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Why Hyperbolic Math
Is Inapplicable to
by Miles Mathis
In this paper I will show direct empirical evidence that disproves a basic axiom of Riemann, Minkowski, and Einstein. It thereby falsifies the tensor calculus and the math of General Relativity. It does not falsify Relativity as a whole, and the intent of this paper is not to attack Relativity. My intent is to correct and fortify Relativity.
The evidence is absurdly simple and may be represented by a situation no more esoteric than a car going in a straight line. The only data I need to present is data showing that a car going in a straight line is capable of accelerating. I hardly think I need to show a graph or to footnote any extensive experiments in support of this. I imagine a reader will give me this data and ask me to proceed. Why does this data disprove the math of General Relativity?
For this reason: in a Cartesian graph representing the acceleration of this car, we would have x and t axes. We would plot x against t and find a curve. Now, I was led to ask why the line on the graph curved and the line on the ground, made by the car, did not. You will say that it is simple. On the graph x and t are plotted against eachother, but the axes are at a 90o angle. This causes the curve. In the physical situation, x and t are not a 90o angle.
That’s right, and my point is already proved.
For the car to be physically able to accelerate in a straight line, it must be true that t and x are varying in that same line. The time variable cannot have any possible angle to x. If the time variable had any angle to x, then the line would curve.
The car is describing a line on the ground. If this is not clear, put paint on the tires and drive for a moment. There will be a line on the ground. I ask you to mathematically explain that line. It is explained in only one way. The x variable and the t variable are in the same line.
You may say that the car is not truly going in a straight line, since the earth curves. But if you say this you are missing my point. When you start talking about the earth curving or the universe being a curved manifold or any of that, you are introducing y and z axes. I am talking only about the x axis, and how t varies relative to it. My proof relies only upon the possibility of driving in a straight line on one axis or in one dimension. If it is possible, then x and t cannot be orthogonal.
Someone versed in pettifoggery but not in logic will now say that it is impossible to drive in a straight line, since the universal manifold is curved in every dimension, according to post-Einsteinian physics. This person will argue that an apparently straight line will look curved from somewhere else, since straightness and curvature are relative to begin with. But this is also a misdirection in argument, since even though time and length are indeed relative, they still have only one value for each event in each frame of measurement. We must choose a frame of measurement, and once we do it is immaterial what someone in another frame thinks. I do not care whether my straight line looks curved to someone else, since I am not using his watch. I am using my watch, which is in my frame; therefore the only x that is important is my own. If I see the line as straight, then it is straight, no matter what you or god think of the matter. This is what relativity implies and must imply, not some mushy inability to measure.
In fact, I don't even have to prove my line is straight at all. Claiming that it looks nearly straight to me is more than enough to destroy hyperbolic math here. If my time moved at a right angle to my x then every accelerating real object I looked at must appear to my eyes to move in a hyperbolic manner. And not hyperbolic in some ultra-slow sense. No, hyperbolic in a quickly curving, immediately obvious sense, since we are assuming that the x and t of the car are changing at values very near to eachother and to the number one. The hyperbola would be a slight curve only if x were very large and t very small, or the reverse. What we have in a straight line acceleration is two x axes and one t axis, all stacked (superimposed), and all in the precise same direction. Any angles, one to the other, would destroy the given situation.
Another problem is encountered when we look at the mathematical definitions of time and motion. Minkowski wants to allow time to move on the imaginary axis, which is orthogonal to the other three. But "motion" is a precise and unalterable thing, mathematically. A motion must either be a velocity or an acceleration, constant or variable. This motion is expressed with distance in the numerator and time in the denominator. But we arrive at a problem if our denominator is orthogonal to our numerator, since we can't express a velocity or acceleration that way. Orthogonal vectors cannot be put into ratios, or fractions. This is a basic rule, and perhaps Minkowski was at too great an altitude to remember it. But it is as true now as it was in the time of Archimedes.
