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Post by mondo on Mar 5, 2023 5:10:28 GMT
I wonder why on page 298 equation 31 says that the metric $d\hat{s} = \frac{ds}{y}$? I have no idea how was it derived. Additionally, on the previous page I don't understand why figure [20] says "shrink by a factor X". This is later also repeated in the text. Where does it come from?
Thank you.
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Post by Admin on Mar 5, 2023 8:58:55 GMT
I wonder why on page 298 equation 31 says that the metric $d\hat{s} = \frac{ds}{y}$? I have no idea how was it derived. Additionally, on the previous page I don't understand why figure [20] says "shrink by a factor X". This is later also repeated in the text. Where does it come from? Thank you. Mondo Read subsection 3 A Conformal Map of the Pseudosphere a few times until you understand it. It answers all your questions. Vasco |
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Post by Admin on Mar 9, 2023 16:47:03 GMT
Mondo
Did you manage to work out the answers to your questions?
Vasco
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Post by mondo on Mar 16, 2023 5:16:51 GMT
Vasco,
sorry for late response I got really busy with work. I reread the section you mentioned and I think I understand equation (31) - it is just a natural consequence of the $X$ relation from the previous page and the fact that $y = \frac{1}{X}$. For my second question, the shrinking is because we divide by $X = e^{-\sigma}$. However, since this is $< 0$ we are actually expending.
I have additional questions:
Why radius $X = e^{-\sigma}$? This is based on equation (30) from a previous page and looks like $R$ is now assumed to be $1$ but why?
Thank you.
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Post by Admin on Mar 16, 2023 9:46:41 GMT
Mondo
See the second paragraph of subsection 3 on page 296.
I think you meant to write "...since this is $<1$" and not "$<0$"
$R$ is just the length of the string and so the curve will be the same shape regardless of the value of $R$. Just think of it as a change of units which makes the subsequent algebra easier to write.
Vasco
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