From: sbs@weyl.bu.edu (Stephen Selipsky)
Newsgroups: sci.astro
Subject: Re: "Modeling" the Expanding Universe? (was Re: That Great Pulsar Timing Flame War)
Date: 26 Jan 93 17:32:09 GMT
Reply-To: sbs@weyl.bu.edu
Organization: Boston University Physics Department


In article <C1GABF.1sK@well.sf.ca.us>, metares@well.sf.ca.us
 (Tom Van Flandern) writes:

|>From: metares@well.sf.ca.us (Tom Van Flandern)
|>Newsgroups: sci.astro
|>Subject: Re: "Modeling" the Expanding Universe? (was Re: That Great Pulsar Timing Flame War)
|>Message-ID: <C1GABF.1sK@well.sf.ca.us>
|>Date: Tue, 26 Jan 93 02:33:14 EST
|>References: <21629@ucdavis.ucdavis.edu> <C18v0D.6K1@well.sf.ca.us> <21736@ucdavis.ucdavis.edu>
|>Sender: news@well.sf.ca.us
|>Organization: Whole Earth 'Lectronic Link
|>Lines: 57
|>
|>
|>Earlier, I wrote:
|>
|>>> 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, I do not see how it is possible for
|>>> local space to be neither expanding nor contracting.
|>
|>and carlip@landau.ucdavis.edu (Steve Carlip) replied:
|>
|>> 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  ...  the field
|>> equations are a complicated set of nonlinear partial differential
|>> equations, for which intuition is not to be trusted.
|>
|>     Well, I agree up to a point.  But we mustn't throw reason out the
|>window.  If solutions do not give reasonable physical behavior, then
|>something is wrong.
|>
|>     We agree that if the matter distribution is homogeneous and isotropic,
|>all of space can expand or contract.  Now consider a homogeneous, isotropic
|>universe plus a single locality in it with an abnormally high matter
|>density.  Let that abnormal matter density approach infinity.  [I consider
|>20 orders of magnitude, as in the case of the solar system, as an
|>approximation of infinity.]
|>
|>     Now as I understand it, the Einstein's Equation is supposed to have
|>solutions that will allow the nearly infinite matter density locality to
|>halt a general expansion and remain flat.  But the condition of local
|>spacetime depends only on the matter distribution in the universe, not on
|>its motion.  Therefore we can make the same statement if the universe is in
|>a state of general contraction: a nearly infinite matter density locality
|>must halt the general contraction and remain flat!  I submit that goes
|>beyond counter-intuitive into the unimaginable.
|>
|>> 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 sincerely appreciate your efforts.  I'm content to give this
|>argument up if a good explanation arises.  You see all the elements of my
|>argument -- which one is wrong?  You've made the best effort yet at
|>addressing my question using my own language and concepts.  But because of
|>the picture I drew above, I'm starting to suspect that there may be no way
|>out of this dilemma despite your valiant efforts.  I'm still open to
|>suggestions.  -|Tom|-
|>
|>-- 
|>Tom Van Flandern / Washington, DC / metares@well.sf.ca.us

I'm a particle theorist not a GR expert, but here's my attempt to combine
nonlinear GR with Newtonian intuition for Tom.  Take a contracting or
at least a closed and later-to-become-contracting universe (the average
density TODAY is just above critical density).  A local region is 20
orders of magnitude denser, and is "decoupled from the general expansion
or contraction" in the sense that geodesics in the local metric diverge
or converge depending not on the external metric, but instead on their
relative velocity (magnitude and angular momentum). Inside objects, there
is enough pressure or stress to push stationary particles off converging
geodesics of the local metric.  The local metric is a solution of Einstein's
equation with source term the LOCAL energy-momentum tensor (local density),
and boundary condition that it match onto the external (closed) Robertson-
Walker metric, at a radius enclosing enough volume that the local dense
region doesn't dominate the mass within that radius.  TODAY this gives
essentially a local Schwarzchild metric, since the boundary condition is
an essentially flat external metric.  Your intuition works fine, and
continues to work when eventually the external universe starts to contract.
Aside from red-shifted galaxies, there are no local consequences until the
contraction has gone 20/3 \approx 7 orders of magnitude.  Then the external
density will approach and exceed that of the local region, and the local
region will be crushed in the general collapse; it is only decoupled when
its boundary "matches on" to an external universe with mass density
negligible compared to the local region's.  When the external mass density
isn't negligible, you still (Birkhoff's theorem) match a local Schwarzschild
solution onto the strongly curved and time-dependent external metric.  In
other words, nothing weird and nonlinear happens; the Solar system just gets
crushed as all that external matter crashes in.  "The fall doesn't kill you,
it's when you hit the bottom".

Regards,  -- Stephen Selipsky