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Originally Posted by Bombadil
So then in all of the analysis’s of the problems that you put forward there is no need for the actual transformations. All that is necessary is that the problem is analyzed in such a way that both observers, that is the rest and the moving observer, obtain the same result when observing a clock that they are at rest with.
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You seem to have missed the entire exercise. When the two observers go to analyze any dynamic (means changing) collection of data referenced via positions in their personal reference frames (think “do physics”), they first need to establish their units of measure in that frame of reference. That means that they must establish both a unit of measure for the coordinates of their frame of reference (a measure of distance) and a unit of measure for time.
Now, the original problem was to obtain a flaw-free explanation of “any dynamic collection of data” in the universe. That means that “both” observers are using exactly the same explanation! Whatever specific phenomena those observers use to establish their units of measure, that phenomena must be an excellent example of that “flaw-free explanation”: i.e., by excellent I mean that the nature of the phenomena behind these measures is presumed to be well understood. The interesting thing here is that, due to the “rules” being enforced by a Dirac delta function, that “flaw-free” solution must be scale invariant. That means that, if the two observers are dealing with “well understood bodies of information which are totally independent of the linear motion of their personal frame of reference”, there exists no such “well understood phenomena”: i.e., a unique linear measure can not be established (the problem is scale invariant and therefore the solution must be scale invariant).
The way out of that problem is the fact of that very scale invariance itself. We have already established that the only possibility which leaves the form of the fundamental equation identical in both frames (under the assumption that a solution does exist) is exactly the transformation deduced by Lorentz and Fitzgerald. That alone is actually insufficient as it does not eliminate the possibility that “no flaw-free solution exists”: i.e., it is possible that the only flaw-free solution demands one unique frame of reference and, if you are not using that frame, your solution is flawed. If true, that possibility would pretty well destroy the field of physics as an exact science.
Thus I finish the essay by presuming that there does exist phenomena capable of establishing linear measure in those coordinate systems which is in accordance with the Lorentz-Fitzgerald solution. Note that I am not presuming that “all phenomena” are in accordance with that solution; however, when I am given that specific established linear measure, the approximate validity of Schrödinger's equation as a solution to my fundamental equation allows me to design objects in an arbitrary “rest frame”. This empowers me to design a “clock” and thus define time in exactly the same way in two non-accelerating frames moving with respect to one another. I then show explicitly that those clocks yield exactly the time relationship required. At this point, I have proved that, not only is the standard relativistic transformation the only possibility but that it is indeed totally consistent with my fundamental equation.
Since via Schrödinger's equation we have a guarantee that objects can exist we can actually move (via acceleration) a specific measuring rod (no matter how arbitrarily that distance is defined) from one frame into the other which is exactly what the marks on that rod in Paris were all about.
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Originally Posted by Bombadil
I suspect that part of the problem here is that while I can understand how the Schrödinger equation is arrived at I have very little idea as to what it suggests other then that it can be used to derive Newtonian mechanics.
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The first issue is that Schrödinger's equation is an equation for determining probabilities for the results of one's measurements. We have, at this point, defined momentum, mass and energy in terms of the various components of the fundamental equation. From that perspective, as a function of “position x”, Schrödinger's equation amounts to a statement as to how one half the square of the momentum divided by mass differs from the energy; the difference being exactly what we have called “the potential energy”. For practical purposes, that is a simple “conservation of energy” equation.
Newton's fundamental equation is F=ma and his concept of momentum is p=mv. Another fundamental concept in Newtonian mechanics is kinetic energy (kinetic means “associated with motion) and, finally, the concept of “potential energy” was introduced in order to maintain “conservation of energy”. Thus the equation of conservation of energy in Newtonian mechanics is
Since velocity is defined to be the time derivative of position, that equation essentially defines the behavior of the normally referred to as Newtonian mechanics. There is an important relationship embedded in the above; Newton's kinetic energy is exactly the square of the momentum divided by 2m. Thus it is that Newton's energy conservation equation is essentially identical to Schrödinger's energy conservation equation (exactly the same terms are related in exactly the same way). As an aside, since acceleration is (again by definition) the time derivative of velocity, Newton's force can be seen as identical to the time derivative of momentum so all of Newton's concepts are also defined in Schrödinger's approach and, since both are essentially using the same “conservation of energy” equation, they both yield essentially identical solutions. The only real difference is that, in Schrödinger's picture, the actual measurements are probabilistic while, in Newton's picture, the actual measurements are discrete and established. Thus, whenever the actual variables in the solutions to Schrödinger's can be seen as established discrete values, the results are essentially, for practical purposes, identical to Newton's. (I say “essentially” because there are problems which can be handled in Schrödinger's picture which are not handleable in Newton's picture; however, even in that case the solution can generally be cast into Newton's picture.)
The only aspect of that fact that I am using is that the behavior of “objects” (collections of fundamental elements which can be seen as physical structures) will essentially obey common sense behavior of standard objects. In particular, the structure I call a mirror assembly can be seen as a classical object in a four dimensional Euclidean space which will obey simple conservation of momentum in that geometry.
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Originally Posted by Bombadil
Even with the sum of the mass being greater then zero will the sum of the mass operators  still sum to zero? |
This would be true except for one subtle fact. The tau dimension is totally fictional and, in the final analysis, the actual distribution of our points in the tau direction can not have any effect on our final “flaw-free” solution. Thus, in a sense, it is impossible for any actual valid data to establish a unique frame of reference in that direction. In a sense, although velocity in the tau direction is defined, real motion in the tau direction is essentially undefinable. The fact that one can not define a rest frame with regard to motion in the tau direction, seems to me to imply that this data need not be so constrained.
I personally think that this is the source of the philosophical problem concerning the lack of universal symmetry between the fundamental elements and the anti-elements of those same elements. Something worth thinking about anyway. (Actually, I have not asserted that I have solved every problem in physics; I have only solved a great many.)
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Originally Posted by Bombadil
For instance supposing that we use a unit rod as a way to measure distance, then in order to actually measure an object we must make measurements on both ends of the rod at what we consider to be simultaneous measurements, which requires that we define what we consider to be simultaneity something which must differ from one frame to anther due to the fact that your clock must behave the same way in any reference frames but that the observation of a clock in a different reference frame will differ. That is, events that appear simultaneous in one frame need not appear simultaneous in any other frames.
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I think you are overlooking a much more fundamental issue here. Everyone is defining “simultaneity” via the assumption that the speed of light is the same in both directions and that is the basis of the problem here. All physicists in the universe could just do all their calculation in the rest frame of the background radiation. Then they would all agree and there would be no problems with identifying “simultaneity”.
The real problem with that solution is that it makes the actual calculations so difficult that no physicist would even consider such a solution. Think about some poor guy doing an experiment in his laboratory calculating the energy levels in hydrogen who is required to use a measuring rod at rest with respect to background radiation. His standard measuring rod would change with the rotation of the earth (since he must use a different speed of light in opposite directions which varies because he is moving). Now the final result would be exactly the same (Maxwell's equations, in fact, insure that) but the intermediate calculations would be horrific.
I'll stop here because I don't think your last paragraph is well thought out.
Have fun -- Dick
