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as a Concept

A Break in the
Pioneer Case

by Miles Mathis

This paper is meant to prepare the reader for a serious analysis of Einstein’s equations. In my experience, most readers enter any analysis with extremely strong prejudices, and they are likely to enter this particular problem with prejudices that are almost religious. No matter how simple and transparent the math is, no one will consider it if they do not imagine the possibility that it may be correct. To provide this possibility, I have constructed here a short overview of the concepts. This will allow the reader to place the math in a proper framework.
       Most will say that they are already aware of the framework—if I have something to say I should just say it. But Relativity has remained uncorrectable for a century due to the fact that no one—not one person—has fully understood the framework, including Einstein himself. If you will allow me, I will show you in the simplest possible language where the framework must be extended and altered. You must understand this framework if my mathematical corrections are to mean anything to you at all. Rest assured that I am not proposing any esoteric ideas or concepts. I am not inventing particles or fields. I will not prove that light can go faster than c or that Relativity is a myth. I am just making small mathematical and conceptual alterations to Special Relativity that will make it even stronger and more useful.
       Let me first say that I know that Relativity is true. The finite speed of light does require us to use transforms. There is no doubt of it. These transforms will give us time dilation and length contraction. Einstein was absolutely correct in that. But his theory, as he presented it, was still flawed and incomplete. Even Einstein knew this. He told his followers explicitly that no theory was ever finished, and especially not his. That is why he continued to work on it until his death. He wasn’t just working on General Relativity, either. He was working until the end trying to understand all the implications of his first postulates. He did not succeed. He left Relativity with many basic errors embedded in it, errors that have not been corrected to this day.
      Before I present the mathematical corrections, it is therefore best to make clear to the reader the conceptual mistakes of Einstein and current theory. Absolutely everyone, Einstein included, thought that the transforms were transforming variables in one coordinate system to variables in another coordinate system. But this is not what the transforms do, mathematically or operationally. In any given experiment, what the transforms do is transform incoming data to local data.
       Let us use the Pioneer spacecraft as our example. If the engineers at JPL apply the transforms to the problem, what exactly are they doing and what are they finding? All would say that they are transforming a moving coordinate system that is far away from the earth to the earth’s system. In some sense this is correct, but it is not precisely correct. It is not correct enough to avoid confusion. What is really being transformed is data from the distant system to the earth’s system. Relativity is primarily a theory of observation and measurement. Therefore all our data is observed or measured data. That is what Relativity means. Newton could gather data without considering who gathered it or where. But Einstein correctly pointed out that we could no longer do this if we wanted to get the right answer. The finite speed of light demanded that we take into consideration who was gathering data and where. If we wanted to get the right answer we either had to make all our measurements near at hand, where the speed of light did not affect our numbers, or we had to do transforms on the numbers. Most data that arrived from any distance would be affected by the E/M waves it was arriving on. We would now have to consider the time separation and the velocity of the object we were receiving data from.
       This means that in any transform we have two sets of numbers. We have our local numbers for things, like the length of a meter rod and the length of a second, and we have the data coming in from a distance. We want to apply our meter rod and second to this incoming data, so that we know what it means, but how do we do it? Einstein gave us transforms and these transforms (usually) work. Great. But with the Pioneer Anomaly the transforms are a tiny bit off. What could be wrong?
       What is wrong is very simple, though it is beastly difficult to discover if you have already made it to the end of your understanding of Relativity and your acceptance of it is set in stone. The only way to find the error is to go back to the very beginning and start over. This is what I did and this is what I discovered:

To do a transform you have to assign a set of variables to your incoming data. You can make them primed variables, for example. This is what Einstein did. But conceptually you aren’t finished. You now have to assign your coordinate system. It isn’t enough to make a variable assignment. You have to also define your new coordinate system. This is where the central error is made. In the Pioneer example, the data is coming from the spacecraft, so Einstein and everyone else defines the primed coordinate system as belonging to the spacecraft. But this is false. What we receive on earth is just data. We are measuring or observing not the spacecraft itself, but information arriving from the spacecraft. In other words, we are seeing how the spacecraft looks to us, not how the spacecraft looks to itself.
       The whole reason we have to do a transform in the first place is that the finite speed of light is skewing the data. The information from the spacecraft is arriving late to us, compared to when it was experienced by the spacecraft, and it is arriving with time periods stretched out and meters compressed and so forth, as we know. We do the transform in order to correct this. The transform gives us numbers that we can compare to local numbers and make sense of. But if the finite speed of light is skewing the numbers, then logically the numbers must have been unskewed back at the spacecraft. The spacecraft is not emitting funky data, we are receiving funky data. It is the distance between us, and the finite speed of light, that is causing the difference. Therefore, the spacecraft, which is no distance from itself and is not seeing itself with light that has had to travel long distances, must be experiencing normal local data.
       This means that the transform is not expressing a difference between the numbers of the spacecraft and the numbers on the earth. The transform is expressing a difference between numbers arriving at the earth on E/M waves and numbers on earth arriving from a negligible distance. To put it another way, x’ is not how the length of the spacecraft looks to the spacecraft, it is how the length of the spacecraft looks to us, from a long distance away. Therefore you cannot give x’ to the spacecraft. You must give x’ to the data only.
       A doubter will say, “But don’t we see things locally with light, too? All our information, local as well as distant, arrives on E/M waves.” True, but the E/M waves do not skew nearby data, since there is no time lag. To get time dilation and length contraction, the light has to travel some appreciable distance or the velocity of the measured object must be very very great. When we define local time and length we do not define it relative to clocks or meter rods that are far away or rushing about. We define it relative to clocks and meter rods that are nearby and at rest relative to us, as observers.

