Re: Galaxies without dark matter halos?

[EMAIL PROTECTED] wrote:

>
> Dag Oestvang  wrote
> >
> >     From this we see directly that whereas V is approximately a
> >     3-velocity given from the Hubble law in the tangent space-time
> >     at the observer, there is no natural way to get V from the
> >     properties of flat space-time itself.
> >
> I don't know what you mean by this.
>
 I will try to clarify: Unless V to a good approximation appears 
 from a cleverly arranged field of 4-velocities in flat space-time 
 (this is exact in the Milne model), there is no natural way to 
 interpret V in this way if the effects on V of curved space-time 
 on parallel transport is not negligible.


 In other words; insofar as the effects on V coming from the
 curvature of space-time can be neglected it is perfectly all
 right to interpret V as coming from a field of 4-velocities in
 flat space-time. But it is not if the size of V depends crucially
 on space-time curvature.

> I claim that I can get V approximately
> by a completely different calculation:
>
> 1. Lay down Riemann normal coordinates in a neighborhood of the observer's
>    location.
>
> 2. Calculate the coordinate velocity of the source in these coordinates.
>
> 3. Interpret the result as the velocity of the source relative to the
>    observer.
>
> Step 3 involves pretending that spacetime is flat.  That's not exactly
> true, but it is approximately true.  Specifically, it involves errors
> of order
>
> epsilon = (source-observer distance) / (spacetime curvature scale).
>
> It's a good approximation when epsilon is small.
>
> If you perform these steps, the result is
>
> V = H_0 d
>
> just as it should be.  Because we've made that approximation, this
> should be interpreted as valid only up to fractional errors of order
> epsilon.
>
> Do you believe me so far?  That is, do you believe that the calculation
> I've described would yield the result V = H_0 d to the stated order
> of approximation?
>
>
     Yes.

> Maybe you believe that this calculation is correct, but you object to
> my interpretation of it.
>
     Exactly. You have constructed a field of 4-velocities in flat
     space-time which yields the correct answer (for small V). But
     for the non-empty FRW models there is some effect on the size of 
     V from space-time curvature on parallel transport in curved 
     space-time. If these effects are not negligible I consider the  
     interpretation of V as coming from a field of 4-velocities in flat
     as space-time spurious and misleading.

> Specifically, maybe I should lay stress
> on the word "natural" in your sentence
>
> : there is no natural way to get V from the
> : properties of flat space-time itself.
>
> Maybe you regard the calculation I just described as "unnatural."

     Well, yes.

> All I have to say about that is that it's precisely as natural as
> what people do all the time when they measure Doppler shifts
> in terrestrial experiments.

  No, whereas for terrestial experiments the effects on V coming 
  from space-time curvature are usually negligible (i.e. gravitational 
  spectral shifts are usually considered negligible), this may not be 
  the case for all the FRW models (e.g. omega > 1).

> >
> >      If this example is meant to be an analogy to the first one,
> >      I claim that it is quite misleading. To have analogous
> >      examples one should consider hovering observers in
> >      example B, or allow for non-comoving observers in
> >      example  A.
> >
> If by "hovering" you mean at rest in the static geometry, that's
> precisely the case I don't want to consider!  I'm trying to point
> out an example of a situation in which
>
> -spacetime is weakly curved
> -an observer measures a spectral shift of a source
> -the natural interpretation of that spectral shift is as a Doppler shift.
>
> I agree that hovering observers are not an example of that.
>
   The reason why I considered hovering observers was that for such
   observers the effects of curved space-time on V is 100%.
   This is not true for near-empty FRW models, but it becomes 
   closer when  omega increases.

> I fear you may think I'm saying more than I am.  All I'm saying is
> that in some fairly ordinary circumstances (the cop with the radar
> gun, or my example B, for instance), we interpret spectral shifts as
> Doppler shifts, because we're willing to pretend spacetime is flat.
> We're "allowed" to do that, as long as we're willing to ignore
> errors of order epsilon.

> >
> >    It is the interpretation in
> >    terms of flat space-time which gets misleading; see above.
> >
> All I'm saying is that it's OK to do this when the errors introduced
> by doing it are smaller than you care about.  (Honest: what I'm
> saying really is that banal.)

   It seems that we mostly agree. This is my position: If the effects 
   of curved space-time on V are negligible then interpretations of 
   spectral shifts as Doppler effects in flat space-time are OK; 
   otherwise not. Do you agree?