Re: big-bang energetics

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's comments about my post:

> >  Global energy
> > conservation is not guaranteed (or even objectively definable) in GR--
> > especially for a metric that is (a) everywhere time dependent, and (b)
> > not asymptotic to a flat Minkowski metric at 'infinity' for a universe
> > that has an always a localized distribution of matter.  Unfortunately,
> > the usual Friedmann-Robertson-Walker metric models for the big bang
> > satisfy neither a) nor b).
>
> That's interesting.

Actually, I misspoke myself here.  I meant to say that, unfortunately,
the usual FRW models satisfy *both* a) *and* b).  IOW, the relevant
models are of the very kind that *doesn't* allow an objective statement
of global energy conservation.  I suspect that you knew what I meant
anyway.

> You've said there is a local continuity equation, which I assume is a
> differential equation which in pedestrian 3+1 coordinates would be roughly
>        - d stuff / d t = div flux

Yep.  It's the *same* relativistically as far as the interpretation is
concerned.  The only difference is that it is written in a covariant
4-vector notation.  In that notation the [stuff density]*c and the
[flux density of the stuff] together form a 4-vector
{stuff,stuff current} and the 4-divergence operator contracts it to a
scalar where the time component is a (1/c)*d/dt operator and the space
components are a generalization of the spatial grad.  It should be noted
that the differentiation operations are a contravariantly raised version
of a covariant derivative (involving the covariant connection) of the
{stuff,stuff current} 4-vector and the continuity equation is:

[4-divergence]({stuff,stuff current}) = 0 .

> So my question concerns *regional* conservation.  Can I construct a pillbox
> and apply Gauss's law to get a regional integral form of the conservation
> law, or is the early geometry to messed up to permit that?

Yes, you can as long as the pillbox (closed volume region) region is
infinitesimal (compared to the curvature scale) in size and contains a
spatial neighborhood so small and you observe the flows across its
boundaries for a short enough time that the time-dependent local
curvature effects of spacetime do not begin to show up.  In the case of
the FRW models that model our current universe this means the "pillbox"
region is smaller than billions of light years across and you observe it
for a time short compared to billions of years.  Remember, sufficiently
locally, GR boils down to SR where the curvature effects of the spacetime
manifold become insignificant.  This is like being able go get away with
using a flat street map of a city, but needing a round globe to map the
whole earth with any accuracy.

> (Thanks for your many thoughtful and thought-provoking posts.)

You're welcome, and thanks for the compliment.  I do wish I could do a
better job of proofreading and avoid so many typos in my posts though.

I've learned a lot from your posts as well.

David Bowman
dbowman@georgetowncollege.edu