universal gravitational constant

the
UNIVERSAL GRAVITATIONAL
CONSTANT
(connecting the number 108 in accelerators
to G using a suggestion by Maxwell)

It occurred to me some time ago that the Universal Gravitational Constant—usually signified by the variable G—might be the key to unlocking the secret to gravity, among other things. It has always seemed puzzling that a constant should have so many unexplained dimensions. A complex constant like that is normally a sign of incomplete theory. All the known concepts are assigned variables and the unknowns are lumped together in a constant. The numerical value is not such a puzzle, since it may just be an expression of incommensurate initial definitions. For instance, we chose the length of the meter and the second and so on pretty much arbitrarily, so it shouldn't be a surprise when all our numbers don't match up at first. But G is not just a number. It has lots of dimensions, L³/MT². Could there be a secret locked up in those dimensions?
I am not the first to ask that question, but no one has yet presented us with any major secrets. Historically, the door for serious questioning was not open that long. Newton's theory became dogma so quickly that very few scientists had the gumption to look hard at it. The ones who did found it mostly convincing or mostly opaque. Since the time of Einstein, no one has taken the constant seriously. It is a piece of discarded and superceded math. To the contemporary physicist, G is about as interesting as the constants of Archimedes or Toltec hieroglyphics. Einstein gave us a new math to express the gravitational field, leaving the mysteries of Newton behind. But Einstein's new math and theory did not dispense with the old mysteries. In many ways it simply changed the text of the mystery. It substituted a new problem for an old one.
This paper is not concerned with critiquing the math of General Relativity. Here it is enough to point out that the mechanism of gravity is admitted to remain a mystery to this day. Relativity describes the gravitational field in ways that are mathematically superior to Newton. It cannot be denied that Einstein at the very least updated the math to include the finite speed of light c—a constant that was unknown in the time of Newton. The finite speed of light implied a difference in measurement of variables between an object and its observer, as well as a difference among various observers, so that even Newton would have admitted the necessity of a mathematical update. By returning to Newton's equations I am in no way questioning the truth or usefulness of Relativity as a whole. I feel I must preface every one of my gravitational and relativistic papers by saying that I am convinced beyond any doubt of time dilation, length contraction and mass increase; I will say it again here. I am returning to Newton's gravitational math not to argue for its historical superiority, but only to answer questions that have remained even after Einstein. No one denies that these questions remain; no one denies that gravity remains mysterious in many ways. Nor will anyone deny that gravity has resisted being incorporated into QED or unification theories.
In this paper I will show that by studying the foundational theory of gravity a bit more closely we can arrive at a better understanding of both mass and gravity. By doing simple algebraic operations on Newton's equations we can derive new knowledge. This knowledge will allow us to discover many things that have so far been hidden. The most important of these is tying the classical equations of Newton to unexplained numbers coming out of particle accelerators. In this paper I will provide the mathematical link between Newton's classical equations of gravity and the equations of mass increase of Einstein. In doing this I will mathematically derive the limit for mass increase for the proton. Until now, this experimental limit has been a mystery. Neither Relativity nor QED has been able to explain the number 108 for the ratio of moving mass to rest mass for the proton. I will derive it with simple high-school algebra and a few simple theoretical postulates.

