Rate of c – Quantum Relativity

The Mathematics of the E_∞ Gravity Mechanism

Abstract

In the earlier paper Speed of c, we examined how variations in permittivity $\varepsilon_0$ alter the value of the speed of light while preserving the phase relationship through constant $\mu_0 \varepsilon_0$ . This companion paper develops the notion of a rate of change in c, derived by considering pointwise deviations in $varepsilon_0$ . The approach is analogous to finite element methods (FEA), in which local perturbations in material properties yield incremental changes in field solutions.

Fundamental Relationship

We recall the basic definition:

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

where:

c is the speed of light in a medium.
$\mu_0$ is the magnetic permeability of free space
$\varepsilon_0$ is the electric permittivity of free space

Differentiating c

Let us consider variations in $\varepsilon_0$ . Differentiating with respect to $\varepsilon_0$ :

$\frac{dc}{d\varepsilon_0} = -\frac{1}{2} \cdot \frac{1}{\sqrt{\mu_0}} \cdot \varepsilon_0^{-\tfrac{3}{2}}$

Or, equivalently:

$\frac{dc}{d\varepsilon_0} = -\frac{c}{2\varepsilon_0}$

This expression defines the rate of change of the speed of light with respect to permittivity.

Point Deviation (FEA-style approach)

If $\varepsilon_0$ changes locally by $\Delta \varepsilon_0$ , then the incremental change in c is:

$\Delta c \approx \frac{dc}{d\varepsilon_0} \cdot \Delta \varepsilon_0$

Expanding:

$\Delta c \approx -\frac{c}{2\varepsilon_0} \cdot \Delta \varepsilon_0$

Thus, even small deviations in permittivity propagate into proportional changes in light speed.

Relative Rate Formulation

It is often more useful to express this as a fractional rate:

$\frac{\Delta c}{c} \approx -\frac{1}{2} \cdot \frac{\Delta \varepsilon_0}{\varepsilon_0}$

This compactly shows that a 1% increase in $\varepsilon_0$ produces approximately a 0.5% decrease in cc.

Discussion

This rate formulation highlights two useful perspectives:

$\frac{\Delta c}{c} \approx -\frac{1}{2} \cdot \frac{\Delta \varepsilon_0}{\varepsilon_0}$

This compactly shows that a 1% increase in $\varepsilon_0$ produces approximately a 0.5% decrease in c.

Linearity in small deviations — the response of c to changes in $\varepsilon_0$ is directly proportional in the small-signal regime.

Analogy to material dispersion — just as optical media slow light according to local ε\varepsilonε, the “vacuum” may itself act as a dispersive medium when its permittivity fluctuates along a path.

This offers a natural mathematical framework for viewing redshift as a cumulative rate-of-change effect along cosmic distances, independent of expansion assumptions.

Mass as Stationary Energy

Energy and mass are two forms of the same underlying entity. Einstein’s famous relation:

$E = mc^2$

defines the rest energy of a particle with mass m. This energy exists even when the particle is not moving — that is, when it is not propagating as a wave.

For a moving particle, the total energy includes kinetic contributions:

$E^2 = (pc)^2 + (mc^2)^2$

where p is momentum.

If $m \neq 0$ , the particle always possesses rest energy $c^2$ .
If m = 0, (a photon), all energy is carried by motion as a wave: $E = pc$ .

Thus, one can interpret mass as energy “stored” locally, not propagating as a wave, while wave energy represents energy in motion. This perspective aligns naturally with the E_∞ framework: stationary energy is a reservoir, and propagating wave energy is a phase-structured release from that reservoir.

In short:

$\text{Mass} \;\; \sim \;\; \text{localized energy, stationary} \quad\quad \text{Wave} \;\; \sim \;\; \text{propagating energy, moving at } c$

This distinction allows the Rate of c framework to relate local field variations $\varepsilon_0$ to both stationary and propagating forms of energy, showing how even “rest energy” can influence observed propagation rates along a path.

Gravity as Rate Modulation of Energy

The mechanism of gravity can be interpreted as the change in the rate of energy propagation caused by local reactance variations in the medium:

$\text{Gravity} \;\; \sim \;\; \frac{d c(x)}{dx} \;\;\; \text{where} \;\; c(x) = \frac{1}{\sqrt{\mu_0 \varepsilon_0(x)}}$

Here:

c(x) is the local speed of energy propagation along coordinate xxx
$\varepsilon_0(x)$ and $\mu_0$ define the local medium reactance
Variations in c(x) produce the observed deflection of energy and mass, manifesting as gravitational acceleration

This reframes gravity as emergent from energy’s interaction with the reactive properties of space, rather than requiring a geometrical curvature of spacetime.

Summary

The E_∞ framework naturally links energy propagation, local field properties, and observation effects:

Redshift becomes a measurement- and path-dependent effect rather than purely cosmological expansion.
Variations in $\varepsilon_0ε$ (and thus c) provide a mechanism for gravity as a local modulation of energy rate.
The same underlying principles connect phase coherence, aperture effects, and observed cosmic signals, unifying seemingly separate phenomena.

The speed of c, rate of c, aperture filtering, phase coherence, and now gravity — forms a coherent, self-consistent picture. E_∞ isn’t just a model; it becomes the thread tying redshift, energy propagation, and gravitational effects into one framework.

E_∞ shows that many “cosmological mysteries” arise from how energy interacts with the structured medium, not from separate postulated forces or expansion terms.

Next Steps

Future work will extend this analysis by integrating pathwise variations:

$\Delta c_{\text{path}} = \int \left( -\frac{c}{2\varepsilon_0(x)} \cdot \frac{d\varepsilon_0(x)}{dx} \right) dx$

which can be compared against observed cosmological redshift data to test if dispersion-driven light speed variation provides a viable alternative or complement to standard models.