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WHY CELESTIAL MECHANICS IS DETERMINISTIC
by Miles Mathis milesmathis.com email:mm@milesmathis.com
This is another paper that would give Laplace fits, were he to read it from the grave. I would pull him in with that title, which he must agree with, and then dash his ghostly little head to the rocks by showing that his math is what caused the need for perturbation theory, chaos theory, and the modern belief in indeterminism.
Laplace is still famous for his quotes on determinism. Laplace's demon is now a cliché, being an allknowing demon that could predict future events by present relations. He could do this only in a mechanically determined universe, where perfect math was possible. Laplace believed in determinism, but he was only able to improve Newton's equation. He was not able to perfect it.
My rigorous analysis of Newton's equation has allowed me to discover that the reason Celestial Mechanics was and still is resisting a precise math is that the equations and theories were never correct to begin with. The theory wasn't even nearly correct, and the math was only an approximation. An incorrect theory and an approximate math are bound to appear to be indeterminate, because they will never get the right answer. They won't get the right answer, no matter how many extra mathematical manipulations you make, because they don't contain the right fields or the right variables acting in the right ways.
An incorrect equation will always appear to be indeterministic, because, unless the fields and variables contain very little complexity, you cannot discover a logical fix that will perfectly fill the gap. If you found a logical fix that perfectly filled the gap, you would have corrected the equation. Your fix in that case is not a fix, it is really the replacing of the incorrect equation with the correct one. But in the case of the threebody problem and all more complex problems, no correction of this sort has been found. Laplace's fix was just a fix, and all perturbation and chaos theory has been a further fix. No one has ever corrected Newton's field and equation from the foundational and mechanical level.
Which is all to say that Celestial Mechanics is not indeterminate, it is incorrect. We have been told that we cannot get the correct answer, even now, because real life problems are chaotic. But the real reason we cannot get the right answer is far simpler than that. The reason we cannot get the right answer is that we do not have the right math or the right theory. What we currently have in Celestial Mechanics and the threebody problem is a partial theory and a few general equations, given us by Newton and Kepler. Then we have ingenious mathematical fixes to those general equations given to us by Laplace and others. Then we have further fixes, and the systematizing of those fixes, by Poincaré and others, under the titles of perturbation theory and chaos theory and so on. Then we have a reshuffling of all those fixes by Einstein, correcting none of them but allowing us to consider the speed of light and the speed of the observer. Unfortunately, the bald assumptions of Newton still lie under all those fixes. And, since those bald assumptions are incorrect, the math cannot be correct. In its current form, it is uncorrectable. No amount of afterthefact pushing can make it correct.
The history of modern physics is a history of hubris. I have already shown that the Copenhagen interpretation of quantum mechanics, which is also indeterministic, is a transparent effort at denial. It is the totemizing of the inability to admit defeat. These quantum physicists could not bear to admit that their theory and math were partial and incomplete, so they assigned the partiality and incompleteness to Nature instead. It is not that they were failing to find the correct equations, we are told, it was that correct equations could not be found. Correct equations were impossible. It is not human equations that are incomplete or incorrect, it is Nature that is indeterminate, fuzzy, illogical, chaotic, and unknowable.
The historical reaction to our failure to solve the threebody problem is a similar instance of hubris and the totemizing of denial. It has been a redressing of our failure as success, a hiding of our mistakes by redirecting into new fields and theories. As in quantum mechanics, our deficiencies are paraded as virtues. We are told that Newton was right, save for relativity, and that Laplace perfected Celestial Mechanics. The remaining error is not a matter of anyone making a mistake, the error is actually assignable to Nature herself, who is chaotic. A perfect math is an indeterminate math. We are so smart we can even measure the indeterminacy. We can devote entire fields to measuring how fuzzy Nature is.
That has been the sales pitch for at least a century now, but it is no longer convincing. It is time to move on. We can get back to work. As I showed in my paper on Laplace, the problem embedded in his equations is that he doesn't have a way to express the charge field that is already inside Newton's equations. He lacks a fundamental degree of freedom, so that his equations must continue to misfire even after all his fixes. It is not that the bodies in motion are indeterminate, it is that his math cannot determine the motions of the bodies. For all its improvements, his math is still failing.
