Re: Galaxies without dark matter halos?

Ted Bunn wrote:

> Dag Oestvang  wrote:
>
> >    The attractiveness of the unified approach to spectral shifts is
> >    that this approach clearly shows how spectral shifts are related
> >    to the geometry of space-time.
> >
> >    In particular, spectral shifts due to curved space-time geometry
> >    should never be thought of as ordinary Doppler shifts in flat space-time.
> >    That is, it is not meaningful to approximate curved space-time with
> >    flat space-time and at the same time keeping spectral shifts due
> >    to curved space-time; such a scheme would be inconsistent.
>
> This is precisely the position that I'm disagreeing with.  Thank you
> for stating it so lucidly.
>
> I claim that it the following procedure is a perfectly meaningful,
> consistent, and moreover extremely useful way to describe the
> expanding Universe on small scales:
>

    Well, in my opinion you are mistaken, see below.

>
> 1. Decide to approximate spacetime as flat over a finite neighborhood.
>

      By this I suppose you mean that space-time curvature can be
      neglected to some specific extent in a neighbourhood of some
      arbitrarily chosen event. However, the size of such a
      neighbourhood is determined by the limits you set on the
      negligence of space-time curvature and vice versa; you cannot
      specify both freely.

>
> 2. Lay down a coordinate system (such as Riemann normal coordinates)
> that "does the best job possible" of approximating curved spacetime as
> flat spacetime over this neighborhood.
>

    But if space-time _is_ curved, no coordinate system can change
    that. Riemann normal coordinates are useful inasmuch that  using
    such coordinates, all connection coefficients vanish at _one  chosen
    event_ .  But the derivatives of the connection coefficients do not vanish
    at this event, so the connection coefficients are in general non-zero in
    any neighbourhood of this event. In fact, parallel-transporting a vector
    along a geodesic connecting some event inside the neighbourhood  to
    the specific chosen event is an operation independent of coordinates. The
    only point with choosing a sufficiently small neighbourhood is that this
    reduces the effects of space-time curvature to the wanted level. This
    is also independent of coordinates.

>
> 3. Calculate the redshift of nearby galaxies using the standard
> Doppler-shift formula.
>

     But regardless of the coordinate system, some of the connection
     coefficients in your neighbourhood will be of size H. And those
     connection coefficients gives you a spectral shift due to space-time
     curvature via parallel transport.

     What you relly have done is this: The 4-velocity of the emitter
     rotates some (small) amount due to curved space-time during the
     parallel transport to the observer. This (small) effect should be
     neglected since that was the point with choosing a small neighbourhood
     in the first place. But rather than neglecting it,  you interpret the
     rotation of the 4-velocity in some fictious space-time which is flat (to
    some unspecified extent) within your neighbourhood.

>
> This procedure works: it gives the right answer, up to errors of order
> (size of neighborhood) / (spacetime curvature scale).  And in this
> approximation, the galaxy's redshift is a Doppler shift.
>

     That the procedure gives the right answer is the best thing that can
     be said about it. However, it is mathematically inconsistent, and the
     physical interpretation of the spectral shift as a Doppler shift in
     flat space-time is at best extremely dubious.

>
> A key principle of general relativity is that it reduces to special
> relativity over small scales.  This is just an example of that.
>

    No, it is not.

>
> I honestly don't understand why this procedure is any different from
> what we do all the time when we use special relativity to analyze
> experiments in terrestrial labs.  We know that spacetime in the
> vicinity of the Earth isn't flat, but we also know that pretending it
> is flat is an excellent approximation over small enough length and
> time scales.  In circumstances in which we're willing to ignore errors
> of order (length scale of experiment)/(spacetime curvature scale), we
> cheerfully pretend spacetime is flat, lay down appropriate
> coordinates, and apply special relativity.  We can do exactly the same
> thing in smallish neighborhoods of an expanding
> Friedmann-Robertson-Walker spacetime.
>

    But if this is done consistently, there will be no cosmological
    spectral shift. That is, since the point with choosing small regions of
    space-time is that this lays quantitative limits on the effects due to
    curved space-time, and since none of the connection coefficients
    which yields the cosmological spectral shift is larger than these
    limits, the only consistent thing to do is to neglect them.

>
> I suspect that some people who say that you can't do this believe
> something like the following: if you follow the procedure I've
> outlined above, you calculate redshifts of zero for the other
> galaxies, because the redshifts themselves are of the same order
> (neighborhood size / curvatur scale) as the errors.

     What counts here is the size of the connection coefficients,
      not the redshifts themselves. Apart from that, this is correct.

> But that's not so.  If you lay down Riemann normal coordinates in a  neighborhood

> of an FRW spacetime, you find that other galaxies are moving away from
> us in those coordinates in accordance with Hubble's law.
>

    I'm afraid that this is inconsistent, see above.