Debate: Lorentz invariance of certain zero angles

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Sorry Tach, I'm done wasting time with you.

You are rude. Based on the repeated refusals to read and try to understand, I would say that you aren't interested in science, you are only interested in being right at any cost.

Prepare your summary.

Find someone else to take orders from you.
One party has requested that summaries be prepared and posted, on the grounds that they wish to discontinue the debate.

The other party has refused to do so, feeling the have grounds to continue the debate.

Given that the second party has been given the opportunity to sum up, and has refused to do so, thread closed.

Further correspondence can be entered with myself (via pm) or admin.
As agreed in the Proposal thread, Tach and Pete will each post one more post - a summary describing their impressions of how the discussion went, the conclusions were reached, and what they learned.

Both parties have indicated that their summaries are ready for posting. After they are posted, the Debate thread will be closed. The Debate is then at an end.
Transforming to S':
$$\vec{T1'}(t',l) = \begin{pmatrix} \frac{r}{\gamma} - vt' \\ l + r\omega(\frac{t'}{\gamma} +\frac{vr}{c^2})\end{pmatrix}$$

$$\hat{P_t}'(t'=t'_0)$$ is parallel to the displacement vector between two points on the rod at $$t'=t'_0$$ with different values of $$l$$:
\vec{T_1}'(t'=t'_0,l=l_0) &= \begin{pmatrix} \frac{r}{\gamma} - vt'_0 \\ l_0 + r\omega(\frac{t'_0}{\gamma} +\frac{vr}{c^2})\end{pmatrix} \\
\vec{T_1}'(t'=t'_0,l=l_1) &= \begin{pmatrix} \frac{r}{\gamma} - vt'_0 \\ l_1 + r\omega(\frac{t'_0}{\gamma} +\frac{vr}{c^2})\end{pmatrix} \\
\hat{P_t}'(t'=t'_0) &= \frac{\vec{T_1}'(t'=t'_0,l=l_0) - \vec{T_1}'(t'=t'_0,l=l_1)}{\left\|\vec{T_1}'(t'=t'_0,l=l_0) - \vec{T_1}'(t'=t'_0,l=l_1)\right\|
$$ \hat{P_t}'(t'=t'_0) = \begin{pmatrix}0 \\ 1 \end{pmatrix}$$


1. So, the recurring issue with your approach is the gratuitous insistence in marking both endpoints of the tangent vector at the same time: $$t=0$$ in S and $$t'=t'_0$$ in S'. There is no justification for this, when you let go of this gratuitous condition, $$\vec{v_P}$$ and $$\vec{T_1}$$ are parallel in all frames.
2. The second issue is an outright error, if you mark the endpoints of the vectors simultaneously in S (at $$t=0$$), you will be marking the endpoints of the transformed vectors at $$t'=t'_0$$ and $$t'=t'_0+\Delta t'$$ respectively. Zero time intervals in frame S transform in non-zero time intervals ($$\Delta t'$$) in frame S'. If you do this correctly you find out that the x component of $$ \hat{P_t}'(t'=t'_0) $$ isn't zero but $$-v \Delta t'$$

While I pointed out the first issue with your solution repeatedly , I have been remiss in pointing out the second issue.

Though we did not agree on the final outcome, the debate was useful in:

-setting up the milestones and solving the issues one by one (bar the one that constitutes the subject of the debate)

-explaining how displacement vectors are differentiated, how partial derivatives work (sadly, there are STILL a lot of errors in your attempts at calculating $$\frac{\partial x'}{\partial \theta}$$, etc.

-giving the opportunity to explain the solution in multiple ways , using different formalisms (polar coordinates, vector algebra, experimental methods), all backed up with a significant amount of math each time

What I was hoping for was for the discussion to be indeed about learning rather than degenerate into insults in the end. Unfortunately, this is exactly what you ended up doing in the end. Your replacing logical arguments with insults is a clear sign that you lost control and, with this, you lost the argument.
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I chose to abandon the debate because I believe Tach is being willfully ignorant. I can no longer maintain the suspension of disbelief required to proceed as if Tach is willing to debate in good faith.

