the observer.

First-Order Shift
Longitudinal Doppler

When the emitter moves directly toward or away from the observer (θ = 0 or π), the coherence cycles compress or expand geometrically.

The projected spatial frequency is given by:

k = k₀ * (1 ± v / ℓ)

This captures the classical Doppler effect as a spatial compression (blue shift) or expansion (red shift).

Second-Order Shift
Transverse Projection

When the emitter moves transverse (θ = π/2), the first-order effect vanishes.

However, the coherence projection still yields a change due to angular misalignment:

k = k₀ * sqrt(1 – (v² / ℓ²))

Expanding this using a binomial approximation:

k ≈ k₀ * (1 – ½ * v² / ℓ²)

This predicts a shift proportional to v² / ℓ², the exact term attributed to time dilation in Special Relativity, here derived purely from geometric misalignment.

Symmetric Bi-directional Doppler Shift

Ives–Stilwell observed symmetric frequency shifts both forward and backward, explained in this framework as follows:

k_forward  = k₀ * (1 + v/ℓ) * sqrt(1 – v²/ℓ²)

k_backward = k₀ * (1 – v/ℓ) * sqrt(1 – v²/ℓ²)

These results reproduce the observed spectral symmetry and shift, without invoking a change in time rate, but rather by spatial projection inefficiency.

Conclusion

The Lilborn Framework successfully re-derives the Ives–Stilwell experimental outcome using a geometric model of coherence. The second-order Doppler effect, long interpreted as time dilation, is shown here to be a structural consequence of coherent projection geometry.

This concludes the full reconstruction. Let this stand as the replacement of relativistic time dilation with the observable geometry of spatial coherence interaction.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams