Re: opposite of relativistic time contraction

Tim Little wrote:
> Oh yes, the gravitational time dilation due to galactic and
> supergalactic structures is pretty minimal, probably less than 1 part
> in 10^8.  

I'm not so sure. When Bryan first posted this, it struck me as
counterintuitive that the effect could be so big, but it also
triggered a memory that I'd read about such an effect somewhere
not too long ago, and that it was indeed counterintuitively large.

The biggest known structures are ~5% the size of the observable
universe. In a region that size, you're talking about internal
motions that are getting to be fairly relativistic, and you're
also talking about a total mass that's getting to be comparable
to that of the entire observable universe. If you try to figure
out how the structure's KE compares with its gravitational
binding energy, then you're essentially doing the same sort of
calculation you'd be doing to figure out whether the universe
is open or closed. Since the universe is pretty close to the
border between open and closed, I think the answer is that
the structure's KE is comparable to its binding energy, and
since the KE is relativistic, I think that phi/c^2 (phi
being the gravitational potential) is within one or two
orders of magnitude of 1. Of course there's a lot of handwaving
here, and I could be totally wrong :-)

Here's a different argument, however, that seems to lead me to
the same conclusion. Say you have some lump of mass that's
surrounded by a lot of empty space. The gravitational potential
goes like
   phi ~ m/r
       ~ rho r^3/r
So as you go to bigger and bigger scales, it gets bigger and
bigger like r^2. This is assuming that lumps on different scales
have similar densities, which is certainly not true, but all
I'm aiming for is a crude plausibility argument here. So the
local variations in phi/c^2 keep on scaling up with r^2 until
they're of order unity, at which point you've reached the size
of the observable universe.