Alex' Mass to Energy Equation - Page 1

Introduction
Einstein derived his Mass to Energy equations from Newton’s equations of velocity and acceleration. I am taking a completely different approach that fixes this transformation process as a totally internal operation.
I have previously introduced the V and W waves as being responsible for the Mass to Energy transformation process but not had a means of showing mathematically how such a force could arise. I have however described this transformation as acting out from the particle core’s centre and then returning continually. The Error Function (see image Error Function 1) is the means I will use to establish a process that is exponential in nature.
I am using the paper “AM375a: The Error Function erf Notes” by Mikko Karttunen. Web: www.softsimu.org November 25, 2007. I trust he will allow me to reproduce large parts of his paper for study purposes here.
Deriving the Error Function as a Mass to Energy Exchange
I am not treating this function, here, as actually, describing error. The function is useful to me, as I will show, in that it:
• Uses an exponential form.
• Integrates the equation allowing for mass and energy to be summed without interruption.
• Has a complimentary form. This allows for mass to counter-balance energy. That is when mass increases then energy decreases and vice versa.
• Has a polar form and so can be used to describe the cross section of a particle.
 
Reinterpreting the Standard Mathematics
It is not surprising that the Error Function has no formal proof attaching its origin to error and is based upon experience of error occurrences that are completely natural! It is very nice for Complex Quantum Mechanics that it has been however. I will suggest here that the probabilities in Classical Quantum Mechanics can be described by the Error Function (erf). In particular I believe this to be true for a photon of light

 (see image Error Function 2)
 
The fact that the erf is even is very useful. In the mass to energy transfer I am describing one half of the integral can describe the movement of energy from the centre of the particle to its surface while the other half describes the reverse process. The fact that they sum to unity in equation (2) is extremely helpful.

 (see image Error Function 3)

Although the intention was not to imply the integral I is complex the squaring of this integral invites that comment.

 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.
 

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!

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

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).

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