Chris,

As I've already explained on this thread, physicists have protested specific points and mathematicians on the newsgroup sci.math have refuted their senseless claims.

Mathematicians outrank physicists.

Yeah, I've already read all of that. I guess basically I'm trying to find out what *really* makes you tick. There are a lot of people on the net who are very stubborn in their ways when it comes to non-intuitive physics, but you've actually put a great deal of technical work into it. So I actually spent a lot of time reading your paper and online discussions with you and others about it.

As far as mathematicians outranking physicists, I have degrees in both, and I don't buy your arguments.

Chris,

You say that you don't buy my argument. Are you sure that you are objecting to the essence of my reasoning or just the way I play with words? What's the first nonsensical equation or inexcusable sentence of my paper? The next update to http://www.everythingimportant.org/relativity/special.pdf will state with greater clarity that section 4 is just a novelty derivation. The real substance of my paper is section 2 and 5. I'm sure that I can justify the semantics that you are tripping over.

Yes, I can do that. I'll start w/ section 2, since you say that's one of the places the real substance of your paper lies.

Please justify your assertion that for the "Shubertian clock" that

T = -x'/u + xi(x)

Here's what I get. If the clock at x = x1 reads T1 when x' = 0 passes then

x(x' = 0) = u*(T-T1) + x1 where u is the relative velocity.

Thus if the clock at x = x1 reads T1 when x' =x' passes by, then

x(x' = x') = u*(T-T1) + x1 + x'.

Solving for T we have

T(x,x') = -x'/u + ( (x-x1)/u - T1)

so your xi(x) has the unique form xi(x) = (x-x1)/u - T1

It sounds like what you're trying to say is that the clocks can actually read whatever we want them to read. I guess that's theorectically possible, but then not all of the clocks would be reading the 'correct' time. There simply isn't consistency otherwise. Sure, you can set all the clocks however you want, but then they aren't telling you anything about time anymore, at least not as a group. I believe your first inexcusable sentence is the "Synchronization is a matter of personal choice." Followed by "it is not a law of phsyics." Techincally that is correct of course, but the implication is that the clocks can still tell me something about the physics of the situation, which is not true. There's no law of physics that says I can't set my upstairs VCR clock to be 10 minutes behind my downstairs VCR clock, but then I'd better not complain if I'm consistently 10 minutes late into any program I tape upstairs but always on time downstairs!

Dear Chris,

Your insistence that my derivation of the Lorentz transformation is "inexcusable" because I begin with an infinite array of clocks without any synchronization to them whatsoever is downright laughable.

Are you sure that you want to state such outlandish criticisms to my paper in full public view?

You missed what I was getting at. Technically, you can set up a bunch of clocks that read all kinds of different times. But doing so gets you absolutely nowhere. The clocks must be synchronized before they can tell you anything about how time works.

I'd like to learn a little bit more about how you understand your synchronization functions. Does xi(x) have to be continuous? Differentiable?

I also have a question about your section 3. In a lot of your rebuttals you said the real derivation does't start until later. So what are you trying to accomplish in section 3? Is it just that if you define a certain xi and zeta than the transformation eqns. take on a Lorentz transformation? If that's all there is than I think that's rather trivial, but I will give you the benefit of the doubt and let you tell me why you have written section 3 and what it has to do with the rest of your paper.

What are you talking about? It gets a thoughtful reader to the Lorentz transformation.


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


There are no conditions to the functions xi(x) and zeta(x'), i.e., these functions can be chosen to be discontinuous at every point. For example, start with a standard Einsteinian synchronization and then add or subtract one hour to every clock depending on if the clock is located at a rational point or not.


The original purpose of section 3 was to illuminate the seeming impossibility of section 4 for a bigoted physicist. It turns out that the basic computation in section 3 repeats itself in section 5.

A thoughtful reader who by some stroke of genius decides to choose
xi(x) = sqrt(1 + u^2/c^2)x/u and
zeta(x') = -sqrt(1 + u^2/c^2)x'/u (where u no longer even represents the relative speed between the 2 rulers if we are to get the right answer). Why the thoughtful reader couldn't have chosen something else (like xi and zeta to be the identity functions) and thus gotten something that was NOT the Lorentz transformation equations, I don't know.


And you got the wrong answer. (Here I will use the notation that's on the link you provided.) The Lorentz temporal transformation is:

t' = (t - vx/c^2)/sqrt(1-v^2/c^2)

If x is a constant, this gives delta t' = delta t / sqrt(1-v^2/c^2)
which is time dilation to which you refer. Your transformation is:
t' = t0 + (t-t0-v(x-x0)/c^2-zeta(x)/c)/sqrt(1-v^2/c^2)
+ 1/c*zeta(x0 + (x-x0-v(t-t0)+v/c*zeta(x))/sqrt(1-v^2/c^2))

Your claim is that this (coupled with a spatial transformation that you also give) is a group (which I haven't checked but I'll take your word for it) and is physically indistinguishable from the Lorentz group. Well let's see about that. Let's let zeta be the identity as I suggested above. Then
delta t' = delta t / sqrt(1-v^2/c2) +1/c( -v*delta t /sqrt(1-v^2/c^2))
= delta t *(1-v/c)/sqrt(1-v^2/c^2) = delta t*sqrt ((1-v/c)/(1+v/c))

Which is not the same as the time dilation given above. Experimental evidence thus suggests your transformation is not physically indistinguishable from the Lorentz transformations, because they give correct results (eg. lifetime of cosmic ray muons) and your transformation does not.