A reader will say, “Maybe, but what difference does it make? You seem to be making very subtle conceptual distinctions, but this is just semantics or metaphysics. Relativity is not philosophy; it is math.”
       Yes, but all applied math must be applied correctly. Incorrect variable or field assignments are not metaphysical errors. They are mathematical errors that will and must lead to mistakes in calculation. This is exactly what has happened with the Pioneer Anomaly.
       This is how the field misassignment has affected the math of Special Relativity:
Two of the fundamental equations and assumptions of SR concern the movement of light in the two fields or coordinate systems.
x = ct
x’ = ct’
The first equation is how light travels relative to us here on earth. The x and t variables are our own local variables. I have no problem with this equation.
       The second equation is how light travels in the other field. But there is no other analogous field, in a strict sense. What I mean is that x’ and t’ are how the spacecraft’s lengths and times look to us. How do we put c into that data, if it just data? In what sense is data a field that light can travel in?
       It’s not. We can’t do it, mathematically or conceptually. We have to find some other way to solve. We know empirically that time dilates and lengths contract, so we can develop simple equations that express that. As time dilates, the period gets longer, by definition. So within the pseudo-field of our data, time and length are inversely proportional.
xt = Kx’t’
       where K is an unknown constant. Now we simply use the two equations we have so far and solve.
x = ct
xt = Kx’t’
ct2 = Kx’t’
x’ = ct2/Kt’
x’ ≠ ct’

Someone may say,
x’ = ct’ may still be true according to your equations if
t2/Kt’ = t’
K = t2/t’2
       But x’ = ct’ cannot possibly be true, since if both x = ct and x’ = ct’ are true then x and t must change in direct proportion. This would be
x = ct
x’ = ct’
c = x/t = x’/t’
       that is a direct proportion. But that contradicts mountains of empirical data. The period does not get shorter as the length gets shorter. Just the reverse. This means that Einstein has accepted as one of his first equations an equation that contradicts all current data and all data that he was trying to explain. He knows that time dilates and that dilation means a larger period. He says it explicitly in the book Relativity (Ch.XII, p. 37), "As judged from K, the clock is moving with the velocity v; as judged from this reference body, the time which elapses between two strokes of the clock is not one second but [γ] seconds, i.e. a somewhat larger time. As a consequence, the clock goes more slowly than when at rest." And yet the equation x’ = ct’ must mean that time speeds up. This is a terrible error.

A reader will say, “With an error of that magnitude, how does he end up getting a transform [γ] that works so well?” He achieves this with a compensation of errors, and by knowing what he needs at the end. The Lorentz transforms already existed—that is why they bear the name of Lorentz and not Einstein. It was empirically known what equations were needed, and Einstein simply supplied the derivation. Lorentz’s math was pure heuristics, since it was applied to the data after the fact. Lorentz’s math was not predictive, it was compensatory. Everyone knows that. But so was Einstein’s math. The transformations are a direct outcome of the Michelson null outcome. The only difference between Lorentz’s heuristics and Einstein’s is that Einstein collected and invented several important axioms. Most of these axioms are true, as I have said above. Einstein made real contributions to the problem and deserves much credit. But Einstein did not get everything right. Both his math and his theory were and remain incomplete. His errors have caused a mountain of confusion and are now causing experimental errors. These errors must be corrected if we are to continue to make progress in kinematics.

The equation x’ = ct’ has never been corrected. It is a fundamental part of gamma and still exists in the derivation of the tensors. Importing tensor calculus into SR did nothing to correct the equation. The only way to correct gamma is to jettison x’ = ct’ and start over. That is what I have done in all my papers. The full derivation of new transforms may be found here.

