Re: Cosmological redshift and Doppler shift

In article <[EMAIL PROTECTED]>,
Dag Oestvang  <[EMAIL PROTECTED]> wrote:

>      But here we part company. That is, I believe that when Omega=1,
>      any sensible interpretation of cosmological spectral shifts as Doppler
>      shifts in flat space-time breaks down at all scales.
>
>      To justify this view it is useful to consider the geometry of the
>      Milne model in some detail. Since this is just the empty FRW model,
>      space-time is flat. In fact, space-time is a piece of Minkowski
>      space-time; namely the region inside the future light cone of
>      some point in Minkowski space-time. Moreover, the hypersurfaces
>      t=constant have hyperbolical geometry and the expansion of
>      the comoving observers is exactly described as a field of 4-vectors
>      in flat space-time.
>
>      Now consider a model where Omega is non-zero, but small (as in
>      example 3 above).  In this case the hypersurfaces t=constant
>      still have hyperbolic geometry. This means that for every point
>      in space-time we can find a neighbourhood where the curvature of
>      space matches the curvature of space in the vincinity of an event in
>      the Milne model to the desired accuracy. Besides, in such an
>      neighbourhood space-time can be considered flat to the desired
>      accuracy. But this means that the expansion of the comoving
>      observers can be described exactly as in the Milne model to the
>      desired accuracy in the chosen neighbourhood; i.e. that V to a good
>      approximation comes from a field of 4-vectors in flat space-time.
>
>      But when Omega increases, the size of the neighbourhood where
>      this description is appropriate, shrinks. That is, if one neglects an
>      effect of curved space-time on V of 10%, say, then the size of the
>      region (centered on the observer) where this limit holds shrinks
>      when Omega increases. And when Omega increases towards 1
>      the size of this region shrinks to zero. 

I'm having a bit of trouble understanding what you're saying here.
You seem to be saying this:

   Given a point in an Omega = 1 FRW spacetime and a level of desired
   accuracy epsilon, there is no finite-sized neighborhood of that
   point in which spacetime can be approximated as flat to an accuracy
   epsilon.

Is that what you're saying?  If so, then that completely explains 
our disagreement.  This statement is wrong.

In fact, this statement is precisely equivalent to

   An Omega = 1 FRW spacetime is not a Lorentzian manifold.

Or, if we define the word "spacetime" to mean "Lorentzian manifold"
(which is essentially what we do in general relativity), to

   An Omega = 1 FRW spacetime is not a spacetime.

More or less by definition, a Lorentzian manifold is one that can be
approximated arbitrarily well on sufficiently small scales as
Minkowski spacetime.  (Compare to Euclidean manifolds: a Euclidean
manifold, by definition, can be approximated arbitrarily well on
sufficiently small scales as flat.)  In the epsilon-delta language
beloved of mathematicians, a Lorentzian manifold must satisfy the
condition that, at any point P, for any epsilon>0, there is a delta>0
such that spacetime deviates from Minkowski by less than epsilon in a
neighborhood of radius delta of P.

To put it more concretely, suppose you lived in an Omega = 1 FRW
Universe at a time 14 billion years after the big bang.  You draw a
sphere of radius one millimeter.  Do you really claim that spacetime
deviates from flat by more than 10% within that sphere?  I swear to
you, on a stack of Misners, Thornes, and Wheelers, that it doesn't.
(If it did, how did anyone ever manage to do experiments confirming
special relativity?)

If your above statement doesn't mean what I wrote above, then
the only other meaning I can ascribe to it is this:

   Given a point in an Omega = 1 FRW spacetime, the
   constant-cosmic-time hypersurface through that point, and a level
   of desired accuracy epsilon, there is no finite-sized neighborhood
   of point P in which the hypersurface can be approximated as flat
   (Euclidean) to an accuracy epsilon.

That's a slightly different statement, but it's still false.  This
statement is equivalent to the statement that the constant-cosmic-time
hypersurface is not a Riemannian manifold, which it is.

Incidentally, when approximating an FRW spacetime as flat in the
neighborhood of a point, the surfaces of constant time do differ from
surfaces of constant cosmic time at second order.  That is, if T
stands for the time coordinate in Riemann normal coordinates that best
approximate spacetime as flat, and t stands for cosmic time, then

T = t + O((r/R)^2),

where the small quantity r/R is the ratio of the distance to the
Hubble distance. 

>      That is, when Omega=1
>      space is no longer hyperbolic but flat. This means that we cannot
>      find any event in the Milne model where the curvature of space
>      matches the curvature of space in a Omega=1 model. 

Two points here:

1. We can't exactly, but we *can* to any desired accuracy, in a small
neighborhood.  (Again, this is the definition of a Riemannian
manifold.)

2. Worrying about whether we can approximate the constant-cosmic-time
hypersurface to arbitrary accuracy is a bit of a red herring anyway.
What matters is whether we can approximate *spacetime* to arbitrary
accuracy.  (It doesn't really make much difference, though, since
we can do both.)

-Ted


-- 
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