Ok, well that's even weirder than what I thought you were thinking before. I thought you thought they at least needed to be continuous. Ok, continuing on...


Ok, well then my next task is to start dissecting section 5 I guess. Although my guess is I'll have the same objections but in a different form.

Section 3 only proves that any two frames of reference, in either a Galilean or Einsteinian universe, can have a Galilean or an arbitrary Lorentzian synchronization scheme. My only point is to demonstrate that the startling derivation of section 4 is possible.


My approach in section 4 and 5 is to go beyond two frames of reference and try to make things work for three frames simultaneously.


No. I got the right answer.


No. 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. You just can't look at one moving frame and compute time dilation for a traveling twin by taking simple derivatives. The correct math is a bit more complicated than that.

That doesn't address my point, but OK.


You are making no sense whatsoever. First of all, I didn't take any derivatives. 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.

Ok, enough with the old. It's time for me to point out the problems with section 5 of your paper, which you claim is the heart of your derivation.

First I have a clarification to ask of you. You say "Time is to be defined with motion..." What the heck does that mean? "...all time computations can be performed with homogeneous functions." First of all, what is a homogeneous function? If I recall correctly, Euler defined this and it means that f(x,y,z,....) is homogenous of order n if
f(Lx,Ly,Lz,...) = L^n f(x,y,z,...). Is that it, or am I thinking of something else? And in any case, why do you assume this? You say it's an axiom, so you can always do that. But what is your motivation behind assuming this? And can you justify why performing all time computations with homogeneous functions is something that applies to our universe?

Next point. (53) states that ti = -xj/uij +L(uij)xi (I'm using L for lambda). Why should this be the case? Why have you assumed that xi(x) has the form L(u)x? Before you said that xi was completely arbitrary. Why are you starting with this form now? Just because it works?

Next point. At the bottom of p. 10 you say "Suppose L is an odd function." Why? Why should it be an odd function? Can you justify this, or is it just another assumption? And if it is just an assumption, how do you know it applies for our universe? Why can't one say 'Suppose L is an even function'?

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.

Your move.

Please forgive me for interpreting your delta as a differential. I was thinking of a generalized formula involving general motion where a differential has to be integrated.

Speaking of misinterpreting equations,


What you have there is just plain wrong. I suspect that you don't understand my basic equations, which define clock time point by point in two inertial frames of reference:
T = -x'/u + xi(x) and T'=x/u + zeta(x').
If you understood these basic equations and the notion that there is an idealized clock at every point in the universe, then you could solve equation (78) for time in terms of x and x' and note that the only difference in clock times between the standard Einsteinian synchronization and my more generalized proposal, clock for clock, is just a fixed desynchrony point by point.



My homogeneous transformation group, equation (78) and (79), is just the homogeneous version of my more general nonlinear transformation.


A proper computation and precise explanation will be included in a future update to my paper. I'm not being evasive. What you ask for is absolutely essential to correctly unravel the popular confusion over the so-called "twin paradox." For now, the most important first step is to compute the constant desynchrony that I mentioned.


The paragraph that you refer to states the answer very clearly: "Recall the Shubertian clock."


A function is homogeneous if it's a homogeneous function of some order. You understand correctly. The equation ti = -xj/uij +L(uij)xi is just the simplest extension of my basic equations T = -x'/u + xi(x) and T'=x/u + zeta(x') to multiple frames of reference.


The advantages of beginning with unquestionable simplicity and defining clock time with a general first order homogeneous function is that it makes relativity accessible to high school students.


If you can find a more general solution to my functional equations, have at it.


My functional equations have no even solutions. That's very easy to prove.


It doesn't fail. You just enjoy extolling a petty linguistic distinction between the Lorentz transformation and a general Lorentzian transformation. If epsilon is zero and if the spacetime structure constant k is zero, then the general Lorentzian transformation reduces to the Galilean transformation. That's well known and widely understood.


The argument against a negative k is extraordinarily precise if you would notice the inevitability of taking a square root of a negative quantity in the formula for adding proper velocities.

Historical footnote: I added an introduction to my relativity paper on June 15th and today I rewrote the section on the definition of time. I hope Chris Osborne respects this improved edition.

Chris,

My paper is now updated with a computation of SR's time desynchrony effect for a nonzero epsilon universe in equations (78) and (79). As you can see, epsilon disappears from the final results.

Now, if you were to follow the same unassailable procedure, which every physics student should understand, and compute the actual time desynchrony using my nonlinear Lorentz transformations, then all the knotted resynchronization effects would likewise vanish.

Your method is obviously flawed.

Epsilon = 0 wasn't my issue. It was k=0, which you have ignored.

Look at all that Eugene. Do you see me talking about epsilon, or about k?