Next point. You correctly note that (71) and the independence of u12 and u23 implies L^2(uij) = 1/uij^2 + k, where k is some constant. So here's my question...what if k=0? The solution is still a solution if k=0. And you never justify why it CAN'T be 0. Let's see what would happen if k=0.
If k=0, then L(uij) = (plus or minus) 1/uij. (take plus like you say)
this implies (back to 2 clocks) T = -x'/u + x/u = (x-x')/u
Now if we take your ad hoc substituion that v = u/sqrt(1+u^2/c^2) we get that u = v/sqrt(1-v^2/c^2).
Thus x' = x - uT = x-vt/sqrt(1-v^2/c^2)
This differs from the actual Lorentz transformation which is
x' = (x - vt)/sqrt(1-v^2/c^2)
Houston, we have a problem!
If k=0, your derivation FAILS. Now all may not be lost if you can justify WHY k does not equal 0. But until you do this it is not a derivation of the Lorentz transformations. You must justify every step for it to be called that.
At the moment you have a very vague argument for why k cannot be negative. (Your argument consists mainly of saying it doesn't make sense, which is not a justification, but...) But nothing about why it must be non-zero.
Look at all that Eugene. Do you see me talking about epsilon, or about k?
That's not really true. I've computed time dilation for a traveling twin (as in the twin paradox) with the arbitrary clock synchronization of exercise 1 and 2 of http://www.everythingimportant.org/relativity/generalized.htm
...
You have only proven that conventional thinking is misguided. The fact that any two frames of reference can have a Galilean synchronization proves that your method is unreliable.
Chris Osborne wrote:
If the standard method of computing time dilation doesn't work in this case then you will surely be able to tell me what does. Tell me then, Eugene, how do we derive time dilation from your form of the transformation equations if we let zeta be the identity function? Show me how to get to the formula
delta t' = delta t/sqrt(1-v^2/c^2)
This formula is unquestionably true because it is supported by experimental evidence, as I stated.
I would like a very detailed, step by step derivation. (Try to use minimal verbiage if you will because you tend to muddle things up when you write.) If you can do it, maybe I will have more respect for you. At the moment though, I think you are clearly losing this debate.
If zeta is the identity function and the translational constants are zero, then my nonlinear transformation reduces to equations (78) and (79) with an epsilon equal to 1/c. My paper proves in section 6 that the value of epsilon doesn't enter the time dilation formula at all for an epsilon universe.
Chris Osborne wrote:
Next point. You correctly note that (71) and the independence of u12 and u23 implies L^2(uij) = 1/uij^2 + k, where k is some constant. So here's my question...what if k=0? The solution is still a solution if k=0. And you never justify why it CAN'T be 0.
I say very clearly on page 8 of my paper that "the value of k can be determined experimentally." Try to comprehend "Lorentz transformations from the first postulate" by A. R. Lee and T. M. Kalotas, published by the American Journal of Physics -- May 1975 -- Volume 43, Issue 5, pp. 434-437. It says exactly the same thing. The abstract states:
Quote:
We present in this paper a derivation of the Lorentz transformation by invoking the principle of relativity alone, without resorting to the a priori assumption of the existence of a universal limiting velocity. Such a velocity is shown to be a necessary consequence of the first postulate, and the fact that it is not infinite is borne out by experiment.
Chris Osborne wrote:
Let's see what would happen if k=0.
If k=0, then L(uij) = (plus or minus) 1/uij. (take plus like you say)
this implies (back to 2 clocks) T = -x'/u + x/u = (x-x')/u
Now if we take your ad hoc substituion that v = u/sqrt(1+u^2/c^2) we get that u = v/sqrt(1-v^2/c^2).
Thus x' = x - uT = x-vt/sqrt(1-v^2/c^2)
The spacetime structure constant k appears in both of my derivations. Because k can't be negative, I write it as 1/c^2. If k=0, then c = infinity and my Lorentzian transformation reduces to the Galilean transformation.
Chris Osborne wrote:
If k=0, your derivation FAILS.
Tell that to the American Journal of Physics.
Chris Osborne wrote:
Now all may not be lost if you can justify WHY k does not equal 0. But until you do this it is not a derivation of the Lorentz transformations. You must justify every step for it to be called that.
Why aren't you complaining to the AJP for publishing the paper, "Lorentz transformations from the first postulate"? That paper derives a general Lorentzian transformation featuring a nonnegative spacetime structure constant k and encourages us to call it the Lorentz transformation.
Chris Osborne wrote:
At the moment you have a very vague argument for why k cannot be negative. (Your argument consists mainly of saying it doesn't make sense, which is not a justification,
The overarching goal of my 14-page paper is to inaugurate the beginning of a great project —to put the logic of special relativity on a more rational and secure mathematical foundation. Hopefully, I will have time to fully elaborate on every minor detail at a later time, such as a clearer and more mathematically satisfying reason for why the spacetime structure constant k can't be negative. There are already proofs of this result in the scientific literature but I'm not satisfied with any of the standard published arguments. If you want mathematical perfection, you will have to wait. Try to be satisfied with what is currently available.
This Shubert derivation is nothing new. Everyone can see that by reading Victor Yakovenko's Derivation of the Lorentz Transformation. The difference is just notational. Yakovenko uses some Greek symbol (I think it is omega) for the Shubert's -1/k. It also shows why k can't be zero or negative.
There's nothing wrong with that paper (I know that guy), or even with a lot of Eugen's math. They're still not derivations of the Lorentz transformations though (even though they both call it that). What that paper demonstrates nicely is that only a Galilean transformation OR Lorentzian type transformations (i.e. Lorentz transformations with a varying maximal speed) are possible under the assumptions of homegeneity of space and time and under speed symmentry between 2 inertial frames. Yankeveno even points out if a is infinity then you just have the Galielean transformations. But you need either the constancy of the speed of light in all inertial frames or the invariance of Maxwell's equations to distinguish between the 2.
Now, Eugene probably wants to say something like "Well why don't you go to Yankevenko then and complain to him about it?" because I know he appreciates that distinction and doesn't intend for it to be taken as a derivation of the Lorentz transformation in a literal sense. Eugene doesn't appreciate this distinction. He thinks his simple assumptions are all you need to derive relativity, and he is wrong.
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