1.2.3 Expansion Velocity: Speed of Light
The expansion velocity of the hyper-spherical universe may be defined as the rate of increase in its radius vectors, which has been shown to increase exponentially over time. This inspires the question - what is this velocity in the present era?
Given that a radius vector, \(R(t) = R_0e^{Ht}\), its expansion velocity, \(v_x\), is simply \(v_x = HR_0e^{Ht}\). In this current era, \(t = 0\) (now), \(e^{Ht}= 1\), and \(v_x = HR_0\), where \(R_0\) is the current length of the radius vectors. Unfortunately, we have no accurate measurements of \(H\), and estimates of \(R_0\) are little more than educated guesses. However, estimates of both \(H\) and \(R_0\) have been made by Cosmologists, including Einstein and LeMaitre.
The value of Hubble’s constant, \(H\), has been estimated by NASA to be, \(2.27 \times 10^{18}\).The magnitude of the radius of the hyper-sphere, \(R_0\) has been estimated to be of the order,\(10^{26}\) meters. These estimates give an expansion velocity of about \(2.27 \times 10^8\) meters per second. This value is surprisingly high and so close to the speed of light, \((3 \times 10^8\) meters per second), that this figure must be considered to be an approximation to that value.
First reactions to the idea that the universe is expanding at the speed of light is that it's impossible. However, given the very large dimensions of the hyper-sphere, \((10^{26}\) meters), it could be possible. Since a radius vector of the hyper-sphere is about \(10^{26}\) meters in length, an increase of \(3 \times 10^8\) meters per second, is equivalent to an increase of about \(10^{16}\) million meters, per year, which is a factor of \(10^{−10}\) of its current length. If this calculation is scaled down to a sphere the size of planet Earth, its equivalent to an increase in radius of about one millimeter per year. With this perspective in mind, the idea that the universe is expanding at the speed of light is less problematical, and needs to be explored further.
Our usual experience of velocity relates to an object moving relative to other objects. In this case, each point in the hyper-spherical continuum is moving relative to all other points individually, even though the total movement of the hyper-sphere is zero - it is simply expanding. It is important to note that each point is a three dimensional element of volume and that its expansion velocity is in a direction perpendicular to all three of its axes. For these reasons, it might be more accurate to view expansion velocity as a property of the energy hyper-sphere which serves only to change its radius but not its location.
It would explain why the speed of light appears in physical processes which have nothing to do with light. For example, if each point in the energy continuum is moving at the speed of light, \(c\), each point will be endowed with momentum. If a small volume of the continuum has energy content, \(E\), then its momentum in the direction of expansion will be, \(E/c\). If the same small volume is measured in terms of its mass, \(m\), then itsmomentum is \(mc\). Since momentum should be the same in both cases, we have that, \(E/c=mc\), in other words, \(E=mc^2\).
One of the mysteries of Special Relativity is that the speed of light is deemed to be constant even when the emitter of the light is moving relative to the receiver, and vice-versa. This paradox is easily explained if the speed of light refers to the velocity of the whole three dimensional domain, in a direction perpendicular to all three axes of its co-ordinate system. This would allow the speed of light to be a property of the coordinate system, so that an object moving within the system with velocity, \(v\), would have a total velocity of \(\sqrt{c^2+v^2} \), for both positive and negative values of \(v\).
It is possible to show that this visualization of expansion velocity is intimately connected with the phenomenon of light. For convenience the direction of expansion will be indicated by using the prefix \(l\) (ell) to distinguish it from the three, \(x; y; z\) axes of the energy continuum to which it is always perpendicular. Hence, we can make use of the identities,
$$\frac{\partial R}{\partial x} = l \nonumber$$ and $$\frac{\partial R}{\partial t} = lc\nonumber$$
where \(c\) is the speed of light, assumed to be constant.
If the time variation of an electric field, \(ξ\), has the form,
$$\frac{\partial ξ}{\partial t} = \frac{\partial ξ}{\partial R} \times \frac{\partial R}{\partial t}\nonumber$$ then
$$c\frac{\partial ξ}{\partial t} = \frac{\partial ξ}{\partial R}lc\nonumber$$
Similarly,
$$\frac{\partial ξ}{\partial x} = \frac{\partial ξ}{\partial R} \times \frac{\partial R}{\partial x}\nonumber$$ then
$$\frac{\partial ξ}{\partial x} = \frac{\partial ξ}{\partial R}lc\nonumber$$
Consequently,
$$\frac{\partial ξ}{\partial t} = \frac{\partial ξ}{\partial x}c\nonumber$$
Differentiating this expression with respect to time and again with respect to x, we have that
$$\frac{\partial^2 ξ}{\partial t^2} = \frac{\partial^2 ξ}{\partial x^2}c^2\nonumber$$
This is the Maxwell's Wave Equation derived without assuming any electrical phenomena except the presence of an Electric Field, \(ξ\). A similar calculation could be done assuming only Magnetic Field, \(B\).
For the reasons given above, in this work the universe is viewed as an energy continuum shaped like an expanding hyper-sphere, each elemental point of which is moving in the direction of expansion at the speed of light.
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