To begin our inquiries, I find it is best to start with a new thought problem. I have solved many other old problems with new thought problems, and I will do so again here. In Newton's original problem there were several unclear points in the definitions or postulates. One of these was whether the distance r included the radius of the large mass. The variable r is supposed to be the distance between two masses, but if the larger mass is very large, its radius comes into question. Another unclear point concerned the smaller mass. In many calculations it is ignored because it is insignificant compared to the larger mass; but you cannot allow it to be zero, for obvious reasons. To avoid getting into the historical discussion of these points, I will offer a thought problem that gets around them completely. In doing so, the thought problem will also bring other things to light.
Let there be two equal spheres of radius r touching at a point. We know that according to the theories of Newton and Einstein there must be a gravitational force at that point, but neither math allows us to calculate it. Newton's math cannot apply since there is no distance between the objects; Einstein's math cannot apply because there is no field at a point. Both theories solve this problem in their own ways, it is true. They add further theory that allows them to calculate in this predicament. In a nutshell they both propose a field centered about a point or a singularity. This causes further problems due to the fact that the objects' gravitational strengths are determined by their masses, and all mass cannot be found at a point. By current theory, mass resides in matter, and matter is made up of atoms. These atoms have real positions: they are found throughout the object—at its outer shell just as at its core. If the mass is a summation of atomic masses, then the force must be a summation of atomic forces. It is difficult to see how the center of force can be behind (in a directional sense) half the mass that causes the force.
We can bypass these further theoretical questions by continuing to propose simple new theory. To do this, let us move our twin spheres s distance apart for a moment. If there is a gravitational force, then after a time interval Δt, this distance will diminish by Δs. Why has the distance diminished? Because a force between the two spheres pulled them closer—this is the classical and current interpretation given to the situation. But can we give it another interpretation? Yes, we can say that both spheres are expanding and that they moved into the distance between them. By the classical interpretation, the centers of the spheres moved toward eachother. By my interpretation, they did not.
With my change in theory, you can see that we no longer have to assign Δs to the diminishing distance between the spheres. We can assign it to a change in the radii of the spheres. This being so, we can move the spheres back together, touching at a point. After a time Δt, the radius of each sphere will have changed Δs/2.

We have changed the idea of gravitational distance in our theory; now let us look at the idea of mass. In article 5 [chapter 1] of Maxwell's Treatise on Electricity and Magnetism, he tells us that mass may be expressed in terms of length and time, in this way: M = L³/T². He derives these dimensions from a simple substitution into two classical equations.
a = m/r²
s = at²/2
m = 2r²s/t²
Notice that L³/T² may be thought of as the acceleration of a volume, or a three-dimensional acceleration. This is very suggestive.
This passing idea of Maxwell caused me to reconsider the concept of mass. His math is true, except for one thing. His first equation is not really correct. As written it should be a proportionality. To be an equation requires the constant G.
a = Gm/r²
The dimensions of G are L³/MT², which gives the mass and acceleration the correct current dimensions. But what if G is a sort of mirage or misdirection? To pursue this further, I went to Newton's gravity equation, like Maxwell had.
F = Gmm/r²
ma = 2Gmm/r²
a = 2Gm/r²
We must have a 2 on the right side, since the force equation for gravity is the force between two masses, but the force that causes an acceleration on the other side of the equality applies to only one of the masses. It is customary to give all the acceleration to one of the masses, but in my thought problem the two equal spheres both accelerate. Now let us apply this equation to our twin spheres touching at a point. There is no distance between the spheres, so r would normally apply to the distance from center to center. But since the spheres are the same size, let us re-assign r to the radius of each sphere. The distance from center to center is then 2r. We have assigned Δs to a change in the radius instead of a change in the distance between the spheres, and this allows us to calculate even when the spheres are touching. For clarity let us make Δs into Δr.
a/2 = Gm/(2r)²
a/2 = Δr/Δt²
Δr/Δt² = Gm/r²
The only remaining problem is the variable r. If the spheres are expanding, then r must be expanding. After time Δt, the radius will be r + Δr. After any appreciable amount of time, r will be negligible in relation to Δr, so that Δr ≈ r + Δr. Therefore we may simply drop the r variable as a variable that approaches zero.
Δr²/Δt² = Gm/Δr
m = Δr³/GΔt²
Now all we have to do is reassign the dimensions of L³/T² to the mass, as Maxwell implicitly suggested. We will drop the dimension M altogether. This gives G no dimensions at all. It is just a number. This is actually much more sensible, since constants with dimensions are a sign of incomplete theory. That is what drew me to this solution in the first place. Newton had to give the dimensions L³/MT² to G only because he had mistakenly assigned mass a new dimension. Mass is not a new dimension. It is reducible to the old fundamental dimensions of length and time.
Our last problem is plugging known values into this new equation. At first it looks like the mass should be changing over time, since the radius is changing. But no. The mass is dependent on Δr/Δt², and that is not changing over time. As the radius gets larger, so does the change in time, so that the ratio is constant. It is a constant acceleration. A constant acceleration gives us a constant mass. Therefore we can plug known values for m into this equation.
m = Δr³/GΔt²
a = 2Δr/Δt²
a = 2mG/Δr²