The specific problem confronting the scientists working on the Pioneer Anomaly is that the motions of their spacecraft are not analogous to the motions Einstein and Lorentz were attempting to explain. Gamma comes very close to correctly transforming the variables in the thought problems given at the time, including the interferometer. But the relative trajectories in these thought problems do not match the relative trajectories in the Pioneer problem. Spacecraft do not recede from the earth in the same way that light travels through an interferometer or what have you. I have shown in all my various papers that trajectory must always be a consideration.
       Following Einstein, current theory accepts that all trajectories are equivalent with regard to Relativity. That is, you do not need to consider direction or angles or curves, only velocity. This is false. I have proven beyond any doubt that direction is of paramount importance. The transforms are completely different for objects approaching than for objects receding. Objects moving at tangents or angles to the line of sight of the observer require more complex transforms. Since the motions of the spacecrafts in question are quite complex, the transforms must be equally complex. Gamma is not sufficient to do the job, no matter how much tensor calculus you bring in to bolster it. In fact, the current transforms are so partial and insufficient that is may be considered a mathematical miracle that they work at all. That they come within any fraction of the correct answer can only be attributed to the great imagination and ingenuity of the mathematicians and engineers who use them.

Someone may ask, “So what does this mean about distant objects? Are they in our field or not? You are saying that the local numbers of the spacecraft are basically the same as ours. Their meter rods and seconds are the same as ours. The time and length differences are only apparent, since it is the data that is skewed, not the spacecraft itself. Doesn’t this put us right back in the universe of Newton? And doesn’t this mean that all the ‘cranks and crackpots’ on the web are right?”
       Not at all. Some of the so-called crackpots have had their points, but from where I am sitting everyone in both camps—believers and non-believers alike—all share the distinction of being pretty spectacularly wrong, including Einstein himself. The single mistake I have related above is of awesome dimension, considering that Einstein had his whole lifetime to spot it and make a correction. None of the great believers or non-believers have ever spotted it either, including Bertrand Russell, Herbert Dingle, and Richard Feynman. According to my limited knowledge, no one living or dead has spotted it or any of the other major algebraic errors that I relate. Therefore, all the finger-pointing and name-calling should immediately end. None of the believers or non-believers are or have been in any position to judge their neighbors.
       For it turns out that the truth is almost precisely in the middle. The non-believers believe that Relativity is false. But in the end, Relativity is simply the necessary existence of transforms. If transforms are necessary, then Relativity is true. Those who believe that transforms are not necessary have either never been on the receiving end of any data or have never done any math, or both.
       As for the believers, they have also strayed far a-field. They have pretended to an understanding they never had. They have tried to force upon us twin paradoxes and varying atomic clocks in airplanes and all manner of other mysteries and mystifications. They have hidden behind imposing maths like the tensor calculus while they could not spot or correct simple algebraic errors. They have filled the blackboard with Hamiltonians and Lagrangians and other multiple abstractions while they have been unable to comprehend simple circular motion or the physical basis of the derivative. They have belittled philosophers and mathematicians while themselves making astounding mathematical and philosophical errors. They have accused others of being metaphysicians while they themselves have become the most naïve idealists imaginable.
       But the answer is that, no, we are not back to Newton, not by the furthest stretch of the imagination. That all data from a distance requires a transform is a huge step forward, and Einstein deserves his place in this. The recognition of a time separation with any distance separation is also quite different from Newton, and crucial. Relativity allows us to do many things in mechanics and kinematics that we could not have hoped to do in the 19th century. It allows us to show the physical genesis of the inverse square law, which Newton could not hope to do. Newton derived the law but could not explain what caused it. Relativity has allowed me to do this. [see Part III of the Third Wave.] No doubt Relativity will allow us to solve many problems that have not even been discovered yet, problems still embedded in classical and current theories.
       The equivalence of all local time is not a return to Newton, since all local measurements are incommensurate when measured from a distance. Newton and Galileo did not believe this and could not have expressed it if they had. In fact, there was no local time for Newton since he saw no use for it. I have defined local time in relation to relative time. Newton had no relative time, therefore he could have no local time. He only had universal time. Newton thought it was “now” everywhere. But we know that this is not true. If we look at Saturn in a telescope, the picture we see is not “now”. It is Saturn some seconds ago. If we look at a distant star, that twinkle is some years old. And so on. Very different from Newton, and true. This is proof of Relativity in itself. Time separation is Relativity. The consequences of that alone are huge, in the history of physics and astronomy.
       In addition, redshifts and blueshifts of data due to relative motion have been known since the 1600’s, when Rømer saw them from Io. I do not mean shifts of light, but shifts of data. The data from Io was seen to vary depending on the relative motions of the earth and Io. This was Relativity, and Einstein’s transforms can be applied to Io to discover the variance. Why? Because any periodic motion can be considered a clock. The eclipses of Io were periodic, so Rømer was seeing a clock in the sky from a distance. The same is true of pulsars, which are clocks in the sky. Their periods are known to vary depending on relative motion. This variance is Relativity. The periods of pulsars shorten if the pulsar is moving toward us; slow down if the pulsars is moving away. This is a Doppler Effect on clocks, which is the same thing as Relativity. And it contradicts the current interpretation of Relativity as being the same for all relative motion. The rate of moving clocks does not always slow down. Pulsars are moving clocks and they speed up when they approach us.

I hope that all this has been of some use to you as the reader, and that you will be able to read my mathematical corrections with a fuller understanding of what we have always been trying to express, and to measure, with Relativity.

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