Some thoughts on the expansion of the universe, and
its effects on the motion of objects in the universe.
In particular, a discussion of what it means to have
a non-steady rate of expansion.
There are two key bits of real data that support the
notion of expansion of the universe:
1) The Hubble expansion, and
2) The cosmic background radiation.
The Hubble expansion can be summarized by saying that
distant galaxies are receding, and the farther ones
are receding faster, in proportion to distance.
v = H * d
The problem is, there are two different ways to "explain"
this data. Call them explanation A and explanation B.
Let's see how they apply to observations (1) and (2).
A1) Suppose I send runners off in every direction. A
few days later I check to see where they are. The faster
ones are farther away. This is perfectly logical. This
has got !!nothing!! to do with expansion of the universe.
It's just a bunch of folks running around.
You _could_ explain the Hubble data this way. You could
say that the observed recession-versus-distance is just
a high-school distance=rate*time problem, wherein the
Hubble parameter H is just the reciprocal of the time
since they started running.
But that is not the only possible explanation; see
item (B1) below.
A2) But before we go, note that the distance=rate*time
explanation could not possibly explain the cosmic
background radiation that we observe. If I take
a bunch of optical-frequency light and shoot it out
into the universe, there will be an expanding sphere
of optical-frequency light. It will not of its own
accord cool to microwave frequencies.
B1) There is the venerable ants-on-a-balloon picture.
If we add air to the balloon, the density of ants must
decrease. (The number of ants is conserved, but the
amount of space in their D=2 universe is increased, so
the density must decrease.) Individual ants don't get
larger in size, but the distance between them increases.
(For simplicity please imagine a _cylindrical_ balloon;
I want something with zero intrinsic curvature.)
Now this picturesque analogy has some strengths and some
weaknesses. One pitfall is that students often imagine
that the ants are somehow stuck to the fabric of
spacetime. (Real ants, after all, have very sticky
feet.) But that's not right. Imagine instead that
the ants are free to follow the usual laws of motion;
some of them are skating around as free particles,
while others are exerting forces on each other, but
in !!all!! cases their D=2 velocity relative to the
"fabric of the universe" is irrelevant, in accordance
with Galileo's principle of relativity.
To make this pitfall explicit: Suppose I have a flat
rubber sheet, with some free particles resting on it.
I then stretch the sheet. The free particles just sit
there. If they occupied a 1 foot by 1 foot region
before expansion, they occupy a same-sized region
afterward. The _average_ density of particles in
the universe decreased, because the size of the
universe increased, but the _local_ density was nowhere
decreased.
Now let's return to the ants on a cylindrical balloon.
The ants were everywhere on the balloon before
expansion, so they should be everywhere afterward.
That means he local density must decrease along
with the average density. This is different from
the flat sheet, for a sneaky reason: it has to do
with the extrinsic curvature of the balloon. The
ants are forced apart as the balloon expands
outward into the embedding dimension. This is very
hard for the ants to understand, because they know
nothing of the embedding dimension, and they know
nothing of the extrinsic curvature of their universe.
They know nothing of the D=3 laws of motion.
All they see is a correction term to the D=2 law of
motion: objects initially at rest do not remain at
rest (almost but not quite).
Imagine two ants joined by a string. The size of
the ants isn't changing, and the length of the string
isn't changing. The expansionist equation of motion
is trying to move the two ants apart, and the string
must have tension in it to prevent this. The amount
of tension is proportional to the mass of the ants,
the length of the string, and the rate of expansion.
Call the present time t=0. For a one-year period
ten years ago (from t=-11 to t=-10) I added air to
the balloon at a steady rate. At all other times,
the size of the balloon has been constant.
During the year of expansion, the density of ants
steadily decreased. During all other times it has
been constant.
At the current time (t=0) the ants look out the window.
Suppose they only observe things within a ten light-year
radius, so we don't need to worry about the effects
of delay due to light-propagation. There will be no
Hubble expansion observed. The more-distant ants will
be more distant, but they will not have greater
velocities.
Distant ants had a Hubble velocity (proportional to
distance) during the year of expansion but not
otherwise. (Of course this applies only to free
ants, not those joined by strings.)
B2) During the year of expansion, any waves in the
ants' world got redshifted to lower frequencies. For
simplicity, consider a standing wave. As the universe
expanded, the nodes of the standing wave got farther
apart.
For supernatural D=3 creatures like us, we can explain
this cooling by saying that the waves have momentum,
and therefore have a pressure, and that this pressure
did work against "the fabric of the universe" during
the expansion. But the ants themselves know nothing
of this; all they see is correction terms to their
D=2 wave equations.
====================
To make the analogies explicit:
In the real universe, typical objects are like
the analogical ants: They do not change size when
the universe expands. Rulers do not get larger when
the universe expands; otherwise we would have no
way to measure the fact that the universe had expanded!
The stars in galaxies are like ants joined by strings:
they _would_ get pulled apart by the expansion, but they
don't, because they are bound by gravity. So even
galaxies don't increase in size.
Clusters of galaxies are arguably just barely bound, or
just barely not bound. Anything with a much lower density
than that won't have strong-enough gravity to resist the
expansion; for more on this see http://mailgate.nau.edu/cgi-bin/wa?A2=ind0102&L=phys-l&P=R16487