Title: Exact Mathematical Solutions for GEONs: 3-D Spherical Confinement and Quantum Energy Levels via the LIFE (Local Intrinsic Field Equilibrium) Framework
Author: Wim Vegt, Physicist & Researcher (Ret.), Eindhoven University of Technology
Introduction:
Wheeler’s GEONs and the Quantization of Light
In standard physics, the quantization of light is almost exclusively associated with Einstein’s concept of the discrete photon. However, in 1955, John Archibald Wheeler introduced a much broader macroscopic concept: the GEON (Gravitational Electromagnetic Entity). Wheeler proposed that an intense electromagnetic wave could theoretically confine itself in a stable, 3-dimensional geometry strictly due to the gravitational pull of its own active field energy (following Einstein’s $E = mc^2$). While Wheeler conceptualized this self-confinement, calculating the exact mathematical boundaries and stable internal states of a GEON has remained a profound challenge.
1. The Geometry of Quantized Confinement
In general physics, quantization is the natural geometric result of wave confinement—much like a stationary sound wave on a guitar string forming discrete nodes and antinodes. For electromagnetic radiation, quantization occurs when a wave is geometrically confined and its wavelength perfectly matches the 3-dimensional boundaries of that confinement in integer (quantum) numbers. In the case of a GEON, the boundary is not a physical mirror or cavity, but the self-generated gravitational field. To realize the "quantization of light" in this context, we must mathematically identify the stable, 3-dimensional configurations where the confined electromagnetic standing waves exist in a perfect zero-force macroscopic equilibrium.
2. 4-Dimensional Equilibrium and the Schrödinger Equivalency
This Mathematica notebook provides the first exact mathematical solutions for the formation of GEONs, utilizing the Localized Intrinsic Field Equilibrium (LIFE) framework. By enforcing a strict 4-dimensional force-density equilibrium (balancing electromagnetic pressure, inertia, and gravitational field density in $N/m^3$), the continuous equations naturally yield discrete, quantized spherical energy levels.
Crucially, the exact solutions to this macroscopic equilibrium perfectly mirror the solutions of the fundamental quantum mechanical Schrödinger wave equation. The calculations demonstrate:
• An exponential decay function dependent on the spherical radius. • A sinusoidal function dependent on time. • A strict 90-degree phase shift between the confined magnetic standing wave and the electric standing wave.
It is this exact 90-degree phase shift in the localized standing waves that mathematically acts as a quantum probability vector, bridging the deterministic equilibrium of the LIFE framework with the probabilistic nature of quantum mechanics.
What This Notebook Demonstrates:
The following calculations provide the rigorous geometric and trigonometric proofs for these stable GEON configurations. I invite the Wolfram Community to run the code, analyze the radial and time-dependent functions, and explore how classical force-density equilibrium naturally gives rise to discrete quantum states.
1] Vegt, W; A Continuous Model of Matter Based on AEONs; Physics Essays volume 8, number 2, 1995; https://zenodo.org/records/19001551; https://research.tue.nl/en/publications/a-continuous-model-of-matter-based-on-aeons/
[2] Vegt W; The Origin of Gravity; Research & Reviews: Journal of Pure and Applied Physics; https://www.rroij.com/peer-reviewed/the-origin-of-gravity-91966.html ; https://www.rroij.com/peer-reviewed/the-origin-of-gravity-91966.html; https://zenodo.org/records/19002089
[3] Vegt W; Enhancing Precision in Electromagnetic Force Density Modulation Using LASER Control; Journal of Laser Aplications; AIP publishing; DOI: https://doi.org/10.2351/7.0001636; https://zenodo.org/records/19009836
[4] Vegt W; Achieving Ultra High Resolution Lithography via Intrinsic Equilibrium and Electron Driven Spin Resonance; https://zenodo.org/records/19020344
[5] Vegt W; A Unified Force Density Framework for Plasma Confinement: Integrating Navier-Stokes with Local Interaction Field Equilibrium (LIFE); https://zenodo.org/records/19024427 Calculations in Mathematica demonstrating the Local Intrinsic Field Equilibrium (LIFE) framework
[6] Vegt W; Mathematical Proof of Arbitrary Wave Propagation and Field Equilibrium using the LIFE Framework; https://community.wolfram.com/groups/-/m/t/2576692?p_p_auth=6wOlNOpR
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