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Originally Posted by Doctordick
Somehow I get the impression that you are still missing the central issue. What we are talking about is a situation where we have an explanation of the universe as we see it. That means our explanation is not bothered by the fact that our reference frame (the coordinate system we use to represent our experiments) is moving with respect to the rest of the universe or not; the explanation includes the issues brought up by that circumstance. This has to do with absolutely any experiment we choose to examine. We choose to examine a few special circumstances because they point out, in detail, the problems with the old fashion Euclidean transformations used by Newton and, in fact, give us exactly what those transformations have to look like. The required transformations are a fact of life; your explanation must obey them and what kind of machinations you have to go through to achieve that is essentially beside the point.
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So the requirement that the Lorenz transformation is part of our explanation is a consequence of being able to construct an object in our explanation where we have defined an object to be a collection of elements that maintain their orientation over changes in t. Would this be equivalent to saying that an observer at rest with an object will always use the same explanation even after they both have accelerated?
As a consequence of defining an object like this, if we take an object and then transform all of the elements that it is composed of so that it is in a moving frame (we accelerate the object for instance) it will appear to contract according to the Lorenz transformation. What seems more important to me is that if we suppose that an observer measures the object while at rest with it and then accelerates with it, since anything that might change the length of the object he is measuring will effect his ruler as well as anything else he might measure, he will still consider the object to have the same measurements.
The actual requirements, though, for a observer at rest with something not necessarily an object, to agree on the measurements before and after an acceleration, and for the object to be Lorenz contracted for a observer that remains at rest and doesn’t accelerate with it is not that it is an object but rather that the construct is unaffected by the rest of the universe. That is, we can still use the same explanation of the construct for an observer that remains at rest with it. This can be accomplished by being able to consider the construct separately of the universe that is of any influences resulting from the Dirac delta function, effecting the construct outside of it can be ignored. This is just saying though, that there is no preferred reference frame in which the fundamental equation is valid in, which is not necessarily a true statement about any explanation but for the time being is considered a useful approximation as it shows that the Lorenz transformation is a consequence of being able to explain things with the use of objects.
Quote:
Originally Posted by Doctordick
The “if we can construct objects then” does not need to be there. We don’t need the existence of “objects” (collections of elements which can be considered as universes unto themselves) in order to create a V(x,t) such that a single element will appear to approximately obey Newtonian mechanics. Given any collection of elements, it is always possible to imagine (or create fictional elements) such that we can see the impact of the rest of the universe (including those fictional elements) as causing an interaction of the form V(x,t). If we cannot create objects, then we cannot create rulers; this is another problem.
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Then is it even possible to create a V(x,t) such that the elements won’t appear to obey Newtonian mechanics? Also, if you are saying that even if we cannot construct objects that Newtonian mechanics will be an approximation to how elements behave I can’t see how we could use Newtonian mechanics and so call it an approximation without the use of a clock which so far requires the use of objects to construct.
Quote:
Originally Posted by Doctordick
I essentially agree with what you are saying except of the fact that I don’t feel that the existence of objects can be considered an axiom. We could certainly conceive of a “universe” without objects and even talk about some of the constraints on explaining that universe; it could even be a subset of our universe which had negligible impact upon that portion of the universe we deal with on a day to day routine. Say the inside of stars (or perhaps the interior of nuclei) or the structure of those great voids between the clusters of stars. We just couldn’t hypothesize rulers or clocks in such a realm.
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It seams clear that such systems will appear to be Lorenz contracted from an outside prospective like a ship going by at relativistic speeds. But in considering possible explanations of such objects, what will happen to the expanding sphere? I know that the fundamental equation is still the equation of an expanding wave, we just can‘t say that it is expanding at a constant speed because we can‘t define the distance that it has expanded or how long it has expanded for.
Quote:
Originally Posted by Doctordick
We cannot solve the general equation anyway so what difference does this make? What I am saying is that the elements of your explanation must obey that equation. I think what you are missing is the fact that most all common explanations are what is called “static”: i.e., none of the fundamental elements change in any way so obedience to the fundamental equation is a trivial issue. Just as the structure of my house obeys Newtonian mechanics: it just stands there without moving.
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But isn’t the only thing that is important is that there is some way that we can map the behavior of the elements in however we are explaining them into a solution to the fundamental equation? How we are explaining them and what we are explaining is of no importance as long as such a mapping exists.
Quote:
Originally Posted by Doctordick
I think it is the issue of the existence of “rulers” without which we couldn’t talk about “coordinate frames of reference”. Without “frames of reference” transformation between frames of reference is a pretty meaningless concept.
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Is there a way though, to define a coordinate system without the use of objects that we can use to define distance? Isn’t it possible that we may be able to use some sort of consistent repeating behavior of elements to set up a coordinate system even if it isn‘t what we would ordinarily consider to be a coordinate system?
