From: Ken Clements (Ken@Innovation-On-Demand.com)
Date: Mon Mar 11 2002 - 12:08:22 MST
I thought I explained this rather carefully, but I guess I was not as helpful as I would have liked. Perhaps I was not clear about the difference between physical space-time and mathematical space. I will try again.
"Dickey, Michael F" wrote:
>
> 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.
>
> Assuming that PI is the ratio of the diameter of a circle to its
> circumference, then altering the curvature of space time would alter that
> ratio, this is a fundamental aspect of general relativity is it not?
No, and it is not just that you stated the ratio backwards, it is much more fundamental. First, you have to understand that a circle is defined as the locus of points that are all equidistant from a given point (the center). Points are theoretical; they are not dots on paper, they are not places in any real space. Points have no width or length or thickness (dimensionless). They are part of the tautological mental
construct we call mathematics. When you "draw a circle" on a piece of paper, the actual mathematical circle is in your mind. What is on the paper is a pattern of ink blobs that all have physical dimensions not associated with mathematical points. If you do a good job drawing, the length of your figure divided by the widest distance across it will be something close to pi, but will not equal it. A circle is not a physical
thing.
You can refer to the ratio of the circumference to the diameter of all circles in all kinds of mathematical spaces. Just because the ratio may exist in some given mathematical space, that does not mean its value is pi. Pi does equal this ratio when we are talking about a flat two dimensional Euclidean space. If you start curving that space, the ratio changes so it is no longer equal to pi.
scerir posted this link: http://math.ucr.edu/home/baez/physics/Relativity/GR/pi.html be sure to take a look. Also see
http://www-groups.dcs.st-and.ac.uk:80/~history/HistTopics/Pi_through_the_ages.html
>
> Imagine a circle drawn on a piece of paper ... The curvature of the
> sphere the circle is drawn on is analogous to the curvature of space time,
> Thus PI changes with the curvature of space time.
I covered the spherical case in my post. Yes, the ratio changes with curvature, but pi does not change. Again, the ratio is simply no longer equal to pi.
> In the absence of a
> gravitational field, I believe there is a net curvature of space time (is
> space time curved or flat?) If that is the case, the the geometry of the
> space time curvature can be altered to change the ratio of a circle to its
> radius and thus hide a number in it (can it be arbitrary?)
>
It's the other way around. In the absence of a gravitational field it is believed there would be *no* curvature, which is not the case because there is a gravitational field. This is physical space-time, not a constructed mathematical space where there are circles.
>
> "Damien, the problem is that, in the book, Ellie does not go out and measure
> some aspect of this universe. Instead, she uses computers to look into the
> mathematical expansion of pi for a hidden message of Creation."
>
> If Elly was figuring this out on a computer, perhaps she modeled a
> mathematically perfect circle and compared its circumference with its
> diameter with the already known curvature of space time, thus making the
> calculation one that is representative of the geommetry of space time, and
> not merely an abstract mathematical construct. Altering such a curvature
> would probably not effect any organisms evolving in the space-time unless
> the curvature was severe.
>
A circular path may exist in physical space-time, but a circle does not. A circle *is* a mathematically perfect circle, and in pane geometry its circumference to diameter ratio is pi. Your suggestion would have been better than what was in the book, as it would have been based on a physical measurement, your "already known curvature." But Carl did not write that. Had he picked that one, it still would not have worked
because the message appears "kilometers downstream of the decimal point" which means the measurement would require at lest several hundreds of thousands of orders of magnitude in resolution. The whole Universe has less than 200 orders of magnitude from top to bottom. (By the way, we know pi to over 200 billion places which, if you wrote down on paper at 4 mm per digit, would run over 800,000 km. Long enough to go from the
earth to the moon and back.)
-Ken
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