Re: Cosmological redshift and Doppler shift

[EMAIL PROTECTED] wrote:

> Dag Oestvang  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.
>

    Indeed. From your comments below it seems that you suffer from a
    fundamental misconception; namely this:

    You seem to think that the fact that space-time can be described as
    flat  in a neighbourhood is a sufficient condition that allows you to
    neglect the contribution to V from space-time curvature in this
    neighbourhood and at the same time keeping V to lowest order using
    the Hubble law.

   This is what you believe, right?

   But this is not necessarily true because V obtained from the Hubble
   law in general contains contributions both from space-time curvature
   and from velocity. In particular, if the contribution from velocity is
   zero, the contribution from space-time curvature obviously dominates
   at any scale. This is what happens in the example of hovering
   observers in Schwarzschild space-time. And this is also what happens
   for an Omega=1 FRW model. The proof of the latter assertion is
  simple: Write down the metric for the Omega=1 FRW model

  ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2]   (1)

  (with the correct form for a(t) inserted). The tangent space-time
  of some event with coordinates (t_1, x_1,y_1,z_1) has the flat
  metric

  ds^2 =  -c^2dt^2 + a^2(t_1)[dx^2 + dy^2 + dz^2] .    (2)

  The comoving observers move orthogonally to the t=constant
  hypersurfaces, and with a trivial rescaling of the spatial
  coordinates in eq. (2) we see that this is equivalent to world
  lines x(t)= const., y(t)=const., z(t)=const. in Minkowski
  space-time equipped with Cartesian space coordinates.
  But then we trivially see that the comoving observers do not
  expand in the tangent space, so the contribution to V coming
  from velocity is 0%; this means that 100% comes from
  space-time curvature at any scale, as asserted.

> 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?

    This is NOT what I am saying; not even close.

    [irrelevant elementary stuff snipped]

>
> 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.
>

     This is still  NOT  what I am saying. In fact you are way off base.

     [more irrelevant stuff snipped]

>
> >      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.)
>

     You are missing the point entirely here; see below.

>
> 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.)
>

    As I said above, the fact that space-time can be approximated as
    flat is not sufficient to sensibly interpret V as coming from velocity
    if this contribution is zero in the first place. Thus one needs a way to
    see how much velocity contributes to V at any scale. To do that one
    needs to understand the Milne model in some detail. In particular, we
    have the following:

     For a field of 4-velocities in _flat space-time_ to mimic a cosmic
    expansion, the hypersurfaces t=constant _must_ have hyperbolic
    geometry.

    This means that if the following conditions are fulfilled in some
    neighbourhood of an event in a FRW-model, the contribution from
    velocity is nonzero in this neighbourhood (no peculiar velocities):

     1) Space-time is flat to desired accuracy in this neighbourhood,

     2) Space is hyperbolic to the desired accuracy in this neighbourhood.

     Now the crucial point is this: IT IS THE CURVATURE OF HYPERBOLIC
     SPACE IN THE NEIGHBOURHOOD WHICH DETERMINES THE
     CONTRIBUTION TO V FROM VELOCITY. This means that it is NOT
     sufficient to approximate space as flat since this yields zero contribution

     to V from velocity (see the proof for the Omega=1 FRW case above).

     You _really_ need to learn this stuff if you want to have a proper
     understanding  of cosmic expansion and its interpretation in the
     setting of the FRW models.