That is the acceleration of each of two equal masses in a gravitational situation. But if we want to give all the acceleration to one of them, holding the other one steady for experimental purposes, then we simply double the value.

a = 4mG /Δr²
If the proton has a radius of 10^-13m*, this yields
a = 8.88 x 10^-12m/s²

We are now in a position to use our new number for acceleration to explain a current experimental mystery. Using the new number to do this will also act as proof of my theory, since it gives us a sort of experimental confirmation.
If the proton has a fundamental spherical acceleration,* then in any one direction it will have a velocity at any given time. If we suppose that the age of the proton is on the order of the age of the universe, then we can estimate the current velocity of the shell of the proton. "Velocity relative to what?" you may ask. "If everything is expanding, then what is our background?" The velocity we will find must be relative to two things. It is relative to the velocity of the radius at t₀, which we define as zero. And, it is relative to the speed of light, c. Einstein defined the speed of light as the universal background, and I continue to accept that definition.
If we accept (one of) the current estimates for the age of the universe as around 15 billion years, then the current velocity of the proton's shell would be 2.1 x 10⁶m/s.
v = at/2 = (8.88 x 10-12m/s² )(4.73 x 10¹⁷s)/2 = 2.1 x 10⁶m/s
That seems ridiculously large at first, except that we have experimental confirmation of that number from accelerators. As I have shown in my paper on accelerators, there is a limit to the speed achieved by the proton. This limit is a final energy of about 108 times the rest energy. Using gamma, this translates to a velocity of .999957c, which is 1.2 x 10⁴m/s short of c. If we theorize that the gap between c and the limit in velocity is caused by a residual velocity or velocity equivalent that the proton already has, then the limit is explained. That is the link between this paper and my paper on the accelerator (last secton).
But there is more. My correction to gamma and to the mass increase equations predicts a limit in velocity for the proton of .9930474c, which is 2.1 x 10⁶m/s short of c. This is an exact match, as you see. If we plug the current form of gamma into my acceleration equation above, we get an age of the universe of only 85 million years. But my correction to gamma gives us an age of 15 billion years. We know that protons must be older than 85 million years. The earth is almost 60 times older than that itself.

Now let me address the first outcries. Some will say that my equations above give us a current radius for the proton of .7 meters. But that is bad math. You have to do a proportionality. As much as .7m is larger than x, x is larger than the proton originally was. If we say that the current proton radius is 10^-13m then the original proton radius would have been
.7/x = x/r
r = 1.43 x 10^-26m

The next problem concerns my claim, just made, that velocity is constant. A velocity, such as the speed of light, will remain constant in an expanding universe simply because time is a function of distance. What I mean is that we define time in relation to distance. If this definitional distance increases, as everything expands, then the definitional period of time will increase proportionally. Distance gets larger, time gets "larger". So the ratio of the two stays the same. Which means that all relative velocities will stay the same.
As an example, we now use the cesium atom to define time. The baseline data in the cesium atom is an oscillation from one energy level to the other, or an atomic wobble. This oscillation is a motion, and all motion implies a distance. If the cesium atom gets bigger, then the distance increases, and the time period increases. Time is dependent on distance. This is even clearer with a pendulum clock. If all material lengths increase, then the length of the pendulum will of course increase, which will increase the length of the second. Time is connected definitionally and operationally to distance, therefore any increase in universal length will cause a proportional increase in universal time. Since velocity is defined as one over the other, velocity will not change. The numerator and denominator both get bigger at the same rate.

Of course there are many other questions that have to be answered, but I have answered them in another place and will not repeat myself here. The purpose of this paper was to connect my use of Maxwell's hint and Newton's constant G to the number 108 in the particle accelerator. I believe I have achieved that, and I will recommend the reader to my theoretical papers on universal expansion and gravity for further explanations.

*In another paper I have found the current estimate of 10^-15 too low, so I don't use it.
**This is actually a Unified Field velocity caused by both acceleration due to gravity/mass and the E/M field. However, this is indeed the velocity we want here, since both fields will affect the limit in an accelerator, and the number 108.

Go to Part II of the Third Wave.

[If this paper was useful or pleasing to you, please consider paying one dollar as a reader's courtesy. This will allow me to continue my research in a completely independent manner.]