Hi Dr. Dick,
Thank you for commenting.
One suggestion: you may still find it of value to look through the rest of my post as some of my comments may still resonate even though you might not follow how I got there.
What actually happened is this:
It took less than 2 hours to follow through your logic (so I thought) first time and it seemed O.K. as far as it went. Then when I typed up the post (that you just responded to) things got more muddled. But I suggest not just look at it linearly; scan the whole post and you may be surprised!
Then I looked again at your thought experiment and it seemed to fall apart quite a lot.
But then I found I got the Euclidean space at the end in a manner which suggested that you said (as you imply yourself) sort of "nothing" (you claim you put forth no theory).
But since then I got a much more spectacular analysis of your thought experiment: not only did I get all your results, including why Minkowski space-time looks like "tunnel vision"; but I got the physicsforums.com 's mentor's "straw metric" comment explainable too! And I got some detailed atomic structure; amazing insight on the nature of "c" and other things; Lee Smolin's quantized area and quantized volume; perhaps also Dr. Ahradsen's "lattice space", and another physicist's discovery.
I go back to very basic ideas as a model of advanced ideas and find it works. The advanced math is "implicate" and does not need to be calculated as its transformations are seen in a say holistic way.
I started my reply at miscellaneous forums here (I give an EXACT explanation of how path-lengths are obtained for high energy particles with short half-lives just as you state!): it may be some days before I type up the full analysis of your experiment but you can find part of it at the other forum here (where I have more confidence about the material being posted O.K.).
P L E A S E do not ignore other parts of a post as soon as you find one or two apparent errors: I do not operate linearly rather I bring the whole picture into focus at once but not all parts are clearly defined to your level (a little more help from you would be an idea here).
I am registered at physicsforums.com and when I have this tidied up hope to post it there as I show a way to accomodate each opinion!
Brief comments on your comments here:
My new look at your experiment is much more basic: one object moves at fixed velocity (or simply a line segment in a graph with x and y axises)
About my two-balls-move-and-collide experiment:
I didn't mean "curve" literally; the lines were straight just a fixed velocities collision scenario. Just my second reference frame I just used "deeper curve" as a way of conveying the left-ball-reference=frame perspective of the way the right ball looked on the graph compared to first viewpoint.
About the graph providing a fixed start and end point: you invited a graphical depiction I thought so one has to draw axises. It did not prevent my taking any start or end point within the graph. If you are going to fix a geometry for your experiment it is no different; you are fixing a beginning and ending?
"Again, I get a distinct impression that you do not understand what I was talking about. By establishing those lines which represent the trajectories of your balls on a piece of graph paper you have already elected a start and finish by not drawing infinitely long lines."
I thought you meant "p" meant (as you stated): "any given line in that experiment may be described by the events that constitute THAT LINE" (not "a point on that line")
If I had used lines wandering off the graph it would have made no difference other than to mean that there was no way of saying that all the pre-set path definition characteristics of the lines (as you said "p" applied to the line) could be reconciled into one viewpoint (but if you say you are using "any standard space-time frame of reference you find convenient" then you have fixed a reconciling "space" for the "p" s.
I used a space-time graph as this was a convenient standard space-time frame of reference.
"I give no information on the forces in the experiment": I just say that whatever forces ARE in the experiment are encoded into these "parameters" that each line can be described by; otherwise the only forces are the ones associated with the idea of a simple collision (if it is not simple this fact is "hidden" in the parameters so does not appear in my representation (I use a parameter-invariant reference frame because if you are talking about an experiment in one geometry that allows ultimate parameter comparisons you couldn't compare them if they had nothing in common?)
I pre-solved any "problem" by encoding it into the comparisons among parameters. I gave the thought experiment a math by-pass to simplify it!
My approach is do the experiment at an underlying level which allows the ideas to be seen by keeping math virtual.