Re: Galaxies without dark matter halos?

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

>> 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.

Rather than following Usenet convention and replying point by point
(which would make for a long and unwieldy post that no one would
want to slog through), I'll try to summarize where we disagree
and explain my position reasonably succinctly.  Please let me know
if you think I've omitted an important facet of the discussion or
misrepresented your position in any significant way.

Everyone agrees on the following:

A comoving observer who looks at a comoving object in an expanding
FRW spacetime will observe that object to be redshifted.

I would like to defend the following proposition:

A. If the distance between observer and observed is much less than the
   scale of spacetime curvature, then it makes sense to describe that
   redshift as a Doppler shift.

(Here "scale of spacetime curvature" means either the horizon
distance or the radius of spatial curvature, whichever is smaller.
I labeled this proposition A because I want to compare it later with
another proposition, which I'll call B.)


Let me make a few general observations first.

- I'm not saying anything the least bit nonstandard about any actual
  measurements.  That is, you and I would actually calculate the
  redshift in the same way.  If I'm saying anything nonstandard at
  all, it's only about which ways to wrap words around the calculation
  are valid.

- The observed redshift is a coordinate-independent quantity (like all
  observed quantities), but the act of interpreting that redshift as a
  Doppler shift is a coordinate-dependent act.  That's not the least
  bit surprising: interpretations are often coordinate-dependent
  things.

- You describe what I'm doing as "mathematically inconsistent."  That
  may be true in some sense.  But only in a pretty benign sense: I
  claim that physicists are mathematically inconsistent in this sense
  six times before breakfast.  To put it another way, "Am I
  mathematically inconsistent?  Very well, then, I am mathematically
  inconsistent.  I am large; I contain multitudes."  (All silly
  literary allusions are confined to this paragraph, by the way.)

Now let me try to explain what I mean with an example.  Suppose
I stand on top of a tall tower and drop a baseball out.  I track
it with a radar gun as it falls to measure its speed.  I claim
the following:

B. If the ball travels a distance that is small compared to the
   curvature scale of spacetime in my neighborhood, then it makes
   sense to interpret the observed redshift as a Doppler shift.

By the way, spacetime is very weakly curved in the vicinity of
Earth's surface, so the assumption happens to be fairly unconstraining
in this case.

The observed redshift will include a gravitational redshift as well
as a Doppler shift, of course, but the latter will be very tiny
under the assumed conditions; calling the observed redshift
a Doppler shift is a kick-ass approximation.

Personally, I have a hard time imagining anyone disagreeing with
proposition B.  If you do disagree with it, then you're much more
of a purist than I am: it seems to me that that would pretty
much mean that you aren't willing to take any sort of 
Newtonian limit of general relativity.  Anyway, if that is your
point of view, then I don't really want to argue about it; I'm
happy to agree to disagree.

If, on the other hand, you accept proposition B but reject proposition
A, then I'd like to know why.  I think that they stand on *exactly*
the same logical footing.  In both cases, the following are true:

- Spacetime is curved, but the curvature is small over the region of
  interest.

- The interpretation involves laying down a particular set of
  coordinates and "pretending" that spacetime is flat in those
  coordinates.  We know that that's not exactly true, but we pretend
  that it is anyway, because of the previous point (the curvature is
  weak).  (To use the hackneyed analogy, this is pretty much like a
  guy on a sphere pretending the sphere is a plane when surveying a
  small region.)

- In a different coordinate system, the coordinate velocity of the
  observed with respect to the observer would be zero.  In such a
  coordinate system, the observed redshift would be interpreted as a
  gravitational redshift.  (In the FRW spacetime, comoving coordinates
  are the obvious example.  In the falling-baseball case, something
  like ingoing Eddington-Finkelstein coordinates would be an example.)

- BUT those other coordinates do a "worse job" of approximating
  spacetime as flat, so IF we've decided to adopt the mindset of
  pretending spacetime is flat over a small region, those are worse
  coordinates to use than the ones in which the shift is a Doppler
  shift.

- By pretending spacetime is flat when it isn't, we're making errors.
  But those errors are quantifiable and small under the specified
  conditions.  Specifically, they're of order (length scale of
  observation) / (curvature scale), which is small by hypothesis.

I honestly don't understand how statements A and B differ in their
"acceptability," so if anyone thinks that B is acceptable but A isn't,
I'd love to hear why.

-Ted


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