Re: Cosmological redshift and Doppler shift

[EMAIL PROTECTED] wrote:

> Dag Oestvang   wrote:
> >[EMAIL PROTECTED] wrote:
> >
> >> Dag Oestvang  wrote:
> >>
>
> >> >      But when Omega increases, the size of the neighbourhood where
> >> >      this description is appropriate, shrinks. That is, if one neglects an
> >> >      effect of curved space-time on V of 10%, say, then the size of the
> >> >      region (centered on the observer) where this limit holds shrinks
> >> >      when Omega increases. And when Omega increases towards 1
> >> >      the size of this region shrinks to zero.
> >>
> >> I'm having a bit of trouble understanding what you're saying here.
> >>
> >
> >    Indeed. From your comments below it seems that you suffer from a
> >    fundamental misconception; namely this:
>
>
> This statement is still wrong, though.

    Understood correctly, the statement is not wrong.

> The only way to prove it, as
> far as I know, is to do the calculation.  Specifically,
>
> 1. Lay down Riemann normal coordinates using your location as the
>    origin.
>
> 2. Find the coordinate velocity of an object at redshift z in those
>    coordinates.
>
> 3. Pretend that spacetime is flat, enabling you to interpret that
>    coordinate velocity as the special-relativistic velocity of the
>    object relative to you.
>

     As I have said before and you have agreed to; the fact that
     it is in general _possible_to interpret V (for small V) in terms
     of flat space-time does not imply that such interpretations are
     meaningful. (Again, recall the hovering observers!) I really
     don't  understand why you insist to make such interpretations
     when you have done nothing to check if they _are_ meaningful.

>
> [Analogy: this is the exact analogue of the surveyor on the sphere
>  laying down a flat grid of coordinates on a small patch of the sphere
>  and using those flat coordinates to measure distances as if he were
>  on a flat surface.]
>
> If you do this, the answer you get is
>
> v/c = z + O(z^2)
>
> That is, the result differs from the right answer by a fractional
> error of order z.  To be specific, I get the right answer for v to 10%
> if I stick to objects with redshifts less than 0.1 or so.  Even when
> Omega = 1.  The size of the region within which my approximation is
> good to 10% does not, as you claim, "shrink to zero" as Omega -> 1.
>

    You are misunderstanding completely what I am saying.
     I am NOT claiming that using the Hubble law does not
    give the right answer for V (for small V). (I even posted
    a calculation which shows that it does! )What  I am saying
    is that if one takes the _tangent spacetime_ of an event
    for Omega < 1, we get a non-zero contribution to V from
    a field of 4-vectors _in the tangent space-time_  (call
    this field a "velocity" field) whether space-time is curved
    or not. This field is a consequence of the fact that space is
    hyperbolic for Omega < 1. But when Omega increases
    space gets flatter and the contribution to V from the
    "velocity" field becomes smaller. Also the FRW space-time
    becomes more curved so the region where the contribution
    to V from "velocity" dominates over the contribution from
    curvature shrinks. And in the limit Omega -> 1 space
    becomes flat and there is no contribution to V from
    "velocity". Thus the curvature of space-time contributes
    100% to V when Omega = 1.

>
> >    You seem to think that the fact that space-time can be described as
> >    flat  in a neighbourhood is a sufficient condition that allows you to
> >    neglect the contribution to V from space-time curvature in this
> >    neighbourhood and at the same time keeping V to lowest order using
> >    the Hubble law.
> >
> >   This is what you believe, right?
>
> Essentially, yes.  And I believe it because I've done the calculation.
>

     I suppose you mean the calculation which shows that the Hubble
     law gives the right answer to lowest order. I repeat: this is
     completely uncontroversial. But the whole point of our discussion
     is whether or not it is meaningful to interpret V as a recession
     velocity in flat space-time for small V. You agreed that this is not
     meaningful if space-time curvature contributes crucially to V.
     And for Omega = 1 space-time curvature contributes 100% to V
     regardless of the size of V. Thus any interpretation of V as a
     recession velocity in flat space-time is misleading according to
     our agreement.

>
> >   But this is not necessarily true because V obtained from the Hubble
> >   law in general contains contributions both from space-time curvature
> >   and from velocity.
>
> I don't think that this statement is true in any meaningful sense.
> If the distinction between "contributions from spacetime curvature"
> and "contributions from velocity" means anything in particular,

    I thought it was obvious what I meant by "velocity". But since it
    wasn't , I explained it above.

>
> >   In particular, if the contribution from velocity is
> >   zero, the contribution from space-time curvature obviously dominates
> >   at any scale. This is what happens in the example of hovering
> >   observers in Schwarzschild space-time. And this is also what happens
> >   for an Omega=1 FRW model. The proof of the latter assertion is
> >  simple: Write down the metric for the Omega=1 FRW model
> >
> >  ds^2 = -c^2dt^2 + a^2(t)[dx^2 + dy^2 + dz^2]   (1)
> >
> >  (with the correct form for a(t) inserted). The tangent space-time
> >  of some event with coordinates (t_1, x_1,y_1,z_1) has the flat
> >  metric
> >
> >  ds^2 =  -c^2dt^2 + a^2(t_1)[dx^2 + dy^2 + dz^2] .    (2)
> >
> >  The comoving observers move orthogonally to the t=constant
> >  hypersurfaces, and with a trivial rescaling of the spatial
> >  coordinates in eq. (2) we see that this is equivalent to world
> >  lines x(t)= const., y(t)=const., z(t)=const. in Minkowski
> >  space-time equipped with Cartesian space coordinates.
>
> This is not the correct procedure for approximating a curved manifold
> as flat.

    But I was not at all "approximating a curved manifold as flat".
    I was just checking formally if the motion of the comoving
    observers _in the tangent space-time_ had an element of expansion
    which could contribute to V. Not surprisingly, they had none. I could
    have done a similar calculation for an Omega < 1 FRW model. In that
    case the comoving observers in the tangent space-time _do_ have an
    element of expansion contributing to V. That is why it is meaningful,
    at least in a sufficiently small region, to interpret V as a recession
    velocity in flat space-time for Omega < 1 FRW models.

    [irrelevant, uncontroversial calculation snipped]