The postulates of Special Relativity (SR)—invariant light speed and frame symmetry—are empirically robust, yet their geometric interpretation has led physics toward abstraction: spacetime curvature, singularities, and unobservable fields. However, as demonstrated in the C.O.R.E. framework, a classical alternative exists: one where light is a continuous wave, gravity emerges from vacuum property variations, and cosmology is static.
In this paper, we show that Lorentz symmetry itself is not fundamental, but emergent from wave propagation in a responsive vacuum medium. Specifically, we derive a Lorentz-like transformation by integrating the eikonal equation along paths with relative motion—demonstrating that relativistic effects arise as the sum of:
This refractive foundation explains why SR works in flat space, why GR doubles its predictions in gravity, and why acceleration fails to mimic gravity—resolving the "half-effect" observed in laboratory experiments [Space Curvature on the Labdesk - Wolfgang Sturm].
In the CUGE framework, mass induces symmetric variations in vacuum permittivity and permeability:
This yields an effective refractive index:
and a coordinate speed of light:
The propagation of a continuous electromagnetic wave (per ASH) is governed by the eikonal equation:
where \( S(r) \) is the phase. The ray equation (from Fermat's principle) is:
where \( \hat{t} \) is the unit tangent to the ray path.
This is the classical analog of geodesic motion—refraction replaces curvature.
The Lorentz factor,
has been historically interpreted as a geometric scaling arising from postulates of special relativity. We show this is incorrect.
Instead, \(\gamma\) emerges as a measure of accumulated phase delay across the two-dimensional transverse plane of a wavefront propagating through a physical vacuum whose electromagnetic properties respond to sources governed by the inverse-square law.
Consider a continuous electromagnetic wave with phase:
Physical reality demands phase continuity: no spontaneous jumps in \(\phi\). When two observers are in relative motion or separated by a gravitational potential, their measurements must preserve the coherence of this phase across space and time.
Now suppose the source of influence (mass or charge) generates a field that spreads uniformly over spherical surfaces. By Gauss’s law, such fields obey an inverse-square dependence:
This field modifies the vacuum permittivity and permeability symmetrically [CUGE, Barbeau 2025]:
Hence, the effective refractive index becomes:
As the wave propagates radially outward, its energy spreads over a surface area \(A = 4\pi r^2\). Any gradient in \(n(r)\) affects the entire 2D cross-section of the wavefront perpendicular to propagation.
When integrating phase shifts — whether due to motion (oblique incidence) or medium strain (refraction) — the result depends on how the field couples to this extended surface.
Thus, the total redshift or time dilation is not a 1D kinematic effect, but a 2D field-integrated delay, given by:
For weak fields and small velocities, this yields:
or under transverse motion:
Crucially, both effects scale quadratically with speed or potential because they represent fractional losses in phase progression across a 2D plane. The term \(v^2/c^2\) is therefore not merely a dimensionless ratio — it quantifies:
✅ The normalized phase delay integrated over the projected 2D area of the inverse-square field.
This explains why gravitational effects are twice their Newtonian predictions: Newton assumed point-like interactions along radial lines. But waves interact across full spherical shells. The factor of 2 is not magic — it is geometry:
👉 Energy spreads in 2D → response integrates over 2D → delay doubles.
Similarly, the expression:
can now be reinterpreted:
Hence:
\(\gamma\) is not a property of spacetime. It is the reciprocal of remaining phase efficiency in a finite-speed, 2D-spreading field medium.
This refractive origin of \(\gamma\) eliminates the need for relativistic mass increase, time dilation, or curved spacetime. What has been mistaken for fundamental symmetry is simply Snell’s Law applied to a universe where all long-range forces spread over \(4\pi r^2\).
Moreover, it predicts a critical distinction:
This split falsifies the strict Equivalence Principle and aligns with experimental anomalies in electron beam focusing (Sturm) and calorimetry (Bertozzi).
In summary:
The Lorentz factor does not originate in abstract postulates about light speed.
It originates in the physical reality of wavefronts delayed across a 2D surface shaped by the oldest law in physics: force diminishes as \(1/r^2\).
This is not reinterpretation.
It is restoration.
In contrast to Einstein’s postulate-driven approach, we begin with a physical requirement: the phase of a continuous wave must be continuous across reference frames.
In the C.O.R.E. framework, light is not a stream of photons, but a continuous electromagnetic wave (per the Atomic Statistical Hypothesis, ASH). Its most fundamental physical property is its phase:
This phase determines:
At the moment of detection, the phase must be consistent. Otherwise, the wave would "break" or fail to transfer energy coherently.
Therefore, phase is a physical invariant: all observers must agree on the phase at a given point in spacetime.
Consider a plane wave in frame \( S \):
In a moving frame \( S' \), the phase must be the same:
Using the dispersion relation in vacuum, \( \omega = c k \), we substitute:
This suggests a transformation that preserves \( x - c t \) and \( x + c t \). Assuming linearity:
Solving with boundary conditions (e.g., \( x' = 0 \implies x = v t \)) and matching Doppler shifts leads to:
The Lorentz transformation is not assumed—it emerges from the continuity of a wave.
Now allow \( n(r) > 1 \). The dispersion relation becomes:
Phase invariance now implies:
leading to a generalized transformation:
When \( n = 1 \), this reduces to SR. When \( n = 1 + \frac{GM}{2 c^2 r} \), the \( n^2 \) term introduces the factor observed in GR.
Thus, the "doubling" in gravitational effects relative to pure kinematics is split: half from kinematics (SR), half from refractive delay (CUGE).
In real gravity:
Total redshift:
matching GR.
Under acceleration, only the kinematic half exists—confirming that acceleration ≠ gravity.
We have shown that the Lorentz transformation is not a fundamental symmetry of spacetime, but an emergent consequence of wave propagation in a medium with variable \( \varepsilon(r) \) and \( \mu(r) \). The full relativistic effect is the sum:
This refractive foundation:
The era of abstract spacetime is ending. The era of classical emergence has begun.