Gradients of c – Charge Admittance

Model the Gradient of c(r)

Abstract

This paper presents a theoretical framework for modeling the spatial variation of the speed of light, c(r), as a function of energy density gradients in the vacuum. Building upon the Charge Admittance (CA) paradigm, we propose that variations in local electromagnetic field properties—permittivity ε(r) and permeability μ(r)—result in a spatially variable light speed. This gradient, dc/dr, is directly linked to gravitational acceleration via the relation g(r)=−1/c(r) dc(r)/dr. This model reframes gravity not as curvature or force, but as an emergent consequence of anisotropic energy propagation within a dynamic ε₀μ₀ lattice. Implications include gravity as an energy-density-driven flow, bypassing the need for mass-based spacetime deformation and opening paths to unify electromagnetic and gravitational phenomena.

Introduction

The speed of light, c, has long been regarded as a fundamental constant. However, in the Charge Admittance model, c is treated as a local property of space, determined by the dynamic structure of the vacuum via its electromagnetic characteristics:

$c(r) = \frac{1}{\sqrt{\varepsilon(r)\mu(r)}}$

In regions of varying energy density, particularly near massive or charged objects, these field properties may shift subtly, resulting in a non-uniform propagation speed of light. Such variation introduces a natural gradient:

$\frac{dc(r)}{dr}$

This gradient is proposed to cause the observable effects attributed to gravity, such as redshift, lensing, and orbital precession. By modeling c(r) and its gradient, we can derive a physically meaningful expression for gravitational acceleration that is consistent with empirical observations while eliminating the need for spacetime curvature or hypothetical dark entities.

Foundational Relation Between Light Speed and Gravity

Under CA, gravity is reinterpreted as a directional energy flow driven by variations in the local speed of energy propagation. The foundational equation linking gravitational acceleration to the gradient of c is:

$g(r) = -\frac{1}{c(r)} \frac{dc(r)}{dr}$

This expression replaces traditional formulations of gravity by grounding the effect in field-based propagation dynamics.

Modeling Light Speed as a Function of Radius

Assume a first-order linear perturbation model for c(r):

$c(r) = c_0 \left(1 - \alpha \frac{R_s}{r} \right)$

Where:

c0: speed of light in vacuum at infinity,
α: dimensionless scaling constant,
Rs: characteristic scale (e.g., Schwarzschild radius).

Derivative of c(r)

Differentiating:

$\frac{dc(r)}{dr} = c_0 \cdot \alpha \cdot \frac{R_s}{r^2}$

Substituting into the gravitational equation:

$g(r) = -\frac{c_0 \alpha R_s}{r^2 \left(1 - \alpha \frac{R_s}{r} \right)}$

Recovery of Newtonian Gravity in the Far-Field Limit

As r≫Rs, the denominator approaches unity:

$g(r) \approx \frac{GM}{r^2}$

$\quad \text{if} \quad \alpha R_s = \frac{GM}{c_0^2}$

This recovers classical Newtonian gravity and links gravitational mass M to a shift in the local light-speed gradient.

Generalized Gravity Density Expression

In practical systems where altitude h above a body is used:

$g(h) = g_0 \left( \frac{R}{R + h} \right)^2$

$\quad \Rightarrow \quad \text{From } c(r) \Rightarrow g(h) = -\frac{1}{c(h)} \frac{dc(h)}{dh}$

This result indicates that both gravity and the speed of energy are manifestations of a single underlying field structure: the spatial variation in electromagnetic energy density.

Implications for Field Theory

Using:

$c(r) = \frac{1}{\sqrt{\varepsilon(r)\mu(r)}}$

$\quad \Rightarrow \quad \nabla c(r) \propto -g(r)$

We imply that gravitational acceleration can be derived from electromagnetic field admittance structure, replacing the curvature of spacetime with the curvature of impedance.

Conclusion

This paper develops a working model for interpreting gravity as an outcome of a spatial gradient in the speed of light, which itself emerges from dynamic electromagnetic field properties. The Charge Admittance model provides a coherent, unified explanation of gravitational effects without resorting to curvature or hypothetical constructs. This approach offers a testable, scalable alternative foundation for gravity, potentially uniting quantum, relativistic, and classical domains under a single energy-field framework.