Time intervals measured by stationary and moving observers

[Hak]

The definitions of "reference point" and "variable point" are that the reference point is where you calculate the derivative and the variable point is not.

Note that you could switch frames and do the calculation in the primed frame; then $v'$ would be the reference point and $v$ would be the variable point. Either choice is fine, but once you've made the choice you have to apply it consistently.

Thank you very much for your reply. What do you mean by "you have to apply it consistently"? Thank you again.

 

[PeterDonis]

What do you mean by "you have to apply it consistently"?

I mean that once you have picked a reference frame, that defines for you which point is the "reference point" and which point is the "variable point". You don't have any choice then about which is which.

 

[Hak]

I mean that once you have picked a reference frame, that defines for you which point is the "reference point" and which point is the "variable point". You don't have any choice then about which is which.

Thank you very much. Sure, I agree. I am still, however, confused. When I had asked which definition of $x$ and $a$ was correct, @PeroK replied "$x = v'^2$ and $a = v^2$", not that the interchanged ones (i.e. $x = v^2$ and $a = v'^2$) were fine too. In the light of what you told me, why are my calculations incorrect in post #21? Have I not been consistent with my choice of reference? Try looking at post #21 and, if you can and want to, let me know. Thank you very much again.

 

[PeterDonis]

When I had asked which definition of $x$ and $a$ was correct, @PeroK replied "$x = v'^2$ and $a = v^2$"

That's because he knew you were using the unprimed reference frame.

not that the interchanged ones (i.e. $x = v^2$ and $a = v'^2$) were fine too.

That's because they're not fine if you have already chosen the unprimed reference frame.

In the light of what you told me, why are my calculations incorrect in post #21? Have I not been consistent with my choice of reference?

Obviously not, because in post #21 you switched reference frames (you didn't realize that's what you were doing when you said $x = v^2$, $a = v'^2$, but it was).

 

[Hak]

That's because he knew you were using the unprimed reference frame.That's because they're not fine if you have already chosen the unprimed reference frame.Obviously not, because in post #21 you switched reference frames (you didn't realize that's what you were doing when you said $x = v^2$, $a = v'^2$, but it was).

Thank you. Yes, I wrote $L' (v'^2)$ instead of $L'(v^2)$, right?

 

[PeterDonis]

Thank you. Yes, I wrote $L' (v'^2)$ instead of $L'(v^2)$, right?

The switching was in defining $x = v^2$ and $a = v'^2$ instead of the other way around.

 

[Hak]

That's because they're not fine if you have already chosen the unprimed reference frame.

I still cannot understand this. I would be grateful if you could explain it again in a way that I can understand it. Thank you very much for your contribution.

 

[Hak]

Obviously not, because in post #21 you switched reference frames (you didn't realize that's what you were doing when you said $x = v^2$, $a = v'^2$, but it was).

I guess I didn't really understand. So you can change the frame of reference, but you can't reverse $x$ and $a$, right?

 

[PeterDonis]

I would be grateful if you could explain it again in a way that I can understand it.

I'm not sure what more I can say. When you define $x = v'^2$, $a = v^2$, you are choosing the unprimed frame. That's just a fact. There is no "explanation" for it. So if you say $x = v^2$, $a = v'^2$, that is inconsistent with choosing the unprimed frame. What more is there to say?

 

[PeterDonis]

So you can change the frame of reference, but you can't reverse $x$ and $a$, right?

Wrong. Reversing $x$ and $a$ is changing the frame of reference. But then you have to start the whole analysis over from the beginning. You can't just reverse them in one equation, that was obtained by a series of approximations that assumed you were using the unprimed frame.

 

[Hak]

When you define $x = v'^2$, $a = v^2$, you are choosing the unprimed frame. That's just a fact. There is no "explanation" for it. So if you say $x = v^2$, $a = v'^2$, that is inconsistent with choosing the unprimed frame. What more is there to say?

I did not understand what calculations I should have done to choose the primed frame of reference. Should I change all the calculations in the unfolding to do that? Thank you very much.

 

[PeterDonis]

I did not understand what calculations I should have done to choose the primed frame of reference. Should I change all the calculations in the unfolding to do that?

See post #40.

 

[Hak]

@PeterDonis Thank you so much for everything, I think I understand. You really gave me a great amount of help. Thank you again.

 

[PeterDonis]

@PeterDonis Thank you so much for everything, I think I understand. You really gave me a great amount of help. Thank you again.

You're welcome!