From: carlip@landau.ucdavis.edu (Steve Carlip)
Newsgroups: sci.astro
Subject: Re: "Modeling" the Expanding Universe? (was Re: That Great Pulsar Timing Flame War)
Date: 24 Jan 93 22:26:22 GMT
Organization: Physics, UC Davis

In article <C18v0D.6K1@well.sf.ca.us> metares@well.sf.ca.us (Tom Van Flandern) writes:

>[...]The Hubble rate appears in Einstein's Equation, but is epoch-
>dependent.

Yes, certainly.  

>Only the second derivative of scale is fixed by the matter
>density, assuming no pressure.  

No. By combining the time-time and space-space components of the
field equations, you obtain an equation for the first derivative of
the scale --- the expansion rate itself --- in terms of the matter
density.  In particular, there are no extra integration constants
to play with.

>  Given that high-enough matter density will always eventually convert
>an expansion into a contraction, and that a non-expanding, 
>non-contracting
>space is unstable without pressure (e.g., the cosmological constant), 
>I do
>not see how it is possible for local space to be neither expanding nor
>contracting.  It appears obvious to me that either a) local spacetime 
>does
>not obey Einstein's Equation; or b) a compensating pressure is 
>required to
>counterbalance the universal expansion.

I'm still not sure why this is confusing, but here's a guess.  I think
you're looking at properties of *a* solution of the Einstein equations
with a *particular* matter distribution, and trying to treat these
as universal characteristics.  You are correct in saying that a
high enough *homogeneous, isotropic* matter density will eventually
convert expansion to contraction.  But on the other hand, there are
certainly solutions to the field equations, such as the Schwarzschild
solution, in which the scale factor is constant.  The geometry of
spacetime depends on the details of the matter distribution, not
just on averages; standard predictions of Friedman-Robertson-Walker
cosmology are trustworthy only to the extent that the approximations
they make about the matter distribution are reliable.  

In particular, your statement that "a non-expanding, non-contracting 
space is unstable without pressure" is wrong; the correct statement 
is only that "a non-expanding, non-contracting space *with a uniform 
mass density* and without pressure is unstable."

It's dangerous to try to pull rules of thumb out of particular
solutions, especially when those solutions have a high degree of
symmetry.  It is as if I said, "Newtonian gravity successfully
predicts that planets should move in elliptical orbits.  But we
know that some comets move in hyperbolic orbits; this is not
consistent, and requires some new ingredient."   

I realize that this is essentially a negative response --- I'm
telling you why your intuition here is wrong, but not suggesting 
an alternative.  I don't work in cosmology myself (at least,
not beyond the cosmology of the first 10^-40 seconds), and my
gut reaction is that the field equations are a complicated set
of nonlinear partial differential equations, for which intuition
is not to be trusted.  If I hadn't read about the results of
(hard) computations, I wouldn't be able to begin to guess whether
small gravitationally bound systems would be coupled to the Hubble
expansion or not.  

Maybe there's someone out there who's worked on this problem and
has a better intuitive sense of the solutions...

Steve Carlip
carlip@dirac.ucdavis.edu