Discussion: Lorentz invariance of certain zero angles

Tach said:
Now, for the calculus part that you seem not to get either:

$$f=f(t,x(\theta))$$

Therefore:

$$\frac{df}{d \theta}=\frac{\partial f}{\partial t} \frac{dt}{d \theta}+\frac{\partial f}{\partial x} \frac{dx}{d \theta}=\frac{\partial f}{\partial t} 0+\frac{\partial f}{\partial x} \frac{dx}{d \theta}$$

Contrary to what you might think, $$\frac{dt}{d \theta}=0$$
Click to expand...

Tach said:
Nonsense, [$$dt/d\theta = 0$$] implies that t is not a function of $$\theta$$
Click to expand...

In the example given by Tach we have something like:

$$y = f(t,x(\theta))$$

where I have introduced y as a new defendent variable. It is confusing to use f both as the name of a function and as the name of a variable.

Tach claims that if $$\frac{dt}{d\theta}=0$$ then $$t$$ is independent of $$\theta$$.

Here's a concrete example. Take:

$$y = f(t, x(\theta)) = t - x = t - 2\theta$$

where I have defined $$f(a,b)=a-b$$ and $$x(\theta)=2\theta$$.

We then have:

$$t = y + 2\theta$$

and

$$\frac{dt}{d\theta} = \frac{\partial y}{\partial \theta} + 2 = -2 + 2 = 0$$

Tach's claim seems to make sense to me.

However, by defining the variable y as above, we have assumed that t and $$\theta$$ are independent from the start, so it's not so surprising to get this result.

I'm not sure whether it is relevant, but it's also perhaps worth considering what happens if we look at a particular value for y. As an example, consider y=3.

We then have

$$3=t-2\theta$$

which is easily arranged to give:

$$t = 3 + 2\theta$$

In other words, the function

$$3=f(t,x(\theta))$$

implicitly makes t dependent on $$\theta$$.

Obviously, in this case $$\frac{dt}{d\theta} \ne 0$$.

---

I don't know if any of this helps.
 
Another possibly-relevant example:

Suppose

$$t= y(x) + z(x)$$

with y and z each depending on x, which does not depend on $$\theta$$.

Then

$$\frac{dt}{d\theta}=0$$

However, it seems clear in this case that t is not constant, since it clearly varies with x.
 
Tach, I understand that you are frustrated that I don't accept all your arguments, but your rudeness is unacceptable.

Please, review your last post.
Politely respond to the questions posed in the previous post.
If you want to defer discussion on any issue, please say so - don't just ignore it.
 
The debate has devolved into illogical and spite, and seems beyond returning to the stated aim of finding common foundations, then building on those foundations to a mutual conclusion.

According to the rules in the proposal thread, this means we skip to the Summary:

  • Tach and Pete will prepare single independent summary posts, describing our impression of how the discussion went, the conclusions we reached, and what we learned.
  • When we both indicate readiness, the two summaries will be posted at around the same time.
  • Following the summary posts, the Debate thread will be closed.

Tach, I'm done.
I can no longer maintain the fantasy that you are debating in good faith.

The next post I make to the debate thread will be my summary, which I will post after you post yours.
 
As indicated above, the debate discussion has stalled.

It has stalled for me, because it's clear to me that I'm wasting my time. Tach is failing to address the arguments I post, and presenting arguments of his own that are blatantly wrong, and blatantly inconsistent with each other.

It has stalled for Tach, because I'm abandoning the debate.

According to the agreed rules, this means we skip to the summary. We each have one final post to make.

Tach, please take your time to prepare your summary post. The thread will be reopened when you're ready. However, if you choose not to post a summary at all, then I'll just post my own and the thread will be locked again.
 
Pete said:
As indicated above, the debate discussion has stalled.

It has stalled for me, because it's clear to me that I'm wasting my time. Tach is failing to address the arguments I post, and presenting arguments of his own that are blatantly wrong, and blatantly inconsistent with each other.
Click to expand...


I see it exactly the other way around, I have repeatedly corrected you in terms of calculating the transforms of displacement vectors, in terms of calculating partial derivatives, in terms of your gratuitous fixation with marking the endpoints of various vectors simultaneously. You never admitted to the errors in your approach, this is why the discussion stalled.
Feel free to post your summary, I was in the process of posting mine when the thread was suddenly locked.
 
No, that was Trippy's initiative. He PM'd me to say he'd done it, and I PM'd him back saying it would have to be reopened when you're ready.
 
Since you're both ready with your summaries, I have reopened the debate thread. Remember, you have both agreed to post just one more post each - the summary post.
 
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