Re: measuring the universe

[John Denker <jsd@MONMOUTH.COM>]

At 12:48 PM 8/12/00 -0600, Jim Green wrote:
> I thought "open" and "closed" were used to describe whether the Universe
> would collapse or not -- ie whether the Universe would expand forever or at
> some point start to contract to an eventual Big Crunch.

Close, but not quite.  Here's the deal:

1) A finite-sized universe (i.e. the kind that astronomers call "closed")
is guaranteed to have a finite time-span as well, according to a wide range
of cosmological models.

You can cook up exceptions, e.g. by fiddling with the "cosmological
constant" which can be used to introduce a bias into Einstein's equations.

2) An infinite-sized universe (i.e. the kind that astronomers call "open")
may go crunch after a finite time, or may continue expanding forever.  It
depends on the initial conditions (or, if you like, present conditions) of
mass density and expansion rate.

> But this aside for a moment --and whether we can actually "see" (by
> whatever means) to the "edge" of the Universe -- doesn't the current
> standard model assume that the Universe is finite in size

No, it certainly doesn't _assume_ that.

To date, measurements of mass density and expansion rate are equivocal.

> -- although I
> don't know what that would mean:  There could be no one standing outside
> the Universe with a large caliper to measure the size.  Or ....

That is a good question.  There is not an easy answer, but there is a good
and definite answer.  The way to approach such questions is as follows:

Suppose you were a d=2 creature inhabiting a D=2 universe.  What could you
observe about your universe _without_ leaving your universe or exploiting
any information from outside your universe?

Imagine an ant crawling on the surface of a balloon.

If your universe is not only finite but small, you could circumnavigate it.

The balloon could be compact in all directions (e.g. a sphere).  Or it
could be compact in some directions and extended in other directions (e.g.
a very long cylinder).

Note that the ant could measure the intrinsic curvature of the sphere by
laying out large triangles and observing that the interior angles add up to
more than 180 degrees.  Similarly it could lay out big circles and observe
that the circumference was less than 2 pi r.

In contrast the ant would observe that the cylinder has _zero_ intrinsic
curvature.  That's news to some people, but it's true.  Note that you can
roll a piece of paper into a cylinder without damaging the paper, but you
cannot deform a piece of paper into a sphere without lots of
damage.  That's because a piece of paper is strong enough to resist
stretching _within_ the D=2 world of the paper, but it is so thin that it
cannot resist curling in the third dimension.  And remember, the ant knows
nothing of the third dimension.

It is also possible that the balloon-universe is expanding.  Somebody could
be blowing air into the balloon.  This does not increase the size of the
ant, but it does increase the size of the ant's universe.  The ant will
observe all other ants moving away because of this, and more-distant ants
will move faster.

Whether or not the balloon-universe is finite in spatial extent, the ant
may be able to predict that the universe will have a finite lifetime.  This
could be done by observing the time-derivatives of the Hubble
expansion.  If the expansion is observed to be slowing sufficiently
rapidly, it is an easy prediction that the expansion will turn into a
contraction, which in turn leads to a big crunch.

Bottom line:  the ant can know quite a bit _without_ asking any questions
about the D>2 space in which the balloon is embedded.  So it is with
us.  Our universe "may" be embedded in a larger space, but we know nothing
about it, and questions about the embedding space are moot.  That is, such
questions are unanswerable and irrelevant.