Alex' Mass to Energy Equation - Page 2

 (see image Error Function 4)
 
The natural transition to a polar form is ideal for use with a particle

 The value for the square root of pi suggests it can translate onto the complex plane if the sign of pi is ignored. This is perhaps the only point lacking substance but it is not an essential component of my interpretation.
 
 (see image Error Function 5)

To derive the V or W waves we can now use erfc (x) with limits between x and infinity. This can be used to describe the energy levels, say, as the energy peaks and reaches the particles surface and then subsides. To complete the V wave we then need to reverse the limits so the total energy curve is U shaped.
Note that we can use the erf (x) function to describe the mass levels at the same time.
Notice that at any one time the mass and energy always sums to unity!

  (see image Error Function 6)
  (see image Error Function 7)

This sets the limits for the functions as containment for the mass to energy transfers that Einstein’s equations cannot do.

 (see image Limits of Error Function)

Comparison with Einstein’s Equations for Energy
At first it appears that Einstein’s equations can only describe external forces at work at the same time upon the particle and they would seem to have little effect, if any, on this internal process.
However it is possible to manipulate Einstein’s Equations to form the basic shape of a V wave (see diagram “Mass to Energy 2” below).

 (see image Mass to Energy 2)

In trans-scribing Einstein’s equation I used equation 30-42 from the book “Essentials of Physics” by Sidney Borowitz and Arthur Beiser Lib Congress # 65-19242 page 539. This states:

E = mc2 = m0c2 / √(1 - (v2/C2))

Where m0 represents the rest mass.
I have had to use x^2 in place of (v^2/C^2) and as the v and c terms are both velocities then they cancel to an integer variable. Also velocity = distance (x) / time so the choice of using the distance x is natural. The Rest mass energy is given as m0*c^2 for the numerator and which I set as unity.
The importance of this result is easily overlooked if the investigator has not got a model in mind. So I am not surprised that no-one has pointed this out before.
Matching Equations for Energy
Although the equations all form a U shape they need manipulation to match more completely. The fact that they do not match perfectly may be explained by noting if the internal and external forces do not match perfectly than as we approach light speed then space will have to deform to compensate for any anomaly.
 

In figure “Mass to Energy 3” I am using the shape for the V wave given by the equation y = e^x + e^(-x). Distances double in 4D so a factor of 2 fits my geometry. I compare this to Einstein’s Modified Equation of y = 1/(1 - (x)^2) and finally the equation derived using the Error Function.
I am particularly interested in these curves between -1