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The Third Wave
A Redefinition of Gravity

Part III
The Inverse Square Law

by Miles Mathis

In this section of the paper I will address an extremely important question regarding the theory of expansion of mass. This question is in regard to the diminishment of the “field” as the square of the distance. Opponents of expansion theory often bring up this problem early on in their argument. They say that expansion theory does nothing to explain the inverse square law. Of course current theory does not explain this either. It is simply a “finding” of current theory, an empirical fact. The force of attraction diminishes with the square of the distance. It is known to be true and so no one even attempts to justify it with a deeper explanation. What explanation could there be? At a certain of level of any theory you stop asking why and just accept that the universe works in that way. This is what both classical theory and General Relativity do at this point. They accept it as an empirical fact and offer maths that express it. If you ask someone who accepts the Standard Model to explain why force obeys the inverse square law, you get no answer. It is equivalent to asking them why electrons exist. They do, that’s all.
      Until recently, I thought this way as well. I was not prone to question the inverse square law because I accepted it like I accept the existence of the Earth and the existence of electrons. You don’t ask why the Earth is here—as a scientific question. You may ask it as a religious question, but science can give no possible answer. Why is the universe here? This is question so far beyond us that even if it is a physical question, it must appear metaphysical at this distance.

However, upon reflection, and after doing some rather simple math, I became grateful that my opponents had (rather rudely) demanded an answer. It allowed me to discover an answer to a question I had not even considered, and this answer allowed me to present one more demonstrable point of superiority of expansion theory to current theory.
      My opponents are correct in one sense: expansion theory does demand an answer to the question a bit more forcefully than current theory. Current theory keeps the question in the shadows. Current theory does not claim to solve gravity with real acceleration of the body outward, so the question does not come up without some digging. The question does not come up, and so the solution does not come up either. The whole problem is just ignored, as it has been ignored since the time of Newton. It is assumed to be unanswerable, so no one does any math to answer it one way or another.

At first glance, it seems to be a big problem for expansion theory. This is because at first glance what you do is imagine two objects close together accelerating at eachother and then imagine two objects very far away accelerating at eachother. There doesn’t appear to be any physical difference. Therefore the inverse square law is unexplainable. However, it isn’t that simple. Unless you have a lot of previous experience with accelerations and with Relativity, it is difficult to see that there is a difference in the two situations and that the difference yields the inverse square law. Most people will have to do the math to see it. And most people who criticize expansion theory are not ready to do the math. They are satisfied with their gut reaction and they do nothing to settle the question one way or the other. For these people I offer the math below.

It is fortunate that I have kept Relativity, because we need some post-Newtonian findings to solve this one. We don’t need a lot of difficult equations, we just need reminding of some primary postulates of Relativity. The first postulate we must be reminded of is that time and distance are theoretically interchangeable, in some ways. What I mean is that a distance separation implies a necessary time separation. In the time of Newton, this was not true. If you were a long way away from point A, that did not mean you were a long time away from point A. Newtonian time was universal time, so that you could think of all points being no time away from eachother. Light was thought to be infinitely fast, so that you could actually see what was going on at a distance immediately, proving the lack of time separation.
      Now we think of time differently. Every distance separation is also a time separation. This being true, we must look more closely at the inverse square law. The inverse square law says that gravitational accelerations vary as the inverse square of the distance. No one has yet applied Relativity to show why this is so, but expansion theory demands that I do it now. First of all, notice that we are comparing accelerations to distances. As distances double, for instance, accelerations must quarter. But to compare distances to accelerations, we must also be aware of the time intervals in the comparison. Acceleration always has a time component, and with Relativity you cannot assume that the times are equivalent from one example to another.
      Consider the comparison of a nearby planet to a distant planet. In comparing accelerations before Relativity, you just put both planets in the same coordinate system and did a straight comparison. You can’t do that anymore, because the more distant orbit has a time separation from the nearer orbit. Let us say the near orbit has a radius of x and the far orbit has a radius of 21x. The distance separation is 20x. The time separation is t = 20x/c. The times at which the two orbits exist are not the same. Therefore you cannot do a straight comparison of accelerations.
      In expansion theory, we give the acceleration to the Sun, not to the orbiting planet. Therefore we are talking about a primary event that is taking place at the Sun. That event arrives at the near orbit x/c seconds after it happens at the Sun, and it arrives at the far orbit 20x/c seconds after that.
x = ct
      But to compare accelerations and distances, you have to compare them at the same time. How do you do that? There is no universal time to set your watches by, so you just pick a time arbitrarily and measure from there. In this problem it is easiest to set your time from the near orbit. It is easiest because it allows us to imagine that the near orbit is the orbit of the Earth and the far orbit is Saturn, say. We are now in a comfortably familiar situation, and we can look at the solar system from a perspective that is our very own.
      The easiest way to get our numbers is to let the event at the Sun reach the orbit of Saturn and then work back. This will help us visualize the whole thing. So at t0 we have an event at the Sun, which is the Sun gobbling up some distance in some dt. We mark that particular distance with a bit of red paint or something so that we can differentiate it from all other similar events.  x/c seconds later the event arrives at the Earth. 21x/c seconds after the event, it arrives at Saturn. But we want to compare two events that happen over the same Δt or at the same dt. Otherwise we can’t do a meaningful comparison. If we compare what is happening at the Earth with what is happening at Saturn 20x/c seconds later, we are not learning anything useful. Or, even if what we learn might be called useful, it can’t be called equivalent to what we are trying to learn and what we have always assumed we were learning. What we have always assumed we were learning with the inverse square law is how two different forces or accelerations looked from the same place. Newton’s equation assumes that all measurements are made from the Earth, or at least from the same place. Newton’s inverse square equation describes what happens when you measure or calculate the acceleration of the Earth from the Earth and the acceleration of Saturn from the Earth, making the measurements at the same time. It does not describe the situation of measuring the acceleration of the Earth from the Earth at one time and the acceleration of Saturn from Saturn at some later time. You must understand this in order to understand the inverse square law. We could measure both events from Saturn as well: I am not proposing a geocentric theory. I am just reminding the reader that both classical mechanics and, even more, Relativity demand precision is choosing points of view and sticking with them.

