3.5.9 Summary: Force, Mass and Momentum
The aim of this chapter has been to show that the dynamics of an elemental volume in the energy continuum, when suitably constrained, conform to Newton's laws of motion. In addition, the relationship between mass and energy, \(E= mc^2\), used in Special Relativity, is also valid for a stationary mass. However, in this new theory, mass is defined in such a way that it is constant for all velocities, which confirms Newton's original concept.
The above relationships have been derived here via the artificial idea of an elemental volume of the energy continuum which has a constant angular momentum. This allowed the derivation to proceed using the familiar concepts of, mass, energy, and momentum. To be useful, these relationships need to be formulated as density of mass, energy, and momentum of a point in the continuum. This will allow the above analysis to be applied to waves of displacement and to the structure of particles.
While the new theory confirms the basic ideas of classical mechanics, it partly disagrees with Special Relativity, particularly in the definition of mass. This is not surprising since the new theory is an objective one, which assigns values to energy density, curvature and expansion velocity which are valid for all observers. On the other hand, Special Relativity is a subjective theory in which observers moving relative to each other with constant velocity perceive different values for mass and energy. In addition, it does not acknowledge curvature into a fourth dimension.
The above analysis may be applied to particles having mass, which in this new theory are viewed as energy structures and therefore, part of the energy continuum, rather than objects separate from it. A later chapter will show that a stationary particle is a spherically symmetrical displacement of the continuum, having high energy density held in place by the local curvature. This results in the particle containing a large amount of energy compared to the surrounding continuum. This is perceived as mass.
Historically, mass was thought to be a substance in its own right, and Newton declared it to be conserved. Subsequently, experiments with radio active elements showed that mass is not conserved, and that mass lost from disintegrating atomic nuclei appears as heat energy. The mass/energy equivalence ratio is given by Einstein's famous equation, \(E=mc^2\), where, \(E\), is total energy, and \(c\) is expansion velocity and also the speed of light.
Geometric Interpretation
If the velocity of the particle is constant, \(v\), then it adds as a vector to the expansion velocity, \(v_x\), which we assume is the usual, \(c_0\), (the speed of light). The resultant total velocity of the particle, \(c_v\) is given by the vector addition, \(c_v^2 = c_0^2+v^2\).
This has important consequences for the concept of momentum which are detailed in the sections below. However, an immediate result of this relationship gives the reason why Maxwell's equations fail to account for the velocity of Electromagnetic (EM) waves emitted by moving objects.
The Maxwell traveling wave equation relates the time and space differentials of an Electric field, \(E\), along an x-axis, as follows.
\begin{align}
\frac{\partial ^2 E}{\partial t ^2} = c_0^2 \frac{\partial ^2 E}{\partial x^2} \nonumber
\end{align}
where \(c_0\) is the speed of light.
To account for the situation in which a moving object emits an EM wave, the speed of light term, \(c_0\) is changed to \((c_0 + v)\), or \((c_0-v)\) where \(\pm v\) is the velocity of the moving emitter depending on its direction of motion. Consequently, the traveling wave equation becomes,
\begin{align}
\frac{\partial ^2 E}{\partial t ^2} = (c_0 \pm v)^2 \frac{\partial ^2 E}{\partial x^2} \nonumber \\
\frac{\partial ^2 E}{\partial t ^2} = (c_0^2 \pm 2c_0 v + v^2) \frac{\partial ^2 E}{\partial x^2} \nonumber \\
\frac{\partial ^2 E}{\partial t ^2} = (c_0^2+ v^2) \frac{\partial ^2 E}{\partial x^2} \pm 2v c_0 \frac{\partial ^2 E}{\partial x^2} \nonumber
\end{align}
which is a traveling wave plus an additional expression which doesnt occur in reality.
The error in Maxwell's wave equation is in assuming that it is a three dimensional wave with a velocity dependent on the medium through which it is traveling. In reality, it is a wave traveling through the continuum with a velocity equal to expansion velocity. We see that the correct expression for a traveling EM wave emitted from a moving emitter is simply,
\begin{align}
\frac{\partial ^2 E}{\partial t ^2} = c_v^2 \frac{\partial ^2 E}{\partial x^2} \nonumber
\end{align}
This is a traveling wave with a speed given by \(c_v =\sqrt{c_0^2+v^2}\).
In the theory of an Expanding Energy Continuum, the universe is regarded as an object in its own right, consisting entirely of energy and shaped like a hyper-sphere. In this context, energy is to be regarded as a kind of fluid, which, due to its own internal pressure is expanding exponentially in the manner already described in a previous chapter. That is, the radius of the hyper-sphere, \(R(t)\), increases exponentially with time, such that
\begin{align}
R(t)=R_0 e^{Ht} \nonumber
\end{align}
where H is the Hubble constant, and \(R_0\) is the radius at a time designated as \(t=0\).
The expansion of radius, \(R(t)\), constitutes an expansion velocity, \(v_x\), given by
\begin{align}
v_x=\frac{\partial R(t)}{\partial t} =H R_0 \nonumber
\end{align}
Because its estimated value is of the same order as the speed of light, expansion velocity is assumed to have the same value as the speed of light, \(c\), that is, \(v_x =c\). This assumption is justified by the presence of \(c\) in situations which have nothing to do with light. (For example, \(E=mc^2\).)
Since \(R(t)\) is increasing exponentially, its time derivative, \(v_x\), will also increase exponentially, so that expansion velocity will increase according to, \(v_x(t)=v_x(0) e^{Ht}\). In numerical terms, expansion velocity will increase by one part in one million every 12,000 years. However, the actual speed of light, that is, the velocity of photons within the energy continuum, depends on the velocity of the emitting source, as in classical physics. In this new theory, the stationary energy continuum takes the place of Newton's absolute space in which all motion is relative. To denote the speed of light in the stationary energy continuum it will be given the symbol, \(c_0\). In other situations, it will be denoted by \(c\). Since \(c_0\) is equal to expansion velocity, it also increases so slowly with time, that it may be assumed constant in the present era.
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