Summary
Initially I believed that the distortion of space had more to do with the mass to energy equations used by both Einstein and myself as both theories deal with the distortion of space. I have come to the conclusion that a more subtle process takes place and the fundamental difference in my taking the internal mass to energy transformation compared to Einstein's external mass to energy process can be fully resolved here.
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 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 error function (see figure 1 – plots of erf[x]) 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.
The fact that the erf is even is very useful. In the mass to energy transfer I am describing one half of the integral in equation (4) 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.
Although the intention was not to imply the integral I is complex the squaring of this integral invites that comment by comparing equations (3) and (4).
The natural transition to a polar form in equation (4) is ideal for use with a particle that is spherical.
The value for the square root of pi in equation (14) suggests it can translate onto the complex plane if the sign of pi is ignored. This point can be compared with my interpretation of i^2 = -1 being a transformation from a circular to a square geometry. This means the work done for the error function would map exactly from 3D to 4D. In fact equation (14) gives the work done for a real V wave as equal to a complex value (ignoring 3D and 4D geometries).
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 in equation (2) the mass and energy always sums to unity! This seems to contradict the V wave but erf(c) + erf(c) is not the same as e^(x^2) + e^(-x^2). There is a subtle difference in the decay rate of these processes and their mass and energy levels.
Technical Point 3.1 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. Also the V wave does not just describe mass to energy proves as does Einstein’s equation but includes a second process,
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<x<+1 as I expect them to correspond to a unit Riemann Circle from my previous work.
However I shall try and get a closer match for the curves.
This is a close match. I note that the Error Function uses powers of x^2 and not x. Simply rewriting x^2 would work but it implies (say s = x^2) that s takes only real values (negative or positive and not complex values).
This figure (Mass to Energy 5) is another curve matching as in figure “Mass to Energy 4”
In figure “Mass to Energy 6” below the match is not so good. This reinforces the need to use powers of x^2 and not x. The Mass to Energy transformation therefore is set in a context in which the particle has a real radius. Although this effect may be small it will increase as velocity reaches light speed.
Distortion of Real Space
In figure “Mass to Energy 7” below the crossing points for the two energy transformation curves are shown.
A close inspection of the energy curves described does show some anomaly. However there may be two explanations. Firstly that space is compelled to deform as a consequence or, secondly, that some other process is necessary to operate at the same time (as it does in the V wave). The fabric of 3D space should resist deformation while the internal mass to energy transformation does not fill the whole particle with energy (say up to 90% diameter?). Beyond this critical point the fabric of 3D space may not immediately deform but the forces upon it should start to initiate this effect. A later examination of the Schroedinger Equation and the V wave suggests that the V wave acts as a free particle with no outside forces upon it (or I might had in a state of equilibrium?) so distorting space by travelling near light speed should upset these results.
Now when x< 0.9326 then the internal energy created by mass transformation (shown by the pink curve) is less than the external energy described by Einstein shown by the blue curve).
When 0.9326 < x < 1.1644 then the reverse situation exists.
So it is a matter of interpretation of what value of x represents light speed.
One possibility is that x = 2 represents this point. However if this is true the balance between the internal energy and the external energy reverses again when x > 1.1644.
I suggest taking the maximum value at x =1.1644 to represent conditions at light speed. This needs more clarity. One consequence of saying this is that beyond light speed (when x > 1.1644) then the energy transformation continues and seems to set in place the restoration of space/time.
In this respect the early choice of using the value x = 2 to represent conditions at light speed allows for continuity with generating the V wave from the Riemann Circle. I suspect that this is a consequence of using the conversion factor of 1.33 and a better fitting of these two energy conditions will clarify the situation better. The best explanation that explains these curves is that they are not entirely responsible for deforming space and that my F wave is involved. If my Grand UnificationTheory is accepted then the values between 2 > x > 1.1644 represents a region of space that is deformed and these curves themselves struggle to show what is happening.
If we set x = 2 as the peak of the F wave in 4D, x = 1 (note not x = 1.1644) as the inflection point or axis of the F wave and x = 0 as the peak of the F wave in 3D then the position is clearer. It is not surprising now that both Einstein’s and my curve (and the difference) are all showing a need for an infinite amount of energy at x =2. This is because the particle would be completely in 4D. Again, it is debatable if the energy actually needs to be infinite or the term “infinite” can be used to describe such a loss. According to the F wave the 3D and 4D energy will balance. I prefer to explain this by saying that this large amount of energy sets in motion a pendulum effect establishing phases for the V wave. It also explains why we can set a finite limit for the peaks of the V wave.
Restoration from an Infinite Asymptote.
The restoration from an infinite asymptote or creating a reaction to an infinite force is an unusual problem. Defining all asymptotes as following the curve y = 1/x as x approaches infinity will not work (at least in this instance) as the entire system represented by each curve has to be perfectly balanced with an opposing force.
The only recourse is to use the fact that f(x)* f(1/x) = 1. The results are shown in figure “Asymptotes 1”.
