Re: Cosmological redshift and Doppler shift

[EMAIL PROTECTED] wrote:

> I'd like to ask anyone who's been following the discussion about the
> low-redshift limit of the cosmological redshift to answer the
> following question.  I'm going to list a sequence of statements, all
> of which I believe.  I'll order them from least to most controversial.
> At what point, if any, do you part company with me?
>
> 1. An observer in flat (Minkowski) spacetime measures the redshift of
>    light from a source and finds z = Delta lambda/lambda = 0.01.  She
>    can use the special-relativistic Doppler shift formula to determine
>    the source's speed relative to her (getting the answer v =
>    0.01c).
>

      Agreed.

>
> 2. An observer in an open FRW spacetime with zero density (Omega = 0)
>    measures the redshift of light from a source and finds z = 0.01.
>    She can use She can use the special-relativistic Doppler shift
>    formula to determine the source's speed relative to her.
>

       Agreed.

>
> 3. An observer in an open FRW spacetime with density parameter Omega =
>    10^(-50) measures the redshift of light from a source and finds z =
>    0.01.  She uses the special-relativistic Doppler shift formula to
>    calculate a speed.  To an excellent approximation, she can
>    approximate spacetime as flat and interpret that number as the source's
>    speed relative to her.
>

       Agreed (see below).

>
> 4. An observer in an open FRW spacetime with density parameter Omega =
>    1 measures the redshift of light from a source and finds z = 0.01.
>    She uses the special-relativistic Doppler shift formula to
>    calculate a speed.  To a good approximation, she can approximate
>    spacetime as flat and interpret that number as the source's speed
>    relative to her.
>

      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. 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. Thus one
      cannot find a neighbourhood of the observer where the expansion
      of the comoving observers can be described as in the Milne model
      to any approximation. This suggests that the interpretation of
      cosmological spectral shifts as Doppler shifts in flat space-time
      is misleading at any scale for a Omega=1 FRW model.

      Thus it seems that the Omega=1 FRW model has some features
      similar to the example of hovering observers in Schwarzschild
      space-time: For both these models one may chose an event where
      space-time is flat to the desired accuracy; yet the effect of
      space-time curvature on V is 100%. This does of course not mean
      that we cannot construct a field of 4-velocities in the tangent
      space-time of the observer using the "Hubble law". But for both
      these models, a sensible interpretation of this field as coming
      from recessional velocities in flat space-time does not exist at
      any scale.