Calculation 4
Time and Radius dependent GEONs with Discrete Spherical Energy Levels
Time and Radius dependent GEONs with Discrete Spherical Energy Levels
1. The concept of “4-dimensional Universal Equilibrium” in the Unification of Electrodynamics with “Quantum Physics”.In physics the quantization of light has always been related to the quantization of Einstein’s concept of photons. In “The Proposed Theory” the quantization of light has been related to a much broader concept also including Wheeler’s concept of GEONs (Ref. [1]).In general when waves have been confined, like a sound wave on a guitar string, the phenomenon of “Standing Waves” (Stationary Waves) appears with the well known aspects of Nodes and Antinodes in all possible Modes.The Equation in The Proposed Theory for Confined Electromagnetic Waves is an equation for “Standing Waves” (Stationary Waves) which appear with the well known aspects of Nodes and Antinodes in all possible Modes, describing the Quantization of Electromagnetic Radiation in GEONs. 2.1 The “Quantization of Light” (Electromagnetic Radiation)In general quantization has been determined by the match of a wavelength on a 3-dimensional geometry. Because only an integer number (Natural Number) of wavelengths matches the 3-dimensional geometry. The quantization of Electromagnetic Radiation is only possible when the electromagnetic wave has been confined due to its own gravitational field and the wavelength has to match a 3-dimensional geometry. This is possible when light “confines” itself in a 3-dimensional stable geometry due to its own gravitational field. According John Archibald Wheeler’s “GEON’s [Ref. 1] (Gravitational Electromagnetic Entities) and Einstein’s relationship (W = m c2 ) every “confined” energy results in inertia. Inertia results in mechanical mass and mechanical mass results in a gravitational field.To realize “quantization of light”, stable configurations have to be found of confined electromagnetic radiation in which the wavelength of the confined electromagnetic energy matches in Natural Numbers the 3-dimensional geometry of the confinement. 2.2 Realization of a stable Electromagnetic Confinement in a 3-Dimensional Configuration matching the wavelength of the confinement in Quantum Numbers. Newton’s third law has been described for the Electromagnetic Field in 3 spatial dimensions, resulting in the “Fundamental Equation for Electromagnetic Gravitational Interaction” in The Proposed Theory: (23)The Gravitational-Electromagnetic Confinement for the elementary structure beyond the confinement of light (electromagnetic radiation) has been presented in equation (24). (24)In which represents the gravitational acceleration acting on the electromagnetic mass density of the confined electromagnetic radiation. A solution for equation (24), describing a Gravitational Electromagnetic Confinement (GEON) within a radial gravitational field with acceleration (in radial direction), has been represented in (25). (25)In which the radial function f(r)(r) equals: (26)The solution has been calculated according Newton’s Shell Theorem.The fundamental boundary condition for the confinement of Electromagnetic radiation (GEONs) is that the energy flow (Poynting vector) equals zero at the surcface of the confinement. This is possible at a 90 degrees Phase Shift between the Electric FieldfField and the Magneticmagnetic Field.Introducing the Quantum Vector Function , (27) Substituting (27) in (25) results in the quantum presentation for the GEON: (28)2.1 GEONs with Discrete Spherical Energy LevelsA solution for equation (24) describing GEONs dependent of time and radius, presenting discrete spherical energy levels, within a radial gravitational field with acceleration (in radial direction), has been represented in (25). (29) (30)Equation (29) presents a Standing (Confined) Electromagnetic Fields Configuration with a phase shift of 90 degrees between the electric field and the magnetic field with the corresponding Nodes and AntiNodes.Fig. 2 Nodal and Antinodal planes for Standing (Confined) Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. An essential requirement for the confinement of Electromagetic Energy is that the Poymtingvector equals zero at the surface of the confinement. For the confinement within a sphere, a standing electromagnetic wave pattern has been required which exists of concentric spheres, at every sphere an antinodal plane for E (or B) with a radius ditance between each sphere of half the wavelength of the confinement (k = n π λ and “n” a natural number and λ the wavelength). Fig. 3 Nodal and Antinodal Spheres for Standing (Confined) Spherical Electromagnetic waves with a 90 degrees phase shift between the Electric field and the Magnetic field. Equation (29)Every concentric sphere represents an anti-nodal surface for the Electric Field or the Magnetic Field. The Poynting Vector at this spherical surface equals zero at any time and at any location at this sphere. The Electromagnetic Energy remains always within this sphere and the next concentric sphere, where all the concentric spheres have a difference in radius of one half wavelength of the electromagnetic radition within the confinement and a different discrete energy level. Every concentric sphere represents an anti-nodal suface of the electric field or the magnetic field.
f[r]=K
-
G1ϵ0μ0
8πr
e
Results force densities in resp r-direction, θ-direction, φ-direction
Choosing for the arbitrary constant K, the value K equals 1 ,represents a GEON with a thousand times smaller radius