Re: Galaxies without dark matter halos?

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[EMAIL PROTECTED] wrote:
> 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.

    In my opinion your examples are quite misleading, and some
    simple calculations, using the Narlikar paper, can make things
    much clearer.  I will explain further below.

> 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.

        If this means "Doppler shift in flat space-time", I disagree.
        See below for justification.

> (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.)

     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.

     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. Rather, in a flat
     space-time interpretation one has to pretend that V is
     obtained operationally by parallel-transporting the 4-velocity
     of a fictious sender located in the tangent space-time.But
     this is not how V is obtained, and any interpretation besed on
     this is rather dubious. On the other hand, if the sender is
     not comoving, it makes sense to interpret his _peculiar
     velocity_ as a ordinary 3-velocity in flat space-time since
     the effect of curved space-time on parallel transport may be
     neglected over small enough distances. (Note that the
     relationship between peculiar velocities is not affected by
     parallel transport in curved space-time.)

     One may do exactly the same reasoning in Schwarzschild
     space-time; see below.

> 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.

       Right.


> - 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.

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

> 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.

      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.

    So let us use eqs. (28)-(30) of the Narlikar paper
    and find V for two hovering observers in Schwarzschild
    space-time. Some simple calculations yield

              1 - exp[v_s - v_o]
   V  =  ------------------------ c                   (4)
             1 + exp[v_s - v_o]

   where
                         2MG
   exp[v(r)] = 1 -  -------------                     (5)
                         c^2 r

   For small v_s  (equiv v(r_s)) we can express it via a Taylor
   expansion and neglect higher order terms

   v_s = v_o + v'_o(r_s - r_o) + higher order terms.   (6)

   Inserting (5) and (6) into (4) we then get

                                     MG(r_o-r_s)
V \approx  0.5(v_o - v_s) \approx  --------------   \approx H_od  (7)
                                      c^2 r_o^2

   where d is the distance in the tangent space-time at the observer, and

                  MG
    H_o \equiv  ----------         (8)
                  c^2 r_o^2

    is a "formal Hubble parameter" at r_o.

    Again, just as for the FRW models, V may be seen as a 3-velocity
    given from a local Hubble law in the tangent space-time of the
    observer. And again there is no natural way to get V from
    operations performed in the tangent space-time. Furthermore,
    any interpretation of V as a recessional velocity in flat
    space-time is obviously absurd. Yet this interpretation has
    exactly the same mathematical justification as the
    interpretation of cosmological recessional velocities. The only
    difference is that cosmological space-times appeal much better to
    an intuitive sense of "flat space-time" than the Schwarzschild
    space-time does.

> 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.

    Any spectral shift in GR may be interpreted as a generalized
    Doppler shift in curved space-time. It is the interpretation in
    terms of flat space-time which gets misleading; see above.

> 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.

    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.

> 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:

    They do not; see above.

> 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.

     I recommend studying the examples I gave (and writing out the
     calculations, it that would help).