From: Lee Corbin (lcorbin@tsoft.com)
Date: Mon Aug 19 2002 - 20:52:32 MDT
Yesterday I wrote some things in this thread about how visualizing
hyperbolic space in terms of Escher's familiar "Angels and Devils"
lithograph can be helpful in understanding cosmological models, and
I'll return to that presently.
First, though, there is still a need to address Emlyn's fundamental
questions, (Sent: Thu 8/15/2002 5:11 PM), such as
> What that model [raisins in dough] implies to me is that there
> would be no integrity in small local systems; all *atoms* would
> move away from each other, with disastrous results.
and
> Does this [a hypothetical consensus among cosmologists that space
> is added between galaxies faster than they're falling towards each
> other] mean that all the local forces, i.e.: the nuclear and
> electromagnetic forces binding the little things, are acting
> against this influx of space too?
First, for the first question. I don't think that there is
any worry of "disastrous results". I confess to having held
this view: think of a long rubber band with some ants
marching along it. Now if you stretch the rubber band, how
inconvenienced are the ants? The answer is, Not at all! At the
microscopic level, imagine that an ant has placed its middle
leg in a certain position on the rubber and now its rear leg
is coming down. Before it does so, however, the rubber band
stretches an infinitesimal bit. The leg then comes down still
at the same distance from the middle leg just as if nothing
had occurred. In this same way, I guess, if it turns out that
the "stretching of space" metaphor holds, the atoms within
molecules continue to vibrate at the distances dictated by the
bond energies, regardless of any underlying stretching of space.
So I would say yes to the second question (or second part), above.
Indeed, if there is this "influx" of space, then the nuclear
and EM forces constantly adjust against it. However, this isn't
completely clear to me by any means, nor am I convinced that we
have really understood what the cosmologists are actually saying;
the "stretching of space", after all, could be but a metaphor
they're using as their technical, mathematical solutions to
Einstein's equation don't inspire in them any better description
for the laymen.
So I'll return to a beginning description of the Escher disk,
upon which I hope to build a better understanding of the
"stretching" of curved spacetime.
The Escher disk that I described yesterday, does depict curvature
from a Euclidean perspective. As you look down upon the myriad
angels, there is indeed a greater number of them than should be
present at a given (Euclidean) radius from the center. But it
turns out, on further thought, that we should still regard this
as a flat space (with complications of curvature to be added
later). (For you mathematicians, this can actually be understood
without even considering the conformal L2 metric, or doing any
calculations. Since geodesics on this L2 are (Euclidean) circles,
consider a small arc centered along a (partial) circle. This is
the small part of the geodesic arc closest to the center of the
Escher disk---in other words, as the geodesic starts from the
edge of the Escher disk (making a right angle to the edge) it
moves closer to the center of the disk before completing the
rest of its journey back to the edge, and I mean a small arc
centered on this whole arc. Now if you consider an apparently
smaller geodesic, it forms a tiny semi-circle at the edge of the
Escher disk, and if its situated symmetrically with respect to
the first geodesic I was describing, then the corresponding
same small centered part holds approximately the same proportion
to its own geodesic arc (the semi-circle).)
The upshot of the previous paragraph is that the size of an angel
is very close to directly proportional to its distance from the
edge of the Escher disk. I believe that this proves that the
space is flat, but I can't articulate exactly why right now.
The next step towards understanding the stretching space metaphor
(hopefully) I credit to my friend Norm Hardy (www.cap-lore.com),
who asks that one should next consider a one-dimensional space,
and be prepared to flip back and forth between two slightly different
conceptions of it. This will also turn out to be a flat space
(unfortunately), but one in which light signals, and time since
the big bang make a lot of sense, and a space for which it is
comparatively easy to think about, and understand the hyperbolic
nature of, and get a good feel for the subtleties of the problems.
Perhaps more tomorrow.
Lee
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