Now we just choose our Δt. We choose the interval when the red paint event reaches the Earth. That interval is 20x/c seconds before the event reaches Saturn. You may say, “If the event hasn’t reached Saturn, how can we compare accelerations and distances?” Fortunately, the Sun was expanding before and after our red-paint event. We have an infinite series of expansions to choose from, you see, all with the same acceleration if measured by the Sun, but gobbling up slightly different amounts of space. Remember that if you go back in time, the Sun will have been a bit smaller, and, although it had the same acceleration, it covered a bit less distance over the same time. That is what acceleration is.
      This means that the event that Saturn is experiencing during our chosen Δt is some event before our red-paint event. So, during our chosen Δt, we experience the Sun gobbling up some Δx; but Saturn is experiencing an earlier Sun gobbling up some smaller Δx. And, if we did the right math, we on Earth would be calculating correctly what Saturn is experiencing during that Δt. If we have Saturn experiencing a smaller Δx over the same Δt, then Saturn will be experiencing a smaller acceleration than the Earth over that interval.

We have already proven half of what we set out to prove. My opponents, judging at a glance, came to the conclusion that all distances would be feeling the same acceleration. I have shown that is not the case, just by applying the most basic finding of Relativity. I have shown that if you increase the distance, you decrease the acceleration; but only if you measure all the distances and accelerations from the same time and place. Now all I have to do is show that the acceleration decreases with the square of the distance. I hope you can see that that is not too hard to do, knowing what we now know.
x = (½)aΔt2
a = 2x/Δt2
      Already we can see that acceleration varies as the inverse square of the time. And we have an equation to relate time and distance.
x = cΔt
Δt = x/c
      So all we have to do is a little switch of variables.
a = 2x/Δt2
a = 2cΔt/(x/c)2
a = 2c3Δt /x2
      c3 is a constant and so is Δt in our problem above. Therefore acceleration must vary as the inverse of the distance. The inverse square law is simply an outcome of the definition of acceleration. We have a time variable squared in the denominator of the basic acceleration equation. Since time and distance have an equivalence and since Relativity allows us to express that equivalence with an equation, it is clear that the acceleration equation must also give us a variance as the inverse square of the distance. Of course the time must be invariant for this to be true, but in our problem above the time was invariant simply because we did all measurements and calculations from the same place. Time is always invariant locally.

There are longer ways to prove this, and you can do other (longer) mathematical proofs if it appeals to you. I think you can see that I chose the most elegant way to get to the end. This allowed me to do a lot of necessary explaining and very little math. Considering the difficulty of the problem, I think this was the best way to go. Pages of differential equations or Lagrangians or what have you would have kept the problem abstract and would have prevented the clear visualizations I have offered here. It is a beastly difficult problem to approach, but it is a simple problem to leave behind you.

