Relations and Connections in the Derivation of the Genesis Formula

By RE CASTEL

The following illustrates the relations and connections leading to the derivation of the genesis formula.

In Galileo's idea of the relativity of motion the transformation equations are as follows.

Basically, the Galilean transformation equations describe velocity additions.

The Galilean equations were acceptable until Maxwell showed that c=lf – which allowed the assumption that the velocity c is the same in all the differing frames of reference (i.e., c=u=u').

Because of the assumption that the velocity of light is the same in all the differing frames of reference, an invariance factor became necessary in the transformation equations to ensure that c=u=u'.

The necessity to have c=u=u' required that rotations on the motion of the electromagnetic phenomenon should be reflected by the transformation equations. (Unfortunately, this idea was never sufficiently treated in the popular explanations). The rotations aspect departs from the velocity additions idea of the Galilean transformation equations.

To reflect the consideration regarding the electromagnetic phenomenon, the value c was introduced into the equation for x'. So, with c=x/t and hence t=x/c, we have

Now, the relation for time was also modified to show t'=x'/c, which departs from the old t'=t assumption. Thus,

Since x'=(x–vt), we have

And because t=x/c, we have

And thus, we have

or simply

The Lorentz invariance factor (1–v²/c²)^–½ was used in order to have the invariant equations that allowed c=u=u'. Thus with γ=(1–v²/c²)^–½, the Lorentz transformation equations are of the following form.

or, to rewrite,

Obviously, the Lorentz invariance factor (1–v²/c²)^–½ does not get to the transformation factor without intermediate approximations and the special treatment for the square root. This is because the Lorentz invariance factor in the Lorentz equations only accounts for a two-dimensional translation of the electromagnetic phenomenon – that is, half of the full tensor.

Also, a fact worth noting, it is not popularly clarified how the transformation factor is arrived at from the use of the invariance factor (1–v²/c²)^–½. The rationale of the application and usage of the Lorentz invariance factor and the Lorentz transformation factor is also not popularly clarified.

On the other hand, the invariance factor (1–v/c)^–1 proponed by the author allows the complete transformation without the intermediate approximations. The new transformation equations take the following form.

The invariance factor (1–v/c)^–1 clearly shows

and

The (1), which is the product of the reciprocals, simply signifies an immediate and complete transformation.

The new invariance factor (1–v/c)^–1 (and hence, consequently, the (1–v²/c²)^–1 transformation factor) allows a relation for a three-dimensional translation of the electromagnetic phenomenon – that is, the full tensor. The two pairs of the v and c vectors along the x-and-y and the x-and-z axes are appropriately represented in the full tensor, which accounts for the three-dimensional rotation and contraction (or attenuation) of the electromagnetic phenomenon that has the transverse electric and magnetic wave components.

The particularly meaningful interpretation of the motion transformation equations becomes obvious by fixing x, y, and z as the static background space dimension, and by assuming the change in t as a separate natural occurrence and an essentially fixed background occurrence, and then by introducing into the equations new variables that represent the actual physical transformations resulting from the motion transformations.

The variables m and E are logically the variables worth introducing into the equations. These variables enter into the equations by substitution – as substitutes either for the x or the t. The substitution gives us the following.

In substituting m for x or t, we have

or substituting E for x or t, we have

It is obvious that the new transformation factor is

Upon a first use of the approximation, we arrive at a transformation factor that actually reflects an appropriate treatment of the transverse-wave vectors in the vectors interaction considered. We have

And this gives us

Thus, solving for the motion transformation that reflect the change in mass, the transformation equation becomes

This shows m' as the initial mass that transforms into the final mass m upon the application of the transformation factor (1–v²/c²)^–1.

Unlike in the Lorentz equations, in the new transformation equations the rationale for the invariance factor is very clear – the reciprocal allows a transformation factor that has essentially the unity value; and so the derivation of the transformation factor is also very clear; and then the use of the new transformation factor, with the introduction of the variables m and E in the transformation equation, is now very clear and meaningful.

The relevant relation relates the velocity or motion transformation that describes the contraction or condensation of motion into mass m or energy E.

Using the transformation factor for the full tensor translation (or motion) of the three-dimensional gravitational translation, the formulation for the final mass m is –

or to rewrite with the commonly known m_o,

The above shows that the final mass m, arising from the velocity-to-mass-energy transformation, is twice the final mass suggested by the Lorentzian half tensor formula m=m_o(1–v²/c²)^–½.

The full tensor formulation and the half tensor formulation are valid formulations. But it must be clarified where each is a valid formulation.

The old formulation accounts for only half of the full tensor translation, since the formulation uses only a pair of the c and v component vectors for the two-dimensional interaction involving only the x and y, with the c along the x and the v along the y for the rotation.

On the other hand, the new formulation accounts for the full tensor translation with two pairs of the c and v vectors upon consideration of the three-dimensional interaction involving the x, y, and z on account of the transverse wave interaction.

The new formulation leads to the relation in which energy increases in increments of mv² and, subsequently, leads to the suggestion of the genesis formula.

Upon the final approximation, we have the relation of the cosmic energy genesis, where m is the final mass and m_o is the initial mass, and where E is the final energy and E_o is the initial energy:

The connection of the idea of gravitational acceleration with the idea of velocity transformation suggests the energy accumulation by the formulation where A is the gravitational acceleration and t the elapsed time in ∆v.

The proposition regarding A is according to the undeniable fact of gravitational acceleration. And the extension of the idea to ∆v (the idea that gravitational acceleration occurs as time passes) is according to the fact of the free flow of time that figures in the formulation to advance the idea of the process of duration as an inherent part of the fundamental existential change.

According to the principle of the relativity of motion, the motion towards a given mass may be viewed in the reverse as the relative motion of the given mass. Thus, the gravitational acceleration is actually also the relative motion of the given gravitational mass. In this case, the motion is a three-dimensional motion represented by the gravitational tensor.

Thus, we have the meaningful genesis formula –

In comparison, Einstein's relation of the change in mass is shown in the following.