This chapter will attempt to describe the mathematics of the theory proposed in the previous chapters. First of all, let us recall some principles that will help us understand what will follow. In Dr. Einstein's General Theory of Relativity, we see two postulates. The first is that all laws of physics are equal in all inertial frames of reference. The second postulate states that the velocity light travels in free space has the same value in all inertial frames of reference. Also, we can recall from general physics the Law of Conservation of Energy, which states that the energy of interacting bodies or particles in a closed system remains constant, though it may take different forms. Keep these in mind, as the following calculations will rely on these three ideas as a basis.
Note that the Law of Conservation of Energy requires that the total final energy in a system must be equal to the total initial energy of the same system. Thus,
where
and
These two equations require refinement when working with relativistic physics. Thus, Einstein developed the following equations to solve the problem.
where
and
.
Considering the wave-particle duality property in quantum electrodynamics, we also have
and
giving us
for v = c.
In the General Theory of Relativity, Einstein describes that as an object accelerates toward the speed of light, it experiences length contraction. The Lorentz transformation formula for length contraction is:
Where Lv is the length of the object at speed v and Lo is the length of the object at speed v = 0. Similarly, in this paper, it is suggested that the length of the object traveling is not contracted, or shortened, it is the distance being traveled, or space itself, that is being expanded. Note that in the equation above Lv ≤ Lo.
Let M be a distance (measured in relevant units) such that Mr is the length of space for an object at rest and Mc is the length of space associated with a given energy (brought on by the same object in motion). Since space contracts for an object in motion, we see that Mr ≤ Mc. This is exactly the inverse of the relationship as defined by length contraction. We can write the above equation for length contraction, and the equation for spatial contraction in the following manner:
by definition, and by the multiplication property, we see
and thus, by the substitution property,
.
Now, let ΔM = |Mc - Mr|, or the physical difference between spatial distances. Using the algebraic process, we will manipulate the above equation to show a new equation for the energy E.
And since the relativistic equation for Kinetic Energy is given above as Ek = mc2 (γ - 1), we can substitute the above and show that
And since the relativistic equation for Kinetic Energy is given above as Ek = mc2 (γ - 1), we can substitute the above and show that
and therefore,
This gives us another equation that we can use to describe energy. However, this equation might be more helpful in describing another phenomenon. Instead of length contraction, we have spatial expansion, and the equation above can help us find out exactly how much space is expanded based on the amount of energy that is present. We can now write the above equation for the energy of a photon as follows:
The above equation shows that there is a clear connection between the world of the small and the world of the very large. The above equation may in fact be the key to unification between quantum electrodynamics and gravitation.
