Galaxies without dark matter halos?

[Mod. note: the post below required substantial reformatting to fix up
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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).

In article ,
writes:

I'd like to ask anyone who's been following the discussion about the
low-redshift limit of the cosmological redshift to answer the
following question. I'm going to list a sequence of statements, all
of which I believe. I'll order them from least to most controversial.
At what point, if any, do you part company with me?

These are all limiting cases, so no problem.

We don't disagree on the numbers nor on when what approximations are
appropriate. My crusade is purely pedagogical. Especially for folks
who don't understand the (special-relativistic) Doppler shift, I think
it is more trouble than it is worth to mention it then say "but in
practice, we can forget this".

I hope we can ALL agree on the statement "if the redshift is large
enough that the relativistic Doppler formula "should" be used, then the
relativistic Doppler formula is completely inappropriate, in our
universe".

In article ,
writes:

I'd like to ask anyone who's been following the discussion about the
low-redshift limit of the cosmological redshift to answer the
following question. I'm going to list a sequence of statements, all
of which I believe. I'll order them from least to most controversial.
At what point, if any, do you part company with me?

These are all limiting cases, so no problem.

We don't disagree on the numbers nor on when what approximations are
appropriate. My crusade is purely pedagogical. Especially for folks
who don't understand the (special-relativistic) Doppler shift, I think
it is more trouble than it is worth to mention it then say "but in
practice, we can forget this".

I hope we can ALL agree on the statement "if the redshift is large
enough that the relativistic Doppler formula "should" be used, then the
relativistic Doppler formula is completely inappropriate, in our
universe".

In article ,
Phillip Helbig---remove CLOTHES to reply wrote:

These are all limiting cases, so no problem.

Great! That's all I've been trying to say.


I hope we can ALL agree on the statement "if the redshift is large
enough that the relativistic Doppler formula "should" be used, then the
relativistic Doppler formula is completely inappropriate, in our
universe".

Absolutely.

Moreover, I want to make it clear that I would only emphasize that the
low-redshift limit of the cosmological redshift can be viewed as a
Doppler shift when talking to a sufficiently sophisticated crowd. In
particular, I wouldn't bring this fact up to anyone who didn't already
clearly understand that you *can't* do this at high redshift.

At some point, I'll try to write in detail about why I think it's
pedagogically useful to think about the cosmological redshift as a
Doppler shift in the low-redshift limit. I don't think I have the
strength to embark on it now, though.

-Ted

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

In article ,
Phillip Helbig---remove CLOTHES to reply wrote:

These are all limiting cases, so no problem.

Great! That's all I've been trying to say.


I hope we can ALL agree on the statement "if the redshift is large
enough that the relativistic Doppler formula "should" be used, then the
relativistic Doppler formula is completely inappropriate, in our
universe".

Absolutely.

Moreover, I want to make it clear that I would only emphasize that the
low-redshift limit of the cosmological redshift can be viewed as a
Doppler shift when talking to a sufficiently sophisticated crowd. In
particular, I wouldn't bring this fact up to anyone who didn't already
clearly understand that you *can't* do this at high redshift.

At some point, I'll try to write in detail about why I think it's
pedagogically useful to think about the cosmological redshift as a
Doppler shift in the low-redshift limit. I don't think I have the
strength to embark on it now, though.

-Ted

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

wrote:

I'd like to ask anyone who's been following the discussion about the
low-redshift limit of the cosmological redshift to answer the
following question. I'm going to list a sequence of statements, all
of which I believe. I'll order them from least to most controversial.
At what point, if any, do you part company with me?

1. An observer in flat (Minkowski) spacetime measures the redshift of
light from a source and finds z = Delta lambda/lambda = 0.01. She
can use the special-relativistic Doppler shift formula to determine
the source's speed relative to her (getting the answer v =
0.01c).


Agreed.


2. An observer in an open FRW spacetime with zero density (Omega = 0)
measures the redshift of light from a source and finds z = 0.01.
She can use She can use the special-relativistic Doppler shift
formula to determine the source's speed relative to her.


Agreed.


3. An observer in an open FRW spacetime with density parameter Omega =
10^(-50) measures the redshift of light from a source and finds z =
0.01. She uses the special-relativistic Doppler shift formula to
calculate a speed. To an excellent approximation, she can
approximate spacetime as flat and interpret that number as the source's
speed relative to her.


Agreed (see below).


4. An observer in an open FRW spacetime with density parameter Omega =
1 measures the redshift of light from a source and finds z = 0.01.
She uses the special-relativistic Doppler shift formula to
calculate a speed. To a good approximation, she can approximate
spacetime as flat and interpret that number as the source's speed
relative to her.


But here we part company. That is, I believe that when Omega=1,
any sensible interpretation of cosmological spectral shifts as Doppler
shifts in flat space-time breaks down at all scales.

To justify this view it is useful to consider the geometry of the
Milne model in some detail. Since this is just the empty FRW model,
space-time is flat. In fact, space-time is a piece of Minkowski
space-time; namely the region inside the future light cone of
some point in Minkowski space-time. Moreover, the hypersurfaces
t=constant have hyperbolical geometry and the expansion of
the comoving observers is exactly described as a field of 4-vectors
in flat space-time.

