Mathematical Formulation of Nucleus-Induced Spacetime Deformation
Abstract
This paper mathematically develops the harmonic gradient hypothesis introduced in the first paper, describing spacetime deformation induced by high-energy density near atomic nuclei. Utilizing differential geometry and harmonic wave theory, we present field equations that describe spacetime's local permittivity variations and resultant orbital resonances. Boundary conditions defining stable electron orbital modes as spatial resonance patterns around nuclei are proposed, establishing a mathematical foundation to unify quantum states with gravitational-like effects at subatomic scales.
1. Introduction
The harmonic gradient hypothesis asserts spacetime deforms harmonically around dense energy nuclei. This paper develops this assertion mathematically, providing equations governing localized harmonic spatial deformation.
2. Mathematical Preliminaries
We begin by defining essential concepts:
- Spacetime manifold: Consider spacetime as a smooth differentiable manifold ( \mathcal{M} ).
- Energy density scalar field ( \rho(x) ): Represents local energy density at a position ( x ) around atomic nuclei.
- Permittivity scalar field ( \varepsilon(x) ): Varies spatially, dependent on ( \rho(x) ).
3. Governing Field Equations
We propose field equations linking energy density to permittivity:
[
\nabla^2 \varepsilon(x) - \frac{1}{c^2} \frac{\partial^2 \varepsilon(x)}{\partial t^2} = \alpha \rho(x)
]
where ( \alpha ) is a proportionality constant related to Planck-scale interactions.
4. Harmonic Solutions and Resonance Conditions
Harmonic solutions of the above wave equation can be found by separation of variables, yielding radial harmonic modes:
[
\varepsilon(r, t) = \varepsilon_0 + \sum_{n=1}^{\infty} A_n \frac{\sin(k_n r)}{r} e^{-i\omega_n t}
]
Boundary conditions for stable resonance modes around nuclei require quantization conditions, ensuring standing waves:
[
k_n = \frac{n\pi}{R_n}, \quad n \in \mathbb{N}
]
where ( R_n ) denotes radii of orbital shells.
5. Electron Orbital Quantization
Stable electron orbitals emerge as harmonic solutions:
[
\frac{d}{dr}\left[r^2 \frac{d\varepsilon}{dr}\right] + \frac{\omega^2 r^2}{c^2}\varepsilon = 0
]
The quantized radii ( R_n ) satisfy boundary conditions imposed by harmonic resonance, aligning closely with observed electron shell structures.
6. Discussion
This mathematical formulation aligns known quantum mechanical structures with harmonic spatial deformation, suggesting gravity-like effects at quantum scales originate from spatial tension resonances.
Next Paper Preview:
The subsequent paper, Electromagnetic Permittivity Variation and Orbital Dynamics, will explore the implications of these equations in detail, predicting and interpreting precise measurable effects in atomic systems.