Re: Galaxies without dark matter halos?

From: [EMAIL PROTECTED]
Newsgroups: sci.astro.research
Subject: Re: Galaxies without dark matter halos?
References: <[EMAIL PROTECTED]> <[EMAIL PROTECTED]>

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

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

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.

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

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.

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

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.

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.  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.

-Ted



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