One other amazing thing to notice before you move on to the next question. Because we solved this problem using Relativity, we find that we cannot reverse it and get the same answer. It is true that Relativity always allows us to find transforms from one system to another, but it does not guarantee that we will find the same numbers or relations if we switch the accelerations. In this problem, if we give the acceleration to the planet instead of to the Sun, the inverse square law cannot be derived!
      If this is not clear, let’s run through the logic and the math. The events must now take place at the planets. We will start with Saturn. Saturn either feels a force from the Sun or simply has an acceleration vector toward the Sun given it by the curved space it inhabits. Either way the primary event is at Saturn. The event is a motion, and in the Standard Model, Saturn is moving due to the field; the Sun is not moving. The same must be said of the Earth. It feels a force or curves in its space, and its event is at the Earth. Now, we want to compare Saturn’s event with Earth’s event, and we must compare them at the same time. This is impossible to do since we now have two “central” events. To find some link between the events, we must find a common background. The light-time between the two events is not enough information to provide a link, since we still don’t know when the two events took place relative to eachother. We can’t yet compare the dt of Saturn to the dt of the Earth. The only way to get a link is to assume that the Sun is influencing them both, which is what all theories of gravitation do and must do. If we believe in GR, we either believe that the Sun is emitting gravitons or that the Sun is warping the space. This puts the central event back at the Sun and allows us to connect up all the orbital events in a common background. In this case we can use the light-time difference between the two orbital distances to give us relative times for the orbital events. Once again Saturn’s event must take place 20x/c seconds after the Earth’s. Let us assume that the Sun is emitting gravitons as our event (it could just as soon be a dt of spacewarp as our event, but the graviton example is easier as a visualization). x/c seconds later the Earth receives its gravitons, and 21x/c seconds after emission Saturn receives some gravitons from the same 360o event. We now have a link and we can compare the two events at the planets.
      If we do all measurements from the Earth, we notice that once again we must be seeing or measuring Saturn’s motions 20x/c seconds in the past. But this time it doesn’t matter. The Sun is not expanding according to the Standard model, so it is emitting the same gravitons now that it was emitting in the past. The Sun’s influence—whatever it is—is not changing over time.
      This is the problem: we cannot get the acceleration to vary over time. According to the postulates of the Standard Model and the postulates of Relativity, distant planets should be feeling the same influence from the Sun, whatever it is, as nearby planets. There is no concept that would explain why the influence would diminish with distance.
      If the Sun is sending out gravitons or warping space, then either the particles or the warp should proceed outwards without diminishment. There is nothing to diminish them, so they should not be diminished. Light proceeds outward from the Sun without diminishment; why should gravitons or warping diminish? Saturn should be receiving the same gravitons or warp that the Earth receives. Why doesn’t it?
      You will say that the equations are the same in the Standard Model as in my math above. That is,
x = (½)aΔt2
a = 2x/Δt2
x = cΔt
Δt = x/c
      Why should the inverse square law work for expansion theory and not for the Standard Model? Because these equations show that if there is a variation in acceleration, it will vary as the inverse square of the distance. But the Standard Model cannot show any variation. According to the standing concepts, there should be no variation in acceleration. Saturn should feel the same influence as the Earth and should have the same acceleration, whether you postulate gravitons or warping. Adding Relativity to the situation does not help, as I have shown, since past influences are just like present influences with a stable Sun.

It is not expansion theory that fails to explain the inverse square law, it is the Standard Model. For the first time in history I have derived the inverse square law from first principles, and from motion alone. And I have shown that the Standard Model cannot explain the inverse square law. In fact, it is logically impossible to derive the inverse square law from the Standard Model. With the Standard Model, the law can only be an empirical fact: one that contradicts the Model itself.

All this stands as empirical proof of expansion theory. We have mountains of physical evidence that the inverse square law is true. All gravitational theory has been built to express this law in more and more precise ways. But the Standard Model cannot explain the law. It can exist only by hiding the conflict between the law and the current math.
      Before Einstein, this problem was insoluble. Without the equation x = ct, no classical physicist could have hoped to solve this problem, or even recognize its existence. The inverse square law could only be accepted as a fact. No possible deeper concepts could explain it. Since Einstein, Relativity has given us a clear method of solving the problem. If Einstein had applied his first concepts to the gravitational problem directly, as I have, instead of getting involved in tensors, he might have solved the problem almost a hundred years ago. For some reason, he didn’t. He quickly became more interested in a complex mathematical representation of the field than in a conceptual clarification of the field. I cannot know why this is so. Possibly he was not able to simplify the field as I have done here. Possibly he saw all I have seen here and suppressed it as politically unfeasible. Mathematicians were powerful and insistent in the early part of the 20th century, and entry into the field was possible with esoteric math, not with “harebrained schemes” like expansion, no matter how much it could explain. The milieu then, as now, demanded cutting edge math and dense and abbreviated text. Perhaps Einstein judged correctly that this would impress where clear and simple math and full explanations would not.
      Whatever the case, General Relativity has successfully hidden the problem for almost a century. The inverse square law is thought to be a Newtonian concept, stale and passé. Force and acceleration have also been superceded as concepts by action variables, and the contradictions of current theory are cloaked in every possible way. The problems with Newton’s theories have been buried, and the one theory that might have solved these problems has only served to bury them further. It is apparently thought by some that using action variables supercedes the old problems. But action is built on a compression of Newtonian variables: it inherits all the problems of its Master. Be assured, gasping under the weight of the tensor calculus and the Hamiltonians and Lagrangians and so forth, this problem survives to the current day. The inverse square law points unerringly at expansion theory and away from the Standard Model.

Click here to go to Part IV.

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