Re: Cosmological redshift and Doppler shift

>
> > >   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 discussion seems to have come to an end without any
     agreement of how to properly interpret small redshifts in an
     Omega = 1 FRW model. Thus, to round off and to back up
     my former arguments, I will post the counterpart to the
     above calculation for an Omega < 1 model.

     Start with the line element  d{\sigma}^2 for hyperbolic
     3-space with unit "radius". Using a standard "angular"
     radial coordinate {\chi}, we then have

     d{\sigma}^2 =  d{\chi}^2 + sinh^2{\chi} [d{\theta}^2 + sin^2{\theta}
d{\phi}^2].    (1)

     Now  the metric for an Omega < 1 model takes the form

     ds^2 = -c^2dt^2 + a^2(t) d{\sigma}^2 . (2)

     We don't need to specify a(t), but note that a(t) is increasing
     monotonously and that {\dot a}(t) < c for t > 0. Furthermore,
     at epoch t_1 we can define another quantity t_2 by

     a(t_1) = {\dot a}(t_1)(t_1 + t_2).  (3)

     Now I claim that  the metric of the tangent space-time at some event at
     epoch t_1 may be written in the form

     ds^2_m = -{\dot a}^2(t_1)dt^2 + {\dot a}^2(t_1)(t + t_2)^2 d{\sigma}^2.  (4)

     Proof:  Make a change of time coordinate by defining

     t' {\equiv} c^{-1}{\dot a}(t_1) [t + t_2].      (5)

     Inserting (5) into (4) we may then easily show that (4) takes the form

     ds^2_m = -c^2dt'^2 + c^2 t'^2 d{\sigma}^2 ,   (6)

     which is the metric of the flat Milne model, as asserted.

     The spatial curvature scalar P(t_1) in space-time matches its counterpart
     P_m(t_1) in the tangent space-time at epoch t_1:

     P_m(t_1) = -6(ct'_1)^{-2} =-6a^{-2}(t_1) = P(t_1).    (7)

     Similarly the Hubble parameters H(t_1) and H_m(t_1) match at
     epoch t_1 (left as an exercise for the reader).

     Now define new coordinates r, {\bar t} by

     r {\equiv} ct'sinh{\chi},         {\bar t} {\equiv} t'cosh{\chi},    (8)

     expressed in which (6) takes its standard form

     ds^2_m = -c^2d{\bar t}^2 + dr^2 + r^2 [d{\theta}^2 + sin^2{\theta}
d{\phi}^2].   (9)

     For purely radial motion equations (5) and (8) yield

     r(t) = {\dot a}(t_1) (t + t_2) sinh{\chi}(t),     (10)

    and since {\chi}(t) = {\chi}(t_1)= const. along the world lines of the comoving
    observers, we see that there is an element of expansion present
    in the tangent space-time. Thus the comoving observers move outwards
    with velocity

    V_m = {\dot r}(t) = {\dot a}(t_1) sinh{\chi} (t_1) = H(t_1) r(t_1),     (11)

    so at least in a neighbourhood of the origin it is meaningful to interpret
    V as coming from motion in flat space-time.

     Now what happens when Omega --> 1? This limit is found by letting
     t_1 --> {\infty}. To have a neighbourhood of constant size r(t_1) we see from
     (10) that we must let {\chi} -->  0 when t _1 -->  {\infty}. But from (11) we
     then see that V_m -->  0  in the neighbourhood, indicating that size of the
region
     where V_m dominates over the contribution to V from curvature, shrinks
     to zero in the limit Omega --> 1.