Laplace's math fails because he approached the problem mainly as a mathematician, rather than as a physicist or mechanic. The biggest problem he worked on, that of the resonance between Jupiter and Saturn, had an obvious clue in it, one that should have allowed him to spot the foundational error. Although the two bodies come nearer in the resonance, they do not collide. Rather, they begin moving apart again. With gravity alone, they could not possibly do that. Laplace should have corrected the fields, and this would have given him the mathematical correction he needed. Instead, he tried to create a math to match the motions directly, with no explanation of the causes of the motions. This made his math much larger than Newton's. Laplace's math, like all math since, has been a math of accretion. New manipulations are pasted over the old, with no corrections of what is beneath. But what was required was a smaller, more elegant math. Laplace needed to pull apart Newton's mass equations to show the fields beneath them. Then these fields could be reassembled in a more transparent manner.
I have made the first and most important steps in that direction. I have separated out the two fields that compose Newton's equations. I have shown that Newton's gravitational equation is really a unified field equation, that it already contains the charge field, and that G is a scaling constant between the two fields. This means that, if we look only at the twobody problem of the Moon and Earth, we now have four fields: the gravity field of the Earth, the charge field of the Earth, the gravity field of the Moon, and the charge field of the Moon. Since the charge field and the gravity field do not change with distance and time in the same way, the unified field cannot be followed correctly by field equations that compress the two into one variable. Newton's equation can be corrected only by reexpanding it, and monitoring both fields separately.
In the threebody problem, we must monitor six fields. In a fourbody problem, we would have eight fields. And so on. All these fields are spherical, and all must be monitored while following the rules of Relativity. This makes the problem very complex, but does not make it indeterminate.
I have not yet solved any restricted threebody problem*, though that may be my next project. But I assume going in that my restrictions should make the problem determinate. In the nonrestricted threebody problem, I would assume that any failure to match equations to data would be caused by factors I had not incorporated into my math, not by chaos.
*[I have now achieved a new limited solution of 3, 4, and 5body problems, correcting Newton and Laplace in a straightforward way, proving my claims in this paper.]
Since we have discovered that Newton's equations were not correct to begin with, I think we should start over from the beginning. We should go back in time and assume once again that Nature is deterministic. We should try to match our new equations to Nature, and assume that any failure to do so is caused by complexity, or by our own disabilities as thinkers. Only after we have spent another century or so in our second attempt to solve the threebody problem should we consider the possibility that Nature is chaotic.
My corrections and clarifications to Kepler's ellipse should also help us in this regard. By making actual corrections to Newton and Kepler, I have dissolved much of the “indeterminacy” that has been swept into perturbation theory and chaos theory. It is gone. Whether we can get rid of all of it is again an open question.
In closing, I will make a short philosophical point. My claim of determinacy in Celestial Mechanics is not a claim that the universe is determinate or that the future is predictable by mathematical equations. I do not go nearly as far as that. I simply believe that physics, as physics, is best done with the assumption of determinacy. It keeps us honest. It prevents us from the sort of totemizing of denial that has defined modern science. It prevents us from quitting when we don't have the answer. It sends us back to the problem again and again, as it should. Beyond that, I see no reason why “pool ball” mechanics should ever be indeterminate. In the threebody problem, we are not concerned with living beings, with minds, with souls, with spirituality, with creation, with choice, or any of those sorts of factors. Intuitively, I don't see where probability or chaos enters the problem. In my professional opinion, the chaos enters on our end. It is our math and theories that are “chaotic”, meaning, in the end, that they are simply wrong. Our own incompleteness does not in any way imply that Nature is chaotic. In fact, if Nature were chaotic, we would have no way of knowing it. To assume Nature is chaotic, we must first assume that our equations are perfect. Our perfect equations do not match data, therefore Nature is chaotic. We cannot and should not assume that our equations are perfect. We cannot, as a matter of science, because we have no evidence that they ever have been perfect and have lots of evidence that they always have been very imperfect. We should not, as a matter of humility. We also should not, as a matter of efficiency or operation. Science cannot advance with such assumptions of perfection.
From every level of logic, intuition, and morality, it is better to assume that we are incomplete and that Nature is complete, rather than the reverse.
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