The final deciding factor was that Tach maintains that parallelism of vectors is lorentz invariant, despite clearly agreeing just a couple of posts previously that vectors parallel in S are not necessarily parallel in S'.

Unfinished business
Undecided issues include:
  • whether Tach's method has any bearing on the orientation of the tangent in S'.
  • whether my calculations (not addressed by Tach, except to point out that it contradicts his preferred conclusion) are valid,
  • whether Tach's calculations of my method are consistent with the rest of his arguments or consistent with the method I proposed
  • Whether the tangent to the wheel in S' is adequately represented by $$\frac{\partial \vec{r}'}{\partial \theta}$$, where r' is the position vector of the wheel elements. (This was originally proposed by Tach. Oddly enough, when I did the calculations it worked out to exactly match the result of my calculations. Tach then decided that no, it wasn't an adequate representation after all.)

How the discussion went
Debate structure
This debate was a kind of experiment to see if a one-on-one structured argument could be more productive than the general forum rough-and-tumble.
It didn't work as well as I'd hoped.
The biggest problem was that it was too hard to maintain the tracking list enough to keep discussion adequately focused. I think the idea could work, but it would ideally have a dedicated tool with automated issue tracking and management.
I think it also needs a committed referee, unless both parties are committed to the concept. I was taken aback when Tach's very first post made it clear that he hadn't read my carefully constructed proposal, and he continued to ignore key elements of the proposal rules right through the thread.

However, the structured approach did allow the discussion to progress past some roadblocks. We were able to step away from issues where we were deadlocked, and focus on other issues. So it wasn't a complete failure.

Debate content
I really hoped that this could be a mutual exploration of ideas, that we could find common ground and learn something new.
However, I found debating Tach to be like nailing the proverbial jelly to the wall. He resisted exploring foundations of ideas, growing frustrated when I pressed him about the source of certain assumptions. And he did not seem dedicated to establishing a locigally consistent platform to build upon. When conclusions were reached that contradicted his assumptions, he did not seem willing to explore the idea that his assumptions might be incorrect, but instead preferred to switch to discussing a different issue, or to add new assumptions to back up his preferred conclusion.
He also maintained the consistent appearance that I was here only to learn, while he was here only to teach. He seems completely unable to consider the notion of reversing those roles, or even of a collaborative relationship (witness the partial derivative debacle).
In the end, it became too frustrating and I abandoned the debate.​

Conclusions reached
We concluded that:
  • A velocity vector transforms like so:
    \vec{v_p}' = \left(\frac{v_{p\parallel} - V}{1-v_{p\parallel}V/c^2} \ , \ \frac{v_{p\perp}}{\gamma(1-v_{p\parallel}V/c^2)}\right)
  • The transformation of a spatial vector depends on how the time of the end points is identified in each frame due to the relativity of simultaneity.
    For example, if the spatial vector in each frame can be matched to the same displacement 4-vector with endpoints simultaneous in S, then the transformation is:
    $$ \vec{r}' = \vec{r} + \vec{V}(\frac{\gamma-1}{V^2}\vec{r}.\vec{V})$$​
  • And not much else, if anything.

What I learned
I learned a lot about the difficulty of conducting a structured argument.
I learned a lot about functions and about notation relating to partial differentiation, although that learning happened [thread=112396]in another thread[/thread].
I learned that transforming spatial vectors required careful clarification of exactly what the vector of interest is in each reference frame.
I learned more about the importance of clarity in general. I think that most disagreements happen because of unclear thinking and communication.
I learned that it is surprisingly difficult to define a tangent plane for a moving object.
I learned about the Hasselkamp transverse doppler effect experiment.
I learned some new tex.​
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