The reaction for Einstein curve is shown as a blue ellipse while the reaction for the V wave energy is shown as a black straight line. I have included the red parabola to show that we must take logarithms of all the terms in the V wave equation. The complex plane has distances that are logarithmic. So this suggests that the reaction must originate there and indeed y=1 sites the complex plane too.
The blue ellipse has an eccentricity of 1.33. This suggests that the correction factor is necessary to avoid achieving a circle. The circle defines the geometry of 3D. However it is now possible to extend our understanding of the mass to energy transfer process. If the blue ellipse was a circle just at the point where x = 0 then the eccentricity would be 1 here. This suggests that the correction factor may increase as velocity and thus x values increase. This would remove entirely the difference between my equation and Einstein’s.
The intersection of the curves calculated by the Graphmatica programme is at x = 0.9324 and y = 1.423.
With this in mind the tangent for the V wave energy curve at x = 0.935 and y = 1.42 is given by the equation y = 1.91x - 0.3793 (as accurate as the Graphmatica programme allows). For Einstein’s energy curve at x = 0.935 and y = 1.42 is given by the equation y = 1.52x - 0.0185. So at this point a new eccentricity for the restoring ellipse is probably between 1.52 and 1.91 (1.72).
In actual fact I discovered that using an eccentricity of 1.31 brings both curves in nearly perfect alignment between 1.4 < y < 2. These results are shown in figure “Asymptotes 2”. So the eccentricity is a critical factor. If we increase the eccentricity factor to, say, 1.72 the equation becomes:
y^2 = 1/(1 - (x /1.72)^2)
These results are shown in figure “Asymptotes 3”.
So allowing for the eccentricity to increase allow the curves to match each other for increasing y values and may be dependent upon the F wave or some other unknown factor to be identified. There does appear to be a limit of 2.0 for the eccentricity. The intersection for the curves if we allow for this eccentricity is at x = 1.9986 and y = 27.1607 beyond this the asymptote will increase very quickly for 1.9986 <x< 2.
This would allow the phase lengths of the F wave to double in size from 3D to 4D along the x axis and its direction of travel.
The Rate of Change in Mass to Energy
The dynamic process of using an elliptical force will balance both equations.
However if I can show how the difference in the two processes in controlled by the elliptical force then I can be much more precise.
In the figure “Mass to Energy Rate of Change 1” I show the two processes without any elliptic correction.
In the figure “Mass to Energy Rate of Change 2” I show the two processes with an elliptic correction. This produces an unexpected result the same dark blue graph is matching the asymptotes of both of the other curves at the same time!
In the figure “Mass to Energy Rate of Change 3” I show the elliptically corrected process in pink against the green V wave energy curve.
My computer is actually struggling to draw the curves but we can see the forces will balance dynamically as I stated. What is unexpected is that this presents a new question of what exactly is the elliptic forces doing.
I believe the answer is that the situation is ‘fluid’ between 3D and 4D. This arrangement is more complicated than can be explained from these last three diagrams alone but nevertheless they show that a perfect balance can be achieved.
I was hoping to show that the rate of change followed Einstein’s equation and so it does according to the graphs but only in undistorted 3D. As 3D is warped into 4D the rate of change and application of the elliptic force depends upon the V wave. This is reasonable when internal and external force considerations are taken into account. Another explanation is that the 2:1 ellipse suggests an orthogonal effect with 3D force acting parallel to or along the minimum elliptic diameter and the 4D force acting parallel to the largest elliptic diameter.
Conclusion
The restatement of the mass to energy exchanges inside a particle can be described more accurately by reference to the V and W waves (the W wave being a small V wave which in combination form a V wave).
The radius in the Error Function integral was summed to infinity while a particle has a fixed diameter. A re-adjustment of these limits does not now seem necessary as the equation holds true for the V wave if we take into consideration the F wave.
Very surprisingly, to me, I found a way to reconcile my Mass to Energy transformations to those of Einstein. That means we can reconcile the interior and exterior processes together if we recall that the V wave does not just describe mass to energy transformation as does Einstein’s but includes the second simultaneous process of to energy to mass transformation.
This result is derived purely from mathematics and not experimental data.
It is also worth noting that the choice of using the variable x is substantiated with the parabolas having asymptotes at x = +2 and also at x = -2.
My previous equation for my V wave now needs correcting so that it conforms to that of the Error Function. This will bring my work completely in line with that of Einstein (albeit that I have used a correction factor of 1.33 that remains unidentified).
The discrepancy between these two energy processes do allow this hypothesis to be integrated with my work for the Riemann Circle.
The mass to energy transfer process is more complicated than these curves suggest but fortunately the F wave acting sinusoidally between 3D and 4D presents us with an explanation that completes the picture.
In the future a further investigation into the eccentricities for my V wave energy equation will yield a better understanding of the rate of change for the eccentricities.
The right hand side of both equations are interesting too as 2y is the differential for x2. This suggest the build up of internal energy as its effective radius increases is indeed accumulative as it balances the external energy.
