Re: Galaxies without dark matter halos?

In article <[EMAIL PROTECTED]>,
Tim S  <[EMAIL PROTECTED]> wrote:

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

That's certainly true, but it's not relevant to the point I was trying
to make.  My argument was based on the fact that the geometry of spacetime
is approximately *flat* (i.e., Minkowski), not FRW.

To be a bit more precise, I'm basing my argument on the utterly banal
observation that everything is linear to first order.  If you want to
study phenomena on length scales much smaller than the curvature
scale, then you can view spacetime as approximately Minkowski, with
small perturbations of order 

epsilon = (length scale of observations) / (curvature scale of spacetime).

When you observe a frequency shift near the Earth's surface,
and you interpret it as a Doppler shift, that's precisely what you're
doing: you're pretending that spacetime is flat and applying special
relativity.  You know that you're making errors because spacetime
is not perfectly flat, but you also know that those errors are
small (because epsilon is small under the circumstances).  In short,

   As long as you're willing to ignore errors of order epsilon, you're
   allowed to pretend that spacetime is flat and interpret spectral
   shifts as Doppler shifts.

I claim that interpreting the redshift of a nearby galaxy in an FRW
Universe as a Doppler shift relies on exactly the same assumptions,
and should be regarded as exactly as valid, as using a radar gun to
measure the speed of a Pedro Martinez fastball.

[OK, I confess I'm exaggerating a bit there.  The quantity epsilon is
 much smaller in the latter case than in the former, so you're making
 less of an error.  But the two cases are exactly the same in
 principle: the act of interpreting the observed spectral shift as a
 Doppler shift (as opposed to a gravitational shift) rests on
 neglecting corrections due to spacetime curvature.  If those errors
 are small, and you're willing to ignore them, you can ignore them.]

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

Maybe I'm missing something, but I don't see the relevance of this
observation.

>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.

Ditto.

I'll change my example to eliminate this feature, if you like.  Put
yourself on a satellite in an elliptical orbit.  Use a radar gun to
measure the speed of another satellite in an elliptical orbit.  Again,
to a kick-ass approximation, the number that you get out can be
interpreted as a Doppler shift (as long as conditions are such that
epsilon is small, which is an easy condition to satisfy in these
circumstances).  Again, the reason that's true is that you can
approximate spacetime as flat + perturbations of order epsilon to a
kick-ass approximation.  Again, the above statements apply equally
well to a small neighborhood of an FRW spacetime.

>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. 

The reason this is "uncontroversial" is precisely because spacetime
is flat to an excellent approximation.  If that weren't true, then
you couldn't even unambiguously defined "relative velocity."  

For instance, suppose you moved the experiment to a region just barely
outside the Schwarzschild radius of a black hole, such that the
quantity epsilon was not small.  Then the question of whether an
observed spectral shift was Doppler or gravitational would not be
"uncontroversial": different people might analyze the situation using
different coordinates (one using Schwarzschild coordinates, one using
Kruskal coordinates, one using Eddington-Finkelstein coordinates).
They'd all agree on the observations, of course, but they'd disagree
on the interpretation.

The reason this doesn't happen for experiments near the Earth's
surface is that the quantity epsilon is small, which means that
there's an obvious coordinate system in which to analyze things
(the coordinate system that makes spacetime look flat).

>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.

I don't really understand why that's relevant, but if it's bothering
you, then let's switch from the dropped baseball example to my new
satellite example above.  Let the two satellites by in very different
orbits (so that their separation is of order the orbital radii).  It's
still much smaller than the curvature scale, so epsilon is still
small, but now tidal effects are important.

>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. 

This sort of statement is precisely the reason I want to harp on this
point.  I think it's a very misleading way to think about the
low-redshift FRW Universe.  One way to understand why is to think
of the Omega -> 0 limit of the FRW spacetime (i.e., the Milne model). 
When Omega = 0, spacetime becomes exactly Minkowski.  It would surely
be absurd to deny that the observed spectral shift of test particles
in that spacetime was a Doppler shift.  Yet as soon as Omega
becomes nonzero (even if it's arbitrarily small), all of a sudden
those observed redshifts are 100% due to "frame dragging" by the
(aribtrarily close to massless) stuff flying through the Universe?
I can't imagine seriously believing that.

>The approximation in which the curvature is
>negligible is precisely the approximation in which the redshift is
>negligible.

That last statement is just factually false.  I promise.  Do the
calculation.  Figure out the coordinate system that best approximates
a neighborhood of FRW spacetime as flat.  Those coordinates will not
be comoving coordinates.  They'll be coordinates in which the galaxies
are flying away from the origin at speeds given by Hubble's law.

-Ted


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