E = mc² Revisited: Vacuum Shielding Stress and the Refractive Foundation of Relativistic Energy

Abstract: We demonstrate that the standard relativistic energy formula $E = \gamma m c^2$ is an incomplete effective description that conflates mechanical and non-mechanical energy storage. Within the C.O.R.E. framework—comprising CUGE (Classical Unification of Gravity and Electromagnetism), REFORM (REfractive Foundation of Relativity and Mechanics), and ASH (Atomic Statistical Hypothesis)—we derive a revised energy partition that explicitly accounts for vacuum permittivity $\varepsilon(r)$ and permeability $\mu(r)$. We show that: 1. The Lorentz factor $\gamma$ emerges from the sum of kinematic (transverse Doppler) and refractive (medium-induced) effects; 2. Total energy input partitions into mechanical kinetic energy and Vacuum Shielding Stress (VSS) energy; 3. In high-energy electron acceleration, >94% of input energy is stored as VSS, not kinetic motion; 4. The local invariance of $c$ arises from atomic scaling with $\varepsilon(r), \mu(r)$, not spacetime geometry; 5. The "half-effect" in acceleration experiments falsifies the strict Equivalence Principle and confirms the physical reality of $\varepsilon/\mu$ variations. This refractive foundation resolves long-standing anomalies in electron calorimetry, reinterprets inertia, and eliminates the need for relativistic mass. Energy is conserved—but not as $\gamma m c^2$.

1. Introduction: The Illusion of Relativistic Mass

The formula $E = \gamma m c^2$, central to Special and General Relativity, treats energy as a geometric property of spacetime. Yet experiments such as Bertozzi's electron acceleration (1964) reveal a discrepancy: calorimetric heat on impact matches total input energy $eU$, but measured velocity implies kinetic energy far below $(\gamma - 1) m c^2$. This suggests that most energy is not kinetic—contradicting the standard interpretation.

In the C.O.R.E. framework, this anomaly is resolved by recognizing that the vacuum is a responsive electromagnetic medium. Mass induces symmetric variations in $\varepsilon(r)$ and $\mu(r)$, altering the propagation of light and the behavior of matter. Crucially, acceleration does not produce these variations—only real gravity does. This distinction underpins a complete revision of relativistic energy.

2. The Refractive Vacuum: $\varepsilon(r)$, $\mu(r)$, and Local $c$ Invariance

From CUGE, a mass $M$ induces:

\varepsilon(r) = \varepsilon_0 \left(1 + \frac{GM}{2c^2 r}\right), \quad \mu(r) = \mu_0 \left(1 + \frac{GM}{2c^2 r}\right)

The refractive index is:

n(r) = \sqrt{\varepsilon(r)\mu(r)} \approx 1 + \frac{GM}{2c^2 r}

Coordinate light speed varies: $c_{\text{coord}} = c / n(r)$.

However, local measurements always yield $c$ because atomic standards scale with $\varepsilon(r)$:

Atomic transition frequency: $ u \propto 1/\varepsilon(r)$
Bohr radius: $a_0 \propto \varepsilon(r)$
Measured wavelength: $\lambda \propto a_0 \propto \varepsilon(r)$
Thus: $c_{\text{local}} = \lambda u \propto \varepsilon(r) \cdot 1/\varepsilon(r) = \text{const}$

This explains gravitational time dilation without spacetime curvature: clocks slow because atomic processes slow in a polarizable vacuum.

3. Deriving $\gamma$ from Phase Continuity (REFORM)

In REFORM, Lorentz symmetry is not postulated but derived from phase continuity of a continuous EM wave (per ASH).

For a plane wave, phase $\phi = \mathbf{k} \cdot \mathbf{r} - \omega t$ must be invariant across frames. In a medium with $n(r)$, the dispersion relation is $\omega = (c/n) k$.

For uniform motion in flat space ($n=1$), phase invariance yields:

\[ k'(x' - ct') = k(x - ct) \]

Assuming linearity and solving with boundary conditions ($x' = 0 \Rightarrow x = vt$) gives:

x' = \gamma (x - vt), \quad t' = \gamma \left(t - \frac{vx}{c^2}\right), \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

But in a refractive medium ($n > 1$), the generalized transformation is:

x' = \gamma_n (x - vt), \quad t' = \gamma_n \left(t - \frac{v x}{n^2 c^2} \right), \quad \gamma_n = \frac{1}{\sqrt{1 - v^2/(n^2 c^2)}}

Thus, $\gamma$ is not fundamental—it is the composite effect of:

Kinematic time dilation (transverse Doppler): $\Delta f/f = -\frac{1}{2} v^2/c^2$
Refractive delay: $\Delta t = \int (n-1)/c \, dl$

In gravity, both contribute equally—hence the "doubling" of GR effects.

4. Energy Partition: Mechanical vs. Vacuum Shielding Stress (VSS)

When an electron is accelerated by voltage $U$, work done is $W = eU$. In CUGE, this energy partitions as:

eU = E_{\text{mech}} + E_{\text{VSS}} = \frac{1}{2} m v^2 + \left(eU - \frac{1}{2} m v^2\right)

This follows from reinterpreting Coulomb repulsion as Vacuum Shielding Stress (VSS)—a second-order effect where environmental attraction partially shields like charges (Schaub & Sturm, 2024).

Bertozzi Reanalysis (Jormakka, 2025):

$E_{\text{meas}}/m c^2$	$v^2/c^2$	$E_{\text{mech}}/m c^2$	$E_{\text{VSS}}/m c^2$
9.0	0.974	0.487	8.513

> 94% of energy is stored in the vacuum, not as motion. Upon impact, all $eU$ is released as heat—matching calorimetry.

