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3.4.2 Properties of the Expanding Energy Continuum

As already noted, in the energy continuum there are only three fundamental variables, Energy Density, \(\rho\), Euclidean curvature, \(\kappa\), and Expansion Velocity, \(v_x=c \), (the speed of light in most situations). These variables are fundamental in the sense that they are the foundations of objective reality and are independent of any coordinate system set up to measure them. Hence, their values are not dependent on the location, or motion, of observers, neither are they subject to the mathematical constraints of Special Relativity.

It is important to accept that these three variables exist at each point in the continuum and at each instant in time, and are the essential properties of the universe defined as a closed volume of energy, shaped like a hyper-sphere. These three variables are independent of each other, but connected by relationships between their values expressed in some consistent system of units. They cannot be derived from some unknown, underlying reality - they themselves are reality.

Since the volume of the energy continuum is finite, the total energy, \(E_u\), contained therein is also finite, and given by
\begin{align} Total\ energy\ E_u = \rho_a V_u\ \ \nonumber \end{align}
where, \(\rho_a\) is average energy density, and, \(V_u\) is volume of the continuum.

If \(E_u\) is constant then there is conservation of energy. However, the calculation \(E_u = \rho_a V_u\) is only valid at each point if energy density has the same value throughout the continuum. Whereas, in the real continuum the energy density may have a unique value at each point. Hence, for conservation of total energy, the values of energy density at every point must be constrained so that total energy summation is constant.

It becomes clear, therefore, that the three fundamental variables of the continuum are not free to vary independently. This can be demonstrated if we imagine the continuum to be divided into a large number of small, equal volumes, \(\Delta Vol\), so that the volume summation
\begin{equation}
\Sigma\ \Delta Vol = V_u
\nonumber
\end{equation}
The energy contained in each \(\Delta Vol\) is given by, \( \rho \Delta Vol\), so that the total energy in the continuum is given by
\begin{equation}
E_u = \Sigma (\rho \Delta Vol)
\nonumber
\end{equation}

Since energy density, $\rho$, varies throughout the continuum and \(E_u\) is constant, an increase in the energy density of one \(\Delta Vol\) must be balanced by a decrease in energy density of other \(\Delta Vol\). This is not possible without some constraint mechanism which ensures that energy density, \(\rho\), expansion velocity, \(c\) and curvature, \(\kappa\), vary in a manner necessary to achieve this balance over the whole continuum. This can be achieved if some other quantity is constant at each point.

In classical mechanics, there are two conservation laws which might apply in the energy continuum. These are conservation of linear momentum and conservation of angular momentum. In the real continuum, an element of volume, \(\Delta Vol\), will contain an increment of energy given by, \(\rho \Delta Vol \), so that
\begin{align}
Linear\ Momentum =& \frac{\rho \Delta Vol}{c} = \frac{\rho}{c} \Delta Vol \notag \\ Angular\ Momentum =& \frac {\rho \Delta Vol}{c\ \kappa}\ = \frac {\rho}{c \kappa} \Delta Vol
\notag
\end{align}

Angular momentum constrains all three fundamental variables so that its constancy provides the overall constraint we are seeking. Hence, at each point in the continuum and at each instant in time, it is postulated that there is a constant quantity, say $\alpha$ (alpha), which has units of angular momentum density, such that
\begin{align}
\alpha = Angular\ Momentum\ Density = \frac{Angular\ Momentum}{\Delta Vol} = \frac {\rho}{c \kappa} \nonumber
\end{align}

