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Did I SAY that I was USING Minkowsi's tensor in my discussion? No. So if I am considering a Euclidean dimension to help explain using a metaphor, then obviously I am making no claim at all that I am using Minkowski metrics. So why do you create that link and use it to insult me?
Several of you certainly love waving your flags. You extract merely one point from long posts, and try to completely counter it with accusatory statements derived from your own extrapolated meaning when I either did not say what you claim or explained earlier. Layra and Morberticus have both chosen to create ways to turn inquiry into insult, which I do not feel I deserve. Enjoy your egos, and may they bring you all the intellectual self-validation you seek. rofl

Hehe, ok I'll bite. I humbly apologize to all of you for my apparently feeble and unsatisfactory input. My attempts have been both 'ignorant' and 'dodgy,' and I beg your forgiveness.
So go ahead, Morbeticus, give us a better metaphor, which is both derived from your own brain/ comprehension (so you're not just a parrot), and which actually lets us conceptualize, in our mind's eye, real-world relativistic dilation effects on a real physical level. Solve our mystery with your mastery, and trump my elementary attempts with your superior one. You have the floor.

The sum efforts of a century of investigation into the field by some of the brightest minds of that period have lead to even more abstract modelling, yet you accuse us here of being too mathematical in our approach. As unsatisfactory as it may be, physics has long discarded the burden of ontology and, rather, seeks merely to describe the universe and not dictate its inner workings. The result is that metaphors are limited in their usefulness: the universe appears to operate according to the workings of a specific mathematical framework; we use the constructs and terms of this framework when discussing how said framework operates.

If you want to confuse then metaphors are precisely what are required. Please do get off your high horse: orthodoxy suggests that a better understanding comes from proper appreciation of the mathematical structure and formalism, I see no reason to buck the trend. If you use a Euclidean metric rather than the Minkowski, then you have a rather large problem in that your dimension now traces out imaginary time. Do we have a good metaphor for complex time?

Ok, I'll bite. I'll try to add my two cents, but knowing that I'm in for a world of corrections and horror faces from the people who actually know what the hell they're talking about, of which I'm not likely one xD

So basically the issue is the lack of a direct link between time dilation as a result of relative velocity, and length contraction as a result of the same relative velocity?
Ok, first off, the concepts we're discussing deal with a simple inertial reference frame. Einstein didn't pose that "c is measured to be the same by all observers", but "c is measured to be the same for all observers in inertial reference frames". Special relativity doesn't deal with accelerations, as far as I know, that's the job of general relativity. This is all just to say, length contraction, if posed by special relativity, can't deal with accelerations or any "force" being applied to this hypothetical pilot.
Length contraction would be the observed effect that, if the pilot were moving at a significant portion of the speed of light, he or she would measure the distances he travelled to be smaller than what he really was, according to an observer on the ground. Similarly, it is also the effect that the plane would seem shorter (but not taller or wider) to the observer on the ground.

I'll again fall back to the well known muon case, and try to explain how length contraction is the logical conclusion if one accepts relativity, and that it is dependant on the relative velocity of an object.
So, muons. Pretty pretty muons. Say a muon is created 15km high in the atmosphere. Let's also say that the muon has a speed of 0.994% c. The reason why we love the muon as a means to explain time dilation and length contraction is that, as an unstable particle, it decays with a half life of 2.2 x 10^-6 seconds. If there was no time dilation, one can see that half of the muons would decay after travelling only a few meters:

(2.2 x 10^-6 seconds) * (0.994% c) = 655.586 meters.

The formula for their decay after travelling the entire 15km, as measured by the muons themselves (it doesn't really matter), would be e^-(ln(2)*Δt/λ), lambda being their half life, delta T being the time after creation as measured by them. So first, we calculate how long it takes them to travel 15km:

(15 x 10^3m) / (0.994 * c) = 5.034 × 10^-5 seconds

With this, we calculate how many of them would have survived (again, if they had travelled 15 kilometers according to them):

e^-(ln(2) * 5.034 x 10^-5 seconds / 2.2 x 10^-6 seconds) = 1.294 x 10^-7

So we'd detect about 0.00001% of the muons. So why do we detect more than this?

Now, we know that their time was dilated according to our clocks, correct? This wasn't as an effect of any force acting upon them, they were travelling with a constant velocity, so they were an inertial reference frame, as were we. So if we know that their time was dilated, then one could say "they lived more than their half life", but I put this in quotes (VERY massive, bold, dense quotes) since the half life of this particle is measured at their rest frame. Half lives are meaningless in actual calculations unless you know the relative velocity the particle is travelling parallel to you, since we know about time dilation. So, back with our previous idea, if they "lived longer", (but they don't know they did!), and they still travelled at the same speed according to us, then we can calculate how large of a distance they actually travelled. We do this by using the oh-so-useful Lorentz factor, which indicates the variation in several of our measurements in relativity. It is given by

γ = 1 / sqrt(1 - v^2/c^2)

I'm assuming you already know that the dilated time, according to us, is gotten by T = T0 * γ, where T is the time as measured by us, and T0 is the time as measured by the speeding particle. The reasoning behind this is the pythagorean theorem. So anyway, here's the part where the link between time dilation and length contraction is clearly stated.
If we say that γ measures how much our time varies from the particle's time measurements, then we can say that the particle "lived longer", but how much longer? Well γλ, of course. So we have that the particle's lifespan, as measured by us, was this:

2.2 x 10^-6 seconds * 1/sqrt(1 - ((0.994 * c)^2)/c^2)) = 2.011 x 10^-5 seconds
(γ = 9.14, quite high)

This is how much we'd see it live. So how much could it travel, according to us, in 2.011 x 10^-5 seconds? Speed * time, so

0.994 * c * 2.011 x 10^-5 seconds = 5992.653 meters, or ~5.99km.

