From: Ken Clements (Ken@Innovation-On-Demand.com)
Date: Mon Mar 11 2002 - 17:10:06 MST
"Dickey, Michael F" wrote:
>
> After re-reading what you had said, I think I understand what the
> disagreement is, since we were both saying the same thing yet drawing
> different conclusions...
>
Go to Google, put in "Pi in curved space" and hit the groups button. You will get a couple of thousand hits. Looking at the first dozen or so, you will see that your question has been quite popular over the years, as have been answers along the argument I used.
>
> >
> > Ken - "Pi is a constant that does not change no matter where you are, how
> > much you weigh, or how fast you are going. It says some things about flat
> > Euclidean space, and to the extent your observations of the space around
> do
> > not match, pi helps you figure out the nature of the curvature"
> >
>
> "Be sure to read the rest of the post; I went into the spherical case."
>
> I did read you original comments about a sphere, and I basically parallel
> what you said.
>
>
Except that you held the radius of the circle constant and let the curvature of the space change to show that the value of the c/d ratio would change. I held the curvature constant to show that in such a space the c/d ratio would depend on the size of the circle. So if you want to use that as the definition of pi, it is no constant, even within that fixed curvature space. If you tell me the curvature of the space you are
in I still cannot give you a numeric value for the c/d ratio of a circle in that space, until you tell me either c or d.
>
> This question relates to whether space-time is curved at all, as far as I
> can tell this question is, as of yet, unanswered in cosmology.
The curvature of space-time is well accepted, with plenty of demonstrations. In the future we may find some corrections to make to General Relativity, but flat space-time is long gone. I suspect you would enjoy some books on the subject such as _Einstein's Universe: Gravity at Work and Play_ by Anthony Zee (1989).
> Would the universe be curved in the absence of matter?
I think this question qualifies for Zen Koan status. Or perhaps it is another way of asking the old "Tree Falling the Woods" question.
>
> Well, in any case thanks for helping to clear that up, it seems we both said
> the ratio changes with the curvature of space time, but differed in our
> interprations of the definition of pi, and thus had a different conclusion
> to the question. Im my google searches I could not find a definition that
> was absolute either way, but I guess definitions are arbitrary social
> constructs anyway. Not being a formally trained mathematician, I couldnt
> say either way what the *official* definition is.
>
You are welcome.
-Ken
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