Assumption? I only assumed Muons start at 10000m above Earth and then only applied Special Relativity - notably, the reciprocity effect of Special Relativity. However, you may be correct - I could have errored somewhere in my analysis of special relativity.
Because, as I have stated before, the muon has been created in the earth's frame and then accelerated by the enormous energy of the gamma ray to 0.988c, while the earth has not. The earth has not changed it's motion through spacetime. The motion of a particle cannot effect distances in other frames. You are misinterpreting the symmetry of the "no preferred frames" statement of relativity.
So you are saying the muon neccessarily undergoes acceleration. So therefore it is inappropriate to analyze the situation from the Muon's perspective, as Special Relativity would say you can if the Muon did not undergo acceleration?
No. Acceleration has nothing to do with it. I think I see the problem with all of this. A problem of scale. Let's try this: P1 and P2 are identical planets with 10000m thick atmospheres. They are some distance away from each other and approaching each other at 0.987c. P1 sees P2 length contracted, including seeing it's atmosphere as 600m thick and vice-versa. When P1 and P2's atmospheres first touch, P1 sees P2's atmosphere touching his at 10000m up, (P2's still appears 600m thick) and vice-versa. Completely symmetrical. Now, shrink P2 down to the size of a muon. P1 still sees it hitting the atmosphere at 10000m, and the little muon people still see P1's atmosphere as 600m thick. So, what I'm saying is that the space between objects in different frames does not change. However the distance between objects that are moving in the same frame appears contracted. So, my animation in the "Visual SRT 1" needs to be corrected as it is misleading. I shall correct it.
I don't think you followed my post - the explaination you just gave does nothing to fix the reciprocity issue here. Edit: That is, from the muon's perspective, shouldn't less time appear to elapse in the Earth frame? If so, then there should equally be length contraction in the Earth frame. And it appears 2inq has one thing right - as long as you don't assume the wrong frame, then everything in special relativity is just hunky-dory.
Change in velocity is all that matters so, when I say acceleration has nothing to do with it, I mean that the fact that change in velocity through spacetime is what's important (and of course this requires acceleration) but the magintude of the acceleration is irrelevant. I think it does. Let's take this one step at a time. Do you agree with the P1, P2 planet scenario above?
Can you repeat the explaination but include calculations along the way. I still don't know where 600m comes from.
Ok, less time passes on the earth clock. In the muon's frame 2.2us elapses as the surface travels the 600m. Yes? And the muon will see say, 0.3us on the earth clock. Ok? In the earth's frame say 1000us elapses as the muon travels the 10000m. Yes? And the earth will see 2.2us on the muon clock. Ok? Problems with this?
Did you analyze my post dealing with the math thoroughly? For the muon to see less time on the Earth clock, it must also see the Earth's frame length contracted.
I'm just using representative numbers here. The actual numbers don't really matter as long as we have the right order of magnitude. I think math details at this point just confuse things. The 600m was one quote for the atmospheric depth in the muon's frame, from some website based on some earth frame altitude that I don't remember.
So, yes, and the earth sees the muons frame contracted. If it had a 10000m thick atmosphere we would see it as 600m (or whatever).
Remember, a frame is just a set of comoving coordinates. So, the earth and the molecules of air are comoving and according to the muon, they are all contracted. The only object comoving with the muon is the muon, and we see that contracted. Yes?
Yes. So it should seem. We? Who's frame are you in now? When analyzing things from the Earth frame, we never refer to anything as seen by the muon, so why are we doing this when analyzing everything from the Muon frame? - That is, refering back to the Earth frame.
Ok fine. If i say the earth sees the muon thusly and the muon sees the earth thusly, Im not mixing frames, just shifting perspective.
That is precisely what I did with my example - and the muon see's the Earth frame contracted to 396m or whatever, no?