Now comes the surprise. What I've just calculated is the distance we'd see it travel, and time time we'd see it live. We already know the distance he saw himself live, which is 2.2 x 10^-6 seconds. It differs by a factor of γ. Now, let's contract length by exactly γ. 5990m / 9.14 = 655m. That was us "contracting" the length, according to the muon's point of view (since our views differ by a factor of γ). Does 655m sound familiar?

It's the same distance that the muon would travel according to his point of view!

So wait, which one is it? Which is the real distance he travelled, 5990m or 655m? The answer is that both numbers are correct, we measure the distance to be larger since we measure his time to be longer (or, alternatively, "his clock is running slower"). Conversely, he measures the distance to be shorter for the opposite reason.

Note that we're never dealing with any acceleration, gravitational force or what have you; this is all toying of variables in a uniform rectilinear motion. Thus we see that without any force being applied, without any "internal" considerations as to the poor muon's structure (if there is such a thing, since this lepton has no internal structure and thus no "uniformity" arguments are applicable), or any other thing. I want to stress this a bit more. There was no acceleration taken into account here. The muon is created with that velocity (by conservation of momentum in the collision between gamma rays from the sun and particles in our atmosphere), not gradually accelerated by any force. The velocity of the muon is constant, and the length between our atmosphere and our detectors is also constant, as measured by us.


If this isn't what was being discussed, however, try to erase this post from your memory, it's what I could decypher from the argument being held here.

Thank you, and Rayquazza, for your input.

Layra, I appreciate your elucidation, and I understand better where you are coming from, and why you guys were poking at me. I was not at all 'trying to impress a girl.' I was mentioning that I am working on a fully integrated conceptual framework. I am aware that there are many incomplete mathematical steps, and understand better your use of the terms 'model' and 'metaphor.'

My intent was to try and help provide a visualization, when instead it seems the request may have been simply for a math explanation. If so, then I went the wrong direction and probably wasted all of our time.

But I did enjoy Rayquazza's post, and believe that he is closer to answering the threadstarter's question in a way they can understand than the mathematical model you put forth. We understand the equations, but your greater formal education on that subject allows you to extract more meaning from the equation than when others look at it. The equations alone do not give most of us a concept of a physical model by which to really 'get' the dilation effect. In pursuit of understanding the physics of this universe on a deeper level than just equations, I at first looked deeply into geometric substructures and background pairs, then proceeded to study a wide variety of disciplines and history and interpreted the metaphorical language of lore from sanskrit and many other sources of data to find links and patterns as they relate to potential comprehensible physical processes, not just mathematical models, for the universe. After traversing many such paths, I am just beginning to re-enter the hard-core physics realm after a long absence. It will be wise for me to brush up so that I may talk in your terms as a peer once again. I have been considering going back and getting a PhD, but it seems I need to take some refresher courses first.

And sure, we'd enjoy seeing equations for some sugar-coated radioactive tomatoes launched out of a rail gun, and how dilation would affect the freshness of ketchup rain. XD

Well if you were trying to unimpress me, you certainly succeeded. I can't see any other reason why you'd even bother to tell me about it, other than your being a conceited p***k.

Now, I do realize that my purely formalistic understanding of physics isn't good for explaining things to people. It is, however, correct, and therefore examples derived from it tend to be correct.
Why the hell were you pulling from Sanskrit? Or anything other than physics? Especially in a physics guild?
But the fact that you're doing so doesn't bother me as much as the fact that you're trying to defend your metaphors in a physics guild. I don't know much biology, but I'm not about to try to defend a stork metaphor in a biology class. Since you know what you've been doing recently, and you know that you're just getting back into physics, you should also try to be a bit more humble. And by humble, I mean listen. When we, and by we I mean everyone, tell you that your metaphor is a crock of s**t, then you should pay a bit of attention. Science isn't democratic, but we all have several years of university level physics behind us.
Since you speak of a PhD, what did you do your undergraduate degree in, and if it was physics, how the hell did you lose so much?

Also, why do you want a concept? I do pure theoretical physics, and get by without a conceptual model because I don't need one. Iguana does the same, only with experimental physics. Concepts only get in the way, because conceptual understanding is just trying to force new models to conform to old vocabularies. You're trying to talk about light in terms of sound and smell, trying to describe blue to a congenitally blind man. I have never, ever, seen a good metaphor or conceptual understanding of the wavefunction, or the Ricci tensor, but both of them exist mathematically, and tell us exactly what is going on physically. Do we need a conceptual understanding of the Ricci tensor, or the wavefunction, or observables-as-operators? No. It works, and that's all that matters.
Conceptual understanding and metaphors are for philosophers. In case you didn't know, I have no respect for philosophers, because they actually believe that their metaphors say something of value.