3.5.1 Structure of Gravitational Particles
The energy continuum has only three degrees of freedom, however, its volume is curved so that, in order to describe this curvature, it is necessary to use a flat, four dimensional, coordinate system. The inherent curvature of the continuum is a consequence of its being shaped like a hyper-sphere, which also ensures that there is no preferred direction. It is reasonable, therefore, to assume that the structure of the gravitational particles, electrons, protons and neutrons within the continuum have no preferred orientation, and hence are spherically symmetrical.
This simplifies the coordinate system necessary to define particle structure, requiring only two location coordinates for each point within it - the radius, \(r\), from the center of the structure, and its displacement, \(N\), into a fourth dimension. This spherical structure can be imagined as a set of concentric, spherical shells each of which is characterized by energy density and displacement at each value of radius. The values of these functions are high near the center, decreasing to normal background levels at remote values of radius. Hence, an energy density function, \(\rho(r)\), and a displacement function, \(N(r)\), are sufficient to describe the structure of a spherical particle.
The energy density function can be derived via the Equation of State, \begin{align} \rho = \alpha_0\ c\ \kappa \nonumber \end{align} In this equation, angular momentum density, \(\alpha_0\), is constant by definition. The expansion velocity, \(c\), is assumed to be constant since any differences in expansion rate within the particle structure would distort its shape. Hence we have that, \(c=c_0\), is constant. This just leaves curvature, \(\kappa\), as the variable which is proportional to energy density, \(\rho\), so that the Equation of State reduces to
\begin{align} \rho = (\alpha_0\ c_0)\ \kappa \nonumber \end{align} It is clear that if energy density is to increase, the curvature, \(\kappa\) will need to increase by the same factor. The curvature in question is that of the spherical shells which are a function of radius; it is standard geometry that the curvature of a sphere of radius, \(r\), is simply, \(2/r\).
Putting this value of curvature into the equation of state gives energy density as a function of radius, \begin{align} \rho(r) = \alpha_0\ c_0\ \frac{2}{r}\ \nonumber \end{align} As radius approaches infinity, the energy density approaches the background level, \(\rho_0\), and curvature approaches that of the continuum, \(1/R_0\). Hence, for this situation we can write, \begin{align} \rho_0 =\frac{\alpha_0\ c_0\ }{R_0} \nonumber \end{align} Dividing this equation into the previous relationship, we find that \begin{align} \rho(r) = \rho_0 \frac{2\ R_0}{r} \nonumber \end{align}
where \(\rho_0\) is the background energy density, and \(R_0\) is the radius of the continuum.
The displacement function, \(N(r)\), of particle structure expresses the relationship between, displacement, \(N(r)\), of each spherical shell in the structure as a function of its radius, \(r\). A constraint on displacement is that it must not affect energy density so that, \(Euclidean\ Curvature\ = \nabla^2 =0\). It was shown in the chapter on gravity, that the constraint, \(\nabla^2 N=0\), is realized in spherically symmetrical coordinates if \(N(r) = A/r\), where \(A\) is the displacement coefficient. For the direction of displacement to be against that of expansion, the displacement parameter should be negative. That is, \(N=-A/r\), and the gradient of displacement, \(A/r^2\), is positive.
These two expressions,
\begin{align} \rho(r) = \rho_0 \frac{2\ R_0}{r}\ \ \ and\ \ \ N(r)= \frac{-A}{r} \nonumber \end{align}
are the required energy density and displacement functions which describe particle structure. It is clear that, in the core zone at the center of this spherical structure, where radius tends to zero, both displacement and energy density will tend to infinity, that is, if the smooth functions of displacement and energy density continue into the center of the particle structure, it will become a singularity. Since, in this work, mathematical singularities cannot exist in physical reality, we assume therefore, that there must exist, within the core zone, a significant non-linearity, which keeps these variables finite.
The most likely reason why this potential double infinity does not exist is that the core zone of the particle is devoid of both energy and intrinsic angular momentum. These two properties become discontinuous at the edge of the core zone where their densities abruptly fall to zero within the core zone. That is, the energy density and angular momentum density functions are truncated at a radius, \(r=r_0\). In order to retain spherical symmetry, it is assumed that the core zone is spherical in shape, with radius \(r_0\). This empty sphere is enclosed by the continuum, but is not part of it.
The necessity for truncation is supported by the fact that the continuum surrounding the core zone has steep curvature which requires extra volume to accommodate it. However, extra volume would require extra intrinsic angular momentum, which, by definition, is constant. Hence, the extra volume of energy continuum required outside the core zone is supplied by the volume of continuum lost from inside the core zone.
Within the core zone, energy density, \(\rho=0\), and intrinsic angular momentum density, \(\alpha_0=0\). These properties of the spherical void suggest that it can be regarded as an particle in its own right. Such a particle does exist, it is known as the neutrino. A more complete description of this particle is given below.
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