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THE EQUATION x’ = x – vt, AGAIN
by Miles Mathis
I presented what I called my final argument against this equation some time ago. But the issue refuses to be put to rest. I have gotten letters from readers for whom my shortest paper was not short and concise enough. My argument has still not been put in its most transparent form, apparently. Beyond that, I have found the equation in a recent paper in American Journal of Physics on the action principle and Noether’s Theorem. The authors claim that action is not invariant in a Galilean transform, and they use this equation as the transform.
This answers my critics who claim the equation is an historical anomaly, used once by Einstein (perhaps mistakenly) but then corrected by him using better math. This is not the case. Feynman and many many others have used the equation when proving Special Relativity. And now we have the equation being used in action equations and Lagrangians, to show that the Galilean system is not invariant with regard to action! Obviously something is very wrong and it has never been corrected.
I will not repeat my arguments here that I sent to American Journal of Physics, since they mirror arguments I have published elsewhere on this site. Those who would like to read this paper may go to http://milesmathis.com/galileo.html. Right now I simply want to further simplify the problem by relating a visualization that I used in a letter replying to one of my readers. She informed me that my contention that origins do not move was false, since it was commonplace, having been axiomatized long ago. I was told that Felix Klein had developed the general symbolism for such transformations, and she implied that the subject was thereby considered closed. This is the visualization I used to counter her remarks. I consider it fatal to Felix Klein and any general symbolism that assumes that origins move.*
“I also still maintain that origins do not move in relative transforms like this. First of all, origins do not move relative to their own systems. I doubt that I have to convince you of that. But more importantly, when one system is moving with regard to another, the time must always be considered. You can't just shift one graph relative to the other, like sliding a drawing over another. That is not a true representation of velocity. Velocity always includes the time variable. Therefore, the velocity graph must "unfold," as it were. It is not the two graphs that move relative to eachother, it is an object graphed that moves. The points on the graphs display increasing intervals relative to eachother as the object moves. But points don’t move. None of them do. It is disastrous to assume that the points on the graphs move. It is even more disastrous to assume that the origins move, since the immobility of the origins is one of the postulates of the situation. The immobility of the other points is a subtle concept that does not really come up except in the most rigorous of analyses. But the immobility of the origins must be clear.
As an example, say you have an airplane moving over a landscape. The airplane flies over a tree. At t_{0} = t'_{0}, the back of the airplane is right over the tree. That point in space right over the tree is the origin. It is the origin of both systems. It is the origin simply because we have assigned it to t_{0} and t'_{0}. If we want to, we can relate the system of the airplane to the system of the earth’s surface, in which case we drop the origin of the earth’s system from that point up in the sky to a point on the earth right below it—at the root of the tree. Then there is just a vertical or yseparation at t_{0} = t'_{0}, and everything else is the same. In the beginning, the t’s and x’s are equal, but the y’s have a definite given separation.
Now, at some later time the back of the airplane is over some other place, therefore most people would say that the origin of the airplane’s system has moved. But the back of the airplane is not the origin of the airplane's velocity graph. The back of the airplane and the origin are two different concepts. They are equal only at t_{0}. Look back at the sentence above. I say, "At some later time the back of the airplane is over some other place." The origin cannot be at some later time, since the origin is always connected, definitionally, to t_{0}. The origin does not move. It is still back there over the tree, at a point in the past."
This whole problem has been caused by a faulty visualization. Most people assume that a relative velocity is caused by one graph moving, as a whole, over or across another stationary graph—like sliding one sheet of paper across another. But this is false. This is not how it works. For instance, take two sheets of graph paper. Put one on the table and fly the second one over it. Most people think you have created a relative velocity. In one sense you have, of course. The second paper is moving relative to the first, so it would be hard to argue that no velocity is involved. But you have created an incomplete visualization, since you cannot represent the passing of time. In the real world, we always live at t_{now}. No matter where you stop that second piece of paper in its flight, it is always at the point “now”. The past always immediately evaporates. This makes it very hard to visualize velocity. We can see things moving, but this is not the same as “seeing” a mathematical representation of movement. A mathematical representation of movement must include the concept that the beginning of the paper’s flight is connected to a definite time, which we call t_{0}. That time does not move along with the paper, because as the paper moves it moves into new times, t_{1}, t_{2}, etc. All parts of the paper, front back and middle, move into new times equally. Therefore the back of the paper is the origin of its system only at t_{0}.
Once this is understood, the equation x’ = x – vt must fall. If these two x variables are understood as points, then a Galilean transform will express their separation at a given instant. The two variables x and x’ must be measured at the same time. But vt cannot provide this separation, since there is no time or velocity at an instant. Even those who think that the calculus can find a velocity at an instant cannot argue that case here, since our equations are algebraic, not differential or integral. The equation x’ = x – vt demands algebraic definitions of time and velocity. In algebra there is no velocity without a Δt, and there is no Δt at an instant.
If the two x variables are thought of as Δx, then the equation is false in that case, too: Δx’ = Δx – vΔt. In any Galilean transform, Δx = Δx’. To find otherwise would be to find length contraction. Length contraction is relativistic. If there is a length contraction, then the situation is not Galilean, by definition.
*I have shown that Felix Klein was one of the first who failed to see the rather obvious mathematical errors in General Relativity, so I have strong reasons for refusing to consider him an expert on any problems relating to these questions.
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