The tensor calculus relies on hyperbolic math, which uses i, the square root of –1. In order to create the field of hyperbolic math, one must assume that the t variable travels orthogonally to x, y, z. That is to say, at a right angle. I have just shown that the t variable does not and cannot travel or vary or change at any angle to x, y, z. Therefore the hyperbolic math is physically false.
You will answer that I have found one exception, but that in general the tensor calculus still may be used. But I have not found an exception. I have just stated the most obvious exception. All cases are exceptions. In no physical case does t travel orthogonally to x, y, or z. All you have to do is actually study the mechanics of any real problem and the mathematical operations that are done to solve these problems. If you do this it quickly becomes clear that t is always internal to x, y, z. It must be since in any particular problem, t is always a second measurement of either x, y, or z.
The math of Riemann and Minkowski was adopted by General Relativity because it seemed to be a poignant way of expressing the problem. Four-dimensional space, where the time variable is treated just like the other three, has turned out to be very seductive to the general imagination. Its mystifications were just the right flavor for the 20th century. But it is time to get back to work. Hyperbolic math is not applicable to physics, since it grossly misrepresents the time variable. If we want to continue to use it for reasons of efficiency, we must do so with full recognition of its conceptual limitations and contradictions. I would suggest, however, that this conceptual falsehood I have just proved is fatal. It has already caused terrible theoretical problems (as I show in my other papers) and it will continue to do so as long as it is used in such a cavalier manner.
As I said in the first paragraph, none of this implies that the other axioms of Special or General Relativity are false. I have falsified only the axiom of hyperbolic math. I have shown elsewhere that Relativity is true, including time dilation, length contraction, and mass increase. There can be no doubt of this, and I am not here to create that doubt. I am here only to point out a mathematical flaw and to recommend the swift correction of that flaw. I recommend the discontinuation in General Relativity of hyperbolic math and of the hiding of the t variable in the concept of i.
Relativity shows the mathematical relation of time to the distance variables, and in this it is not wrong. There is a certain mathematical equivalence between time and distance, as is shown in the equation x = ct. Einstein and Lorentz saw this equation as representing the speed of light in a given coordinate system. Although that assignment of the equation is wrong—since it contradicts postulate 2 of SR—it can be used as a direct transform between time and distance, where c is the very simple transform. But in no case does time travel orthogonally or externally to x, y, z. This should have been known from the start, since the other postulates of Relativity imply it very strongly. If x and t are so easily transformable, in such a simple equation as x = ct, then the two variables cannot be traveling at complex angles to one another. We simply cannot use hyperbolic math with confidence in physical situations.
You will say that hyperbolic math is perfectly applicable, since it allows us to get the right answer. That is to say, it yields the correct number. Even if that were true (it isn't), I reply that applied mathematics must be more than getting the correct number. If applied mathematics does not also give us the proper concepts, then it has failed. The 20th century has been very short on mechanical explanations, and the mathematics has taken the place of the mechanics. That is to say, the mathematical concepts have become conflated with the physical concepts. That is precisely why string theory is postulating curled up dimensions and other esoterica. Theoreticians have long since begun to let the math lead them into physical assumptions and physical theories.
Minkowski allowed the time variable to be orthogonal to the other variables simply to keep his math symmetric. He did not care a fig for any physical implications, and said so. But his cavalier attitude has allowed many modern physicists to assume that a rather meaningless mathematical postulate is the same as a physical reality. Many if not most contemporary physicists believe that time really does "travel at a right angle" to x, in a mechanical or kinematic sense. This belief has spawned many other theoretical absurdities and contradictions, and has led directly to a muddling of their understanding of many concepts, including "dimension", "event", and so on. Applied mathematics must return to a scrupulous accounting of concepts, in order to prevent any more mathematical concepts from crossing over into physical theory unnoticed and ultimately unanalyzed.
Link to my paper on non-Euclidean math and complex numbers for a fuller critique.
For much more on this problem, you may now visit my multi-part analysis of the Einstein field equations, which revisit this problem in finer detail, including a line-by-line critique of Einstein's 1916 paper.
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