Mathematical Derivations in Electromagnetic Permittivity Variation and Orbital Dynamics (EPVOD)
Abstract:
This paper formalizes the mathematical structure underlying the EPVOD framework. It establishes how variations in vacuum permittivity and permeability under high-energy-density conditions alter orbital mechanics, electromagnetic field propagation, and gravitational coupling. We present the derivation of field equations, resonance conditions, and effective metrics emerging from energy-density-induced spacetime deformation.
1. Introduction
We assume that spacetime behaves as a deformable medium, where local electromagnetic properties (specifically vacuum permittivity (\varepsilon_0) and permeability (\mu_0)) vary in response to energy density gradients at subatomic scales. This introduces a harmonic spatial structure superimposed on standard quantum and relativistic field frameworks.
2. Modified Maxwell's Equations in Variable ( \varepsilon, \mu ) Background
Starting from classical form:
[ \nabla \cdot (\varepsilon(r) \mathbf{E}) = \rho ]
[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} ]
[ \nabla \cdot \left( \frac{\mathbf{B}}{\mu(r)} \right) = 0 ]
[ \nabla \times \left( \frac{\mathbf{B}}{\mu(r)} \right) = \varepsilon(r) \frac{\partial \mathbf{E}}{\partial t} + \mathbf{J} ]
We define permittivity/permeability as harmonic scalar fields:
[ \varepsilon(r) = \varepsilon_0 \left[1 + \delta \cdot \cos(\omega r + \phi)\right] ]
[ \mu(r) = \mu_0 \left[1 + \delta' \cdot \cos(\omega r + \phi')\right] ]
These modulate wave propagation and introduce frequency-dependent refraction and phase shift effects.
3. Orbital Stability in Harmonic Spacetime Potential
Model orbital shells as stable nodes in a radial harmonic potential:
[ V(r) = -k \cdot \frac{1}{r} + \alpha \cdot \cos^2(\omega r) ]
Solve Schrödinger-like radial equation:
[ \left[-\frac{\hbar^2}{2m} \nabla^2 + V(r) \right] \psi(r) = E \psi(r) ]
Yields quantized orbits defined not only by principal quantum number (n) but by harmonic resonance index (m), which correlates with energy-density-driven permittivity wavefronts.
4. Coupling to General Relativity: Metric Perturbation
In weak-field limit, we apply metric perturbation:
[ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}(r) ]
With stress-energy tensor coupling:
[ T_{\mu\nu}^{(EM)} = \varepsilon(r) (F_{\mu}^{\ \alpha} F_{\nu\alpha} - \frac{1}{4} g_{\mu\nu} F^{\alpha\beta} F_{\alpha\beta}) ]
Feed into Einstein's equation:
[ G_{\mu\nu} = 8\pi G T_{\mu\nu}^{(total)} ]
Yields curvature sourced by energy-dense harmonic electromagnetic fields.
5. Implications for Photon Propagation
Modified wave equation in permittivity gradient:
[ \nabla^2 \mathbf{E} - \mu(r) \varepsilon(r) \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 ]
Predicts refraction, dispersion, and localized lensing near nuclei.
6. Quantized Conditions and Derived Constants
From resonance conditions:
[ \omega_n r_n = n\pi ]
Link to Bohr radius:
[ r_n = \frac{n^2 \hbar^2}{m e^2} \rightarrow \varepsilon(r_n) \text{ modifies } e^2 \text{ term dynamically} ]
Yields corrections to fine structure constant:
[ \alpha_{eff}(r) = \frac{e^2}{4\pi \varepsilon(r) \hbar c} ]
7. Conclusion
The EPVOD framework supports testable predictions based on harmonic structure and variation in vacuum properties. These affect orbital dynamics, field propagation, and coupling to gravitational effects. Further computational modeling can refine these predictions against empirical quantum electrodynamic and gravitational datasets.
Appendix A: Relational Constancy of (\varepsilon), (\mu), and the Speed of Light
In this framework, vacuum permittivity (\varepsilon) and permeability (\mu) are not independent; their product must maintain consistency with the speed of light (c):
[ c = \frac{1}{\sqrt{\mu \varepsilon}} ]
Thus, while both (\varepsilon) and (\mu) may vary locally in response to high energy density (e.g., near the nucleus), they must do so in a mutually compensatory way to preserve causality. The ratio (\mu / \varepsilon) defines the impedance of space and constrains allowable electromagnetic wave propagation.
We posit that the radial variation of (\varepsilon(r)) and (\mu(r)) maintains:
[ \mu(r) \varepsilon(r) = \text{const.} = \frac{1}{c^2} ]
Consequently, fluctuations in (\varepsilon) around condensed matter dictate proportional adjustments in (\mu), preserving the wave equation's causal structure. These harmonic variations correspond to electron orbital shells, implying that the discrete orbital structure of matter arises from standing wave patterns in spacetime geometry, modulated by electromagnetic properties.