From: bweiner@ruhets.rutgers.edu (Benjamin Weiner)
Newsgroups: sci.astro
Subject: Re: "Modeling" the Expanding Universe?
Date: 28 Jan 93 03:42:55 GMT
Organization: Rutgers Univ., New Brunswick, N.J.

This is a long post, and it took some effort, so read it carefully
before tearing it to shreds!  Thank you.

schumach@convex.com (Richard A. Schumacher) writes:
>Just so I'm clear on this: 

>If one adds two test particles to the universe, the general
>expansion does NOT by itself cause the test particles to move
>apart. Subsequent movement of the test particles depends on
>their mutual gravity and the gravitational or electromagnetic
>forces imposed on them by other matter/energy in the usual,
>non-cosmological ways. 

I think I would agree with part of this but there is some murkiness
which is, among other things, the cause of your objection to the
expanding-balloon picture, so here goes:

Suppose I consider our universe but pretend that it is fairly
homogeneous, i.e. neglect the effects of individual galaxies'
potential wells, etc.  It's expanding with some linear velocity-
distance law and the expansion is slowing down.  Now I add two
test particles, one here, and one far away, let's say at the location
of some galaxy XXX at z=0.05.  (The galaxy itself has nothing to do
with it, I'm just trying to give an idea of the distances involved:
large, but considerably less than the Hubble radius; with H_0 = 75,
the distance between the particles is 200 Mpc.)

Comoving observers on our galaxy and the distant galaxy will
measure their relative speed of separation to be v = cz = 15,000 km/s.
So in order to put in two test particles which are stationary w.r.t.
each other, they can't be stationary w.r.t. their local comoving
observers.  Exhibiting a regrettable anthropocentric bias, I'll 
assume we have particle I stationary at our Galaxy and particle II
stationary w.r.t. our Galaxy but _at galaxy XXX_, which means a
Glorbian that lives in galaxy XXX sees particle II whipping by at
15,000 km/sec!

Now, down with ethnocentricity!  Look at it from the Glorbian's point 
of view.  It sees us, particle I, and particle II all zooming away,
at 15,000 km/sec.  The Glorbians, however, also know general relativity,
and so they know that our recession velocity will gradually decrease, so
what happens?  In fact, particle II will eventually catch up to us.

Try a quasi-Newtonian analysis, which should be valid in this case because
1. the distances involved are small w.r.t. the Hubble radius, and
2. inside a spherical volume, we can neglect the gravitational effects of 
an external spherically symmetric universe (Birkhoff's thm. - see Weinberg):

Consider us sitting at the center of a sphere of r_0 = 200 Mpc; inside
the sphere, there is a constant density of matter, rho, but the matter
is moving outward: matter at radius r away from us has velocity v = Hr.
It is not too hard to see that this case is homogeneous and isotropic,
but you'll have to doodle a few pictures; for example, an observer A
sitting 100 Mpc "east" of the center, looking at an observer B sitting
100 Mpc "north" of the center, does in fact see her moving away at
v_AB = H r_AB = H * 141 Mpc, directed radially, as required by homogeneity
(the universe has to look to A like it's expanding radially outward from
him, just as it looks to us like it's expanding away from us.)

Okay, now that it seems like this is a legitimate picture of our universe[*]
consider the particle II, at radius r_0 = r(t_0), speeding away from
the Glorbian galaxy which is receding from us, so particle II is
stationary with respect to us.  We can neglect the effect of the universe
outside r_0, but inside r_0 there is a sphere of matter, mass
  M(t_0) = 4pi/3 * rho(t_0) * (r(t_0))^3
so particle II feels a gravitational acceleration, _towards us_, of
  a(t_0) = - G M(t_0) / (r(t_0))^2 .
As time goes on, particle II falls towards us, so r(t) decreases, and
rho(t) decreases due to the universal expansion, but the acceleration
remains toward us, and so particle II eventually reaches us, although
the specifics of r(t), rho(t), etc. will depend on the specifics of the
universal expansion (and possible contraction).

*(Exercise: calculate the critical density rho in terms of H.  rho is the
density for which the separation velocity v goes to zero as r (and time)
go to infinity.  You should get the same as the textbook general relativity
answer.)

Probably somebody is screaming bloody murder right now, because it seems
completely illegitimate for me to assume that the mass in the sphere
centered on us counts and the rest of the universe doesn't!  I know,
I used to feel that way about this approximation, but try this: put the
center at the Glorbian galaxy, with both us, at radius r = 200 Mpc,
and particle II, at r = 0, moving outward at velocity v = Hr.  Using
the same approximation, but centered at the Glorbians, you can calculate
the acceleration of particle II relative to us, and get the same answer 
as before.

>Distant objects will in general 
>move away from the test particles, because the distant objects
>inherited their velocity from the big bang and continue to move
>with that velocity (slowing due to mutual gravity, to which the
>test particles add trivially). One can say that space itself is
>expanding, but this expansion does not drag particles along
>with it; it might be better to say that the expanding distribution
>of original matter/energy creates new space as it expands. 

It definitely doesn't drag particles.  You can view as creating new
space, though I prefer to think of stretching the space which is
already there.

>A big, misleading flaw in the "expanding balloon" or "rising loaf"
>pictures is that added test "particles" (dots painted on the balloon,
>raisins stuck into the loaf) WILL move away from each other.

No, because if you paint a dot onto the balloon, you're implicitly
making it stationary in the frame of the comoving observers: the "dots"
which were already on the surface of the balloon.  To get dots
stationary w.r.t. each other, you can paint one on, but then the other
one has to start off in motion, moving across the surface in order to
compensate for the expansion (stretching) of the ballon's surface.

Personally, I prefer the balloon analogy, because an ordinary loaf
has an edge, whereas the balloon, like the universe, has no edge.
Unfortunately the balloon is finite where the real universe can be
(and probably is) infinite.  I haven't seen too many balloons with
hyperboloidal geometry ...

>Right?

Sort of.  I hope these examples help people understand this problem;
it may take a little while, so if you're confused print it out, and
do some doodles and calculations to see what's going on.  _After_
that, you can flame me.
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