Galaxies without dark matter halos?

In article ,
Serenus Zeitblom wrote:

Give it up, Ted!

I certainly concede that I haven't been very convincing or clear, and
for the time being I will give it up. At some point, I may try again,
and if so, I'll put a lot more effort into laying out the explanation
in a clearer and more systematic way.

-Ted


--
[E-mail me at , as opposed to .]

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.

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.