Author: Wim Vegt, Physicist & Researcher (Ret.), Eindhoven University of Technology
Introduction
In 1861, James Clerk Maxwell published his foundational theory of electromagnetism, providing the mathematical basis for idealized plane and spherical waves. However, standard classical electrodynamics struggles to fully model the localized, non-diffusing propagation of complex, arbitrary wave patterns—such as a concentrated laser beam or the highly detailed projection of an arbitrary slide or film onto a screen.
To perfectly model the propagation of these unique, non-plane wave patterns without structural degradation, we must look beyond standard wave equations and examine the continuous balance of spatial forces.
The LIFE Framework and 4-Dimensional Equilibrium
In this notebook, I present a mathematical solution using the newly formalized Localized Intrinsic Field Equilibrium (LIFE) framework. The LIFE framework is deeply rooted in Isaac Newton’s foundational concept of harmony and equilibrium (his Third Law), but it extends this principle into a rigorous 4-dimensional spacetime equilibrium.
In the LIFE framework, the propagation of an arbitrary image is not just a passing wave, but a dynamic fluid-like state. The mathematical proofs below demonstrate a perfect, continuous equilibrium between all electromagnetic force densities and spatial inertia force densities (N/m³) during propagation.
Bridging Classical and Quantum Mechanics
By expanding this equilibrium into the fourth dimension (time), the LIFE framework naturally converges with the fundamental, relativistic quantum mechanical Dirac equation. This provides a universally applicable mathematical description that remains valid at relativistic velocities, superseding the limitations of the low-velocity Schrödinger equation.
What This Notebook Demonstrates:
The following Mathematica calculations provide the exact geometric and algebraic proofs showing how the LIFE framework successfully describes the propagation of an arbitrary transverse light profile. I invite the Wolfram Community to evaluate the code, manipulate the variables, and observe how this intrinsic equilibrium maintains the structural integrity of the projected image where classical Maxwell equations fall short.
References:
[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/19067591
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
[7] Vegt W; Mathematical Proof of Light Propagation and Gravitational Redshift via the LIFE Framework; https://community.wolfram.com/groups/-/m/t/2576537?p_p_auth=gVJYf1N4
[8] Vegt W; Exact Mathematical Solutions for Toroidal Confinement of MASER Radiation in Tokamaks https://community.wolfram.com/groups/-/m/t/3115543?p_p_auth=8VRRtDct
[9] Vegt W; Mathematical Proof of "Slow Light" in Silicon Crystals via Lattice Resonance and the LIFE Framework https://community.wolfram.com/groups/-/m/t/3571371?p_p_auth=Yid6Zlnp
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