The math is simple, trivial even, but the meaning gives a normal person headaches. What are we to make of "slowing clocks, contracting length" and other oddities that come once you accept the basic premise that c is a constant for every inertial observer.
Of course I have time, and since I started thinking about "Spring Theory" I decided to revisit classics see if I could make progress. Working on the Feynman book, I rediscovered basic SR.
I still stumble somewhat at the Dingle paradox, which is a version of the twin paradoxes. For those initiated with the paradoxes, I am OK with twin paradoxes (which involve acceleration) but still struggle with inertial versions of the twin paradoxes. (2 clock in inertia with V relative intersect, what do they read, assuming they were synchronized at some point, what do they read?). So I went over to sci.physics.relativity. The old usenet group. Funnily the "simple question" generated about 200 answers, which you can read here .
The answers were confusing, some plain wrong, taken as a whole completely contradictory, some saying I got the result right but the interpretation wrong, some saying I got the result wrong and the interpration wrong, some just don't waste time and call me a motherfucker, most just say I am stupid for not understanding the basics and so the joke goes on with SR. What is "trivial, basic" concepts still generate this amount of controversy 100 years from the inception of theory. In all fairness there are also well meaning folks that actually provided real information.
Clocks and Clicks
I recently I a bit of a breakthrough thanks to the Feynman book. I could finally put my finger on a few things.
In his book "Feynman lecture" (vol 1, 15-6) Feynman covers basic SR. In typical Feynman fashion it is highly readable and entertaining. He clearly presents C constant for all observers as a postulate of the framework. Says that with C postulated this way one must adopt the Lorentz transformation of coordinates as the proper transformation. Shows Lorentz has the Galilean transformation as limit as v tends to 0.
In there he has a very interesting paragraph about Clocks that Tick. Basically he imagines a setup where the "clock" is keeping count of the number of "clicks" a light beam bouncing off parallel mirrors separated by a distance d, makes. See the scanned image straight from the book (that is a bad bad thing to do). Imagine a photon going back and forth and count every time it bounces at the source.
If the mirrors are moving, the photons are making a zig-zag between the mirrors. (the mirrors move while the photons travel). So the photons will take more time to go back and forth. If we admit that the speed of light is C in the rest frame then the time it takes for the photons is longer as we are doing a zigzag. The "moving time" corresponds to the time it takes for a photon to cover the zig-zag, as seen from the rest frame.
It is trivial to see that the distance travelled by the photon is the hypothenuse of a triangle that has d on one side and v*t-m/2 on the other. So the distance is longer. OK, then the time it takes is longer as well because the distance was longer. You work out the math, I will spare you the math, which is frankly boring and straightforward and you find that the time it takes for the photon in the moving framework to reach the mirror is t-rest(1/sqrt(1-v/c^2)). Meaning the EXACT FORM of lorentzian transform in this case.
Reflections on Ether and Muon decomposition
So what is very interesting is that this explanation by Feynman, does give us a sense of why a "clock" like this would show lorentzian contraction. He uses this to say "every clock behaves like this" (a bit of a non-sequitur, but whatever, great prop). Of note it depends on the orientation of the mirrors, if you rotate them by 90 degrees, then there is no zig-zag and we would show the same time than at rest. But it is essentially a very "classical" explanation. If fact you can almost picture an ether, where the ACTUAL distance has increased and where the speed is limited to C and you get actual time dilation. In other words, a muon moving at the speed of light would indeed take the proper SR time to decompose (more than at rest) as it falls to the earth. I find that absolutely fascinating because in fact MUON DECOMPOSITION IS COUNTED AS ONE OF THE BIG EXPERIMENTAL VALIDATIONS OF SR. But taken as is, it can also be accounted by simple "classical" views.
Of course a big difference is that unlike SR, this theory only works ONE way. SR posits that the result be true for any inertial movement. In other words, the muons travelling would see the muon at rest decomposing more slowly due to the symmetry. Otherwise the travelling muons would know they are travelling (they decompose more slowly) which is in violation of the SR postulate.
In short, while amusing and entertaining (at least to me) the above only replicates half the predictions of SR, but, as far as I know, we have not measured the point of view of the muon.
Musing about time
But the big shortcut I do in reasoning here, is assuming that "time" is defined by the number of clicks. I don't know why or how, but let's assume that the smallest of oscillations is used as a 'reference' for all other oscillations. In this case time is defined in a relative way with respect to this "basic oscillation". It is the "reference". Well then time is an emergent property. You count other events with reference to this basic event. And if your basic event shows "time dilation" then you time reference has just dilated.
So what is evolving is the 'philosophical' definition of time. Time, far from being an ontological reality, is just a highly localized entity that entirely depends on the movement of the particle in consideration. If the muon is moving, it bases it's "time" on the basic oscillation and that oscillation just showed "time dilation" due to movement to a relative "rest media".
Every point of matter will therefore have a time. Related to its speed relative to a resting media. If we imagine that for some reason the most basic oscillation is always perpendicular to the movement (something to do with less resistance in that direction compared to the direction of propagation, I guess) then it is straightforward to see that your "moving time" follows exactly a Lorentzian transform. I find that thought particularly amusing.
The problem with velocities
If it wasn't speculative enough, let me dive some more into this "time". There is no time. Time is an emergent construct. There is no time "dimension", we are just counting clicks, and if you are moving it takes more 'time' for your clicks to happen. Or more strictly for every 1 click of the main clock you will have a fraction of 1 click (gamma from relativity) for the moving clocks. Simple.
The problem with velocities is that it is defined as a ratio of "distance by time" but time itself is an emergent property and I suspect that the way we "intuitively" understand velocity is erroneous because we assume a universal "time". But is a locally emerging construct and every moving body has a different time.
At least muons agree :)
A word to the wise
Please do not take the above too seriously. I don't take it seriously at all, altho I must admit to being tickled pink. In essence it is a simple way to pass time and entertain myself. For example the lack of symmetry (it assumes a medium, ergo a privileged frame) is anathema to most modern science. I just find it amusing that one of the canonical experiments (mentioned by Feynman himself as "proof SR works!") the decomposition time of muons as seen here is in fact trivially explained by classical mechanics. It also implies things about what "time" really is. An emergent property of movement.