The total angular momentum, \(A_u\), of the whole continuum is the summation of angular momentum density over the entire volume, \(V_u\). Assuming that the continuum is a perfect hyper-sphere, expanding uniformly, (\(c\) and \(\kappa\) are constant), into the fourth dimension, we have
\begin{align}
A_u =&\ \Sigma\ \alpha = \Sigma\ \frac{\rho\ \Delta Vol}{ c\ \kappa} \nonumber \\
Since\ c\ and\ \kappa \ are\ constant\ and\ E_u =&\ \Sigma\ \rho\ \Delta Vol \nonumber \\
A_u =& \frac {E_u}{c\ \kappa} \nonumber
\end{align}
At each point in the ideal continuum and at each instant in time, expansion velocity, \(v_x= c = HR \), and curvature, \(\kappa =3/R \), so that \(c\ \kappa = 3H\), where \(R\) is the radius of the continuum and H is the Hubble constant. Hence
\begin{align}
A_u = \frac{E_u}{3 H} \nonumber
\end{align}

Given that H and $E_u$ are both constants, in the ideal continuum \(A_u\) must also be a constant. Assuming that \(H, E_u, A_u\) do not change over time, we can refer to them as 'eternal constants'. We postulate that these three, eternal constants are also valid in the real continuum.

It is important to note that the angular momentum in classical physics is a vector property of rotating bodies. In particle physics it is a scalar property of particles where it is known as intrinsic angular momentum, (or 'spin' even though there is no evidence that anything is spinning). It is in this latter role that we interpret the concept of angular momentum density, and use the term 'spin' to confirm this.

In this theory, intrinsic angular momentum (spin) is a property of the whole energy continuum, and its value, \(A_u\) is an eternal constant. It's density has the same value, \(\alpha\), at each point in the continuum. It is related to the three fundamental variables as follows.
\begin{align}
spin\ density\ \alpha= \frac{\rho}{c\ \kappa} \ \
hence,\ energy\ density\ \rho = \alpha\ c\ \kappa \nonumber \end{align}
This equation constrains the values of expansion velocity, curvature and energy density at each point in the energy continuum and at each instant in time. The constant, \(\alpha\), has units of spin density, and is constant at each point throughout the entire continuum. However, over time, it decreases as the volume of the continuum increases via expansion, as follows.
\begin{align}
\alpha =&\ A_u \div V_u \nonumber \\
\alpha =&\ \ \frac{A_u}{V_{u0} e^{3Ht}}\ = \ \frac{A_u}{V_{u0}} e^{-3Ht} \nonumber \\
\alpha =&\ \ \alpha_0\ e^{-3Ht} \nonumber
\end{align}\\
In the current era, \(\alpha= \alpha_0\), so that the relationship, \(\rho = \alpha_0\ c\ \kappa \), applies at each point in the continuum. This is effectively an EQUATION of STATE of the energy continuum. The pair of variables, \(c\ \kappa\), are derivatives of radius, \(R\), so that if \(R\) is a scalar, then so are \(c\ and\ \kappa\). For this reason, this equation of state is a scalar equation, which considerably simplifies calculations.

In regions of the continuum where gravity is negligible, (inter-galactic space), the three variables take on basic, background values which we may assume to be constant in the current era since they vary by one part in one million over about 12,000 years. The ideal equation of state then becomes
\begin{align}
\rho_0=\alpha_0\ c_0\ \kappa_0 \nonumber
\end{align}
where the suffix 'o' denotes background values of these variables in the current era.

Since the value of \(c_o\ and\ \kappa_o\) are such that \(c_o \times \kappa_o = 3H\), the value of spin density, \(\alpha_o\) is given by \(\rho_o/3H\). Since energy density, \(\rho_o\) and Hubble constant, H, are known only approximately, an approximate figure for spin density can be found. The density of matter in intergalactic space has been estimated to be about \(9\ 10^{-27}\) Kg per cubic meter, which translates to energy density of about \(8\ 10^{-10}\) Joules per cubic meter. Recent measurements of the Hubble constant range between \(2.1\ 10^{-18}\ and\ 2.5\ 10^{-18}\) per second. Estimates using the present theory of an expanding energy continuum give values \(\rho_o =7.489\ 10^{-10}\ and\ H= 2.644\ 10^{-18} \). This results in a value for spin density \(\alpha_o=0.944\ 10^8\)

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