Re: Galaxies without dark matter halos?

on 25/10/03 9:28 pm, [EMAIL PROTECTED] at
[EMAIL PROTECTED] wrote:

> 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.)

<snip further clarification>

> 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.

I can't speak for Dag Oestvang, but your comparison seems to present the
following difficulties:

1) The geometry of spacetime in the immediate vicinity of the earth is not
even approximately FRW.

2) You and the baseball are (a fortiori) not even approximately comoving.

Crucially, the following more specific statements are true:

1') The geometry of spacetime in the immediate vicinity of the earth is (to
a very good approximation) static.

2') The baseball is following a geodesic, but you aren't: you have a proper
acceleration of 9.8 m s^-2 due to the upward force exerted by the tower.

The bulk of the baseball's redshift as measured by you is uncontroversially
due to the non-zero relative velocity of the ball wrt you. You mention an
additional 'gravitation redshift', but I maintain that this redshift isn't
gravitational in the strict sense, but is actually _accelerational_. That
is, it's the extra redshift due to acceleration that would also be measured
by an accelerating observer in flat Minkowski spacetime.

In short, gravity is completely irrelevant in example B. The whole thing
could be done without significant alteration in SR. Note the irrelevance of
tidal effects, which are the identifying mark of GR.

By contrast, cosmological redshift is essentially gravitational; it cannot
be reproduced in flat spacetime. The role of the relative motion of the
galaxies is to produce frame-dragging, a GR effect, and it is the
frame-dragging that is responsible for the 'expansion of space' and hence
the cosmological redshift. The approximation in which the curvature is
negligible is precisely the approximation in which the redshift is
negligible.

I suppose it is possible that the following idea might be made to work,
although it doesn't feel right to me:

If we assume that our FRW model is a spatially flat one, then the spatial
slice at any given moment of cosmological time is Euclidean, and hence has
an unambiguous notion of distance between us and the distant galaxy. We can
then calculate the rate of increase of distance between us and the galaxy as
a function of cosmological time (or our proper time, which hopefully isn't
too different), and get a relative velocity, to which we could attribute the
redshift. (Note that this description is coordinate-free.)

However, conceptually this feels dubious to me. With a genuine doppler
shift, the redshift is basically determined at the point that the light is
emitted (assuming the observer doesn't accelerate in the meantime). With the
gravitational redshift, the redshift depends on what happens to space during
the time of flight of the light. That, as it were, is 'when' the wavelength
gets 'stretched'.

I also haven't done the calculations to check that this would come out
numerically right; if it didn't, that would of course throw off the whole
idea anyway.

<snip>

Tim

[Mod. note: it does come out numerically right: d(proper
distance)/d(cosmological time) is equal to the naively calculated
recession speed in the limit z -> 0. Personally, I think this isn't a
coincidence -- mjh]