Re: Galaxies without dark matter halos?

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

>     Using eqs. (19)-(21) of the Narlikar paper some simple
>     calculations give the norm V of the 3-velocity (obtained from
>     the parallely transported 4-velocity of the sender )as seen
>     from the observers local inertial rest frame:
>
>             [a^2(t_o)/a^2(t_s)]-1
>    V =  ---------------------------- c                             (1)
>             [a^2(t_o)/a^2(t_s)]+1
>
>     where a(t) is the scale factor and the subscripts stand for
>     "observer" and "sender", respectively. (I do not use c=1 as
>     is done in the paper.) Notice that that V depends on a^2
>     rather than a.
>
>     When  t_o - t_s is small we can write a(t_s) as a
>     Taylor expansion and neglect higher order terms
>
>   a(t_s) = a(t_o) + [\dot a (t_o)](t_s - t_o) + higher order terms. (2)
>
>     Inserting (2) into (1) then yields
>
>             \dot a (t_o)
>V  \approx  --------------(t_o - t_s)c  \approx  H_0(t_o-t_s)c = H_0d (3)
>               a(t_s)
>
>     where d is the distance _in the tangent space-time_ (i.e. flat
>     space-time) at the observer.

I completely agree with this.  (I haven't checked the calculation
leading to equation (1), but it's certainly plausible to me.)

>
>     From this we see directly that whereas V is approximately a
>     3-velocity given from the Hubble law in the tangent space-time
>     at the observer, there is no natural way to get V from the
>     properties of flat space-time itself. 

I don't know what you mean by this.  I claim that I can get V approximately
by a completely different calculation:

1. Lay down Riemann normal coordinates in a neighborhood of the observer's
   location.

2. Calculate the coordinate velocity of the source in these coordinates.

3. Interpret the result as the velocity of the source relative to the
   observer.

Step 3 involves pretending that spacetime is flat.  That's not exactly
true, but it is approximately true.  Specifically, it involves errors
of order

epsilon = (source-observer distance) / (spacetime curvature scale).

It's a good approximation when epsilon is small.

If you perform these steps, the result is

V = H_0 d

just as it should be.  Because we've made that approximation, this
should be interpreted as valid only up to fractional errors of order
epsilon.

Do you believe me so far?  That is, do you believe that the calculation
I've described would yield the result V = H_0 d to the stated order
of approximation?

Maybe you believe that this calculation is correct, but you object to 
my interpretation of it.  Specifically, maybe I should lay stress
on the word "natural" in your sentence

: there is no natural way to get V from the
: properties of flat space-time itself. 

Maybe you regard the calculation I just described as "unnatural."
All I have to say about that is that it's precisely as natural as
what people do all the time when they measure Doppler shifts
in terrestrial experiments.  If a cop measures your speed with a radar
gun and pulls you over, he's interpreted the redshift he measured
as a Doppler shift.  He was able to do so because he was willing
to approximate the weakly curved spacetime in the vicinity
of his radar gun as flat spacetime.

If you're willing to insist that the cop has to parallel transport your
four-velocity along the null geodesic to his location before he
can pull you over, then I guess you can insist that I should do the
same when I observe the redshift of a nearby galaxy.  (In that case,
please let me know if you ever try to contest a ticket in traffic
court; I'd love to see it!)


[...]

>      Interpretations should also be made coordinate-independent as
>      much as possible.

Sometimes.  Not always.  For the cop pulling over the speeder, it
would be very inconvenient to have to write the ticket in a
coordinate-independent way.  Since he happens to be in a situation
where there's a convenient approximation (namely that spacetime is
flat), it may be preferable to interpret his results in a specific
coordinate system that takes advantage of that simplification.

>
>      If this example is meant to be an analogy to the first one,
>      I claim that it is quite misleading. To have analogous
>      examples one should consider hovering observers in
>      example B, or allow for non-comoving observers in
>      example  A.

If by "hovering" you mean at rest in the static geometry, that's
precisely the case I don't want to consider!  I'm trying to point
out an example of a situation in which 

-spacetime is weakly curved
-an observer measures a spectral shift of a source
-the natural interpretation of that spectral shift is as a Doppler shift.

I agree that hovering observers are not an example of that.

I fear you may think I'm saying more than I am.  All I'm saying is
that in some fairly ordinary circumstances (the cop with the radar
gun, or my example B, for instance), we interpret spectral shifts as
Doppler shifts, because we're willing to pretend spacetime is flat.
We're "allowed" to do that, as long as we're willing to ignore
errors of order epsilon.

>
>    Any spectral shift in GR may be interpreted as a generalized
>    Doppler shift in curved space-time. 

True.

>    It is the interpretation in
>    terms of flat space-time which gets misleading; see above.

All I'm saying is that it's OK to do this when the errors introduced
by doing it are smaller than you care about.  (Honest: what I'm
saying really is that banal.)

>    I'm no "purist". However, when forming opininions regarding
>    geometrical models, I prefer that my understanding is based
>    on geometry as far as possible. One example of this is what
>    the unified model of spectral shifts makes so clear; namely their
>    geometrical relation to curved space-time.  On the other hand
>    I find your position to be a little unclear and confused. For
>    example, the geometrical properties we are discussing has
>    nothing whatsoever to do with taking Newtonian limits.

Let me make clear: I love the unified picture of spectral shifts!  I
hadn't seen the Narlikar paper until a couple of weeks ago, and I find
this way of looking at things very natural and insightful.  But often
in life it's nice to have more than one way of looking at things.  In
particular, it's often nice to see how an exact but sometimes
complicated theory (say, general relativity) reduces approximately to
a simpler, more familiar theory (say, special relativity).  In the
limit where epsilon is small, that's what happens in the FRW case.

Let me explicitly admit that this only works as an approximation.
The exact procedure is exactly as you've described it.  But
approximations are nice.

-Ted


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