Thus, $E = \gamma m c^2$ overestimates mechanical energy. The correct expression is:

eU = \underbrace{\frac{1}{2} m v^2}_{E_\text{mech}} + \underbrace{E_\text{VSS}}_{}

where $eU = (\gamma - 1)mc^2$ is the total input energy. This reveals that the Lorentz factor $\gamma$ quantifies the **total system energy cost**, not just kinetic energy. The fraction of this energy that manifests as mechanical motion is given by the efficiency ratio:

\eta(v) = \frac{E_\text{mech}}{(\gamma - 1)mc^2} = \frac{\frac{1}{2} v^2/c^2}{\gamma - 1}

As $v \to c$, $\eta(v) \to 0$, demonstrating that $\gamma$-scaling is dominated by non-mechanical vacuum stress.

5. The Half-Effect: Falsifying the Equivalence Principle

Sturm's 2022 experiment shows:

Vertical acceleration: $\Delta E/E = a h / c^2$
Horizontal acceleration: $\Delta E/E = \frac{1}{2} a h / c^2$

CUGE Explanation:

Scenario	Kinematic Shift	Vacuum Shift ($\varepsilon,\mu$)	Total
Real gravity	$-\frac{1}{2} gh/c^2$	$-\frac{1}{2} gh/c^2$	$-gh/c^2$
Horizontal accel.	$-\frac{1}{2} ah/c^2$	0	$-\frac{1}{2} ah/c^2$

Acceleration cannot alter $\varepsilon(r), \mu(r)$—only mass can. Thus, only half the gravitational effect appears. This falsifies the strict Equivalence Principle and confirms the physical reality of vacuum property variations.

6. Implications

6.1. Inertia as Electromagnetic Resistance

From CUGE Section 8, the resistive force on an accelerating charge is:

F_{\text{resistive}} = -\frac{q^2}{6\pi \varepsilon_0 c^2} \left( \frac{\nabla \varepsilon}{\varepsilon_0} \right) a

This suggests inertial mass emerges from interaction with $\nabla \varepsilon$—a Machian, electromagnetic origin of inertia.

6.2. No Relativistic Mass

The concept of velocity-dependent mass is obsolete. Mass $m$ is constant; apparent "mass increase" is energy stored in vacuum stress.

6.3. Quantum Thresholds Depend on $\varepsilon(r)$

Work function $\phi \propto 1/\varepsilon(r)^2$. Near mass, $\varepsilon(r)$ increases → $\phi$ decreases → easier photoelectric emission. This predicts:

Atomic clocks slow in gravity,
Photoelectric cells become more sensitive—opposite trends testable in space.

6.4. Unified Origin of Relativistic Effects

All "relativistic" phenomena reduce to:

\text{SR} = \text{Transverse Doppler} + \text{Snell's Law in } n=1

\text{GR} = \text{Transverse Doppler} + \text{Snell's Law in } n(r) > 1

7. Conclusion

The formula $E = \gamma m c^2$ is a useful approximation but physically incomplete. In the C.O.R.E. framework, energy is partitioned between mechanical motion and vacuum stress, with $\gamma$ emerging from wave propagation in a medium whose properties $ (\varepsilon, \mu) $ are altered by mass—but not by acceleration.

This refractive foundation:

Resolves Bertozzi's calorimetry anomaly,
Explains the half-effect and falsifies the Equivalence Principle,
Derives Lorentz symmetry from phase continuity,
Eliminates relativistic mass and singularities,
Unifies gravity, inertia, and electromagnetism.

Energy is conserved—but not as spacetime geometry. It is stored in the electromagnetic vacuum, and that changes everything.

References

David Barbeau. CUGE: Classical Unification of Gravity and Electromagnetism, 2025. 🔗 https://ai.vixra.org/abs/2507.0112
Alfred Schaub and Wolfgang Sturm. The Antigravity on the Lab Bench. viXra:2410.0176, 2024. 🔗 https://vixra.org/abs/2410.0176
David Barbeau. C.O.R.E.: Classical Origin of Reality and Emergence, 2025. 🔗 https://rxiverse.org/abs/2508.0003
David Barbeau. REFORM: REfractive Foundation of Relativity and Mechanics, 2025. 🔗 https://rxiverse.org/abs/2508.0021
Wolfgang Sturm. Space Curvature on the Labdesk. viXra:2207.0014, 2022. 🔗 https://vixra.org/abs/2207.0014
A laboratory experiment demonstrating directional dependence in relativistic redshift under acceleration, revealing the “half-effect” and providing empirical evidence that acceleration does not fully mimic gravitational effects.
Jorma Jormakka, Aalto University. Calculation of the longitudinal mass from Bertozzi's experiment, 2025 🔗 https://www.researchgate.net..
Jorma Jormakka, Wolfgang Sturm. Can Relativistic Mass or Weakening of Force be Measured with a Vacuum Tube?, 2025. 🔗 https://vixra.org/abs/2509.0022

Scenario	Kinematic Shift	Vacuum Shift (\(\varepsilon,\mu\))	Total
Real gravity	\(-\frac{1}{2} gh/c^2\)	\(-\frac{1}{2} gh/c^2\)	\(-gh/c^2\)
Horizontal accel.	\(-\frac{1}{2} ah/c^2\)	0	\(-\frac{1}{2} ah/c^2\)