Now consider a model where Omega is non-zero, but small (as in
example 3 above). In this case the hypersurfaces t=constant
still have hyperbolic geometry. This means that for every point
in space-time we can find a neighbourhood where the curvature of
space matches the curvature of space in the vincinity of an event in
the Milne model to the desired accuracy. Besides, in such an
neighbourhood space-time can be considered flat to the desired
accuracy. But this means that the expansion of the comoving
observers can be described exactly as in the Milne model to the
desired accuracy in the chosen neighbourhood; i.e. that V to a good
approximation comes from a field of 4-vectors in flat space-time.

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. That is, when Omega=1
space is no longer hyperbolic but flat. This means that we cannot
find any event in the Milne model where the curvature of space
matches the curvature of space in a Omega=1 model. Thus one
cannot find a neighbourhood of the observer where the expansion
of the comoving observers can be described as in the Milne model
to any approximation. This suggests that the interpretation of
cosmological spectral shifts as Doppler shifts in flat space-time
is misleading at any scale for a Omega=1 FRW model.

Thus it seems that the Omega=1 FRW model has some features
similar to the example of hovering observers in Schwarzschild
space-time: For both these models one may chose an event where
space-time is flat to the desired accuracy; yet the effect of
space-time curvature on V is 100%. This does of course not mean
that we cannot construct a field of 4-velocities in the tangent
space-time of the observer using the "Hubble law". But for both
these models, a sensible interpretation of this field as coming
from recessional velocities in flat space-time does not exist at
any scale.

wrote:

I'd like to ask anyone who's been following the discussion about the
low-redshift limit of the cosmological redshift to answer the
following question. I'm going to list a sequence of statements, all
of which I believe. I'll order them from least to most controversial.
At what point, if any, do you part company with me?

1. An observer in flat (Minkowski) spacetime measures the redshift of
light from a source and finds z = Delta lambda/lambda = 0.01. She
can use the special-relativistic Doppler shift formula to determine
the source's speed relative to her (getting the answer v =
0.01c).


Agreed.


2. An observer in an open FRW spacetime with zero density (Omega = 0)
measures the redshift of light from a source and finds z = 0.01.
She can use She can use the special-relativistic Doppler shift
formula to determine the source's speed relative to her.


Agreed.


3. An observer in an open FRW spacetime with density parameter Omega =
10^(-50) measures the redshift of light from a source and finds z =
0.01. She uses the special-relativistic Doppler shift formula to
calculate a speed. To an excellent approximation, she can
approximate spacetime as flat and interpret that number as the source's
speed relative to her.


Agreed (see below).


4. An observer in an open FRW spacetime with density parameter Omega =
1 measures the redshift of light from a source and finds z = 0.01.
She uses the special-relativistic Doppler shift formula to
calculate a speed. To a good approximation, she can approximate
spacetime as flat and interpret that number as the source's speed
relative to her.


But here we part company. That is, I believe that when Omega=1,
any sensible interpretation of cosmological spectral shifts as Doppler
shifts in flat space-time breaks down at all scales.

To justify this view it is useful to consider the geometry of the
Milne model in some detail. Since this is just the empty FRW model,
space-time is flat. In fact, space-time is a piece of Minkowski
space-time; namely the region inside the future light cone of
some point in Minkowski space-time. Moreover, the hypersurfaces
t=constant have hyperbolical geometry and the expansion of
the comoving observers is exactly described as a field of 4-vectors
in flat space-time.

Now consider a model where Omega is non-zero, but small (as in
example 3 above). In this case the hypersurfaces t=constant
still have hyperbolic geometry. This means that for every point
in space-time we can find a neighbourhood where the curvature of
space matches the curvature of space in the vincinity of an event in
the Milne model to the desired accuracy. Besides, in such an
neighbourhood space-time can be considered flat to the desired
accuracy. But this means that the expansion of the comoving
observers can be described exactly as in the Milne model to the
desired accuracy in the chosen neighbourhood; i.e. that V to a good
approximation comes from a field of 4-vectors in flat space-time.

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. That is, when Omega=1
space is no longer hyperbolic but flat. This means that we cannot
find any event in the Milne model where the curvature of space
matches the curvature of space in a Omega=1 model. Thus one
cannot find a neighbourhood of the observer where the expansion
of the comoving observers can be described as in the Milne model
to any approximation. This suggests that the interpretation of
cosmological spectral shifts as Doppler shifts in flat space-time
is misleading at any scale for a Omega=1 FRW model.

Thus it seems that the Omega=1 FRW model has some features
similar to the example of hovering observers in Schwarzschild
space-time: For both these models one may chose an event where
space-time is flat to the desired accuracy; yet the effect of
space-time curvature on V is 100%. This does of course not mean
that we cannot construct a field of 4-velocities in the tangent
space-time of the observer using the "Hubble law". But for both
these models, a sensible interpretation of this field as coming
from recessional velocities in flat space-time does not exist at
any scale.

In article ,
Phillip Helbig---remove CLOTHES to reply wrote:

I've now finished reading the paper I mentioned a couple of posts back
in this thread, astro-ph/0310808. The authors mention that the
cosmological time dilation---(1+z), independent of the cosmological
parameters---does not agree with the special-relativistic prediction
EVEN TO FIRST ORDER (Eqs. 3--5). If so (and if not please point out
their error), then special-case cosmological models such as the Milne
model (i.e. lambda=0, Omega_matter=0; obviously a limiting case) cannot
be expressed in the language of SR and agree with observations.

Equation (4) is wrong. Either that, or the authors mean something
quite different by it than what it seems to mean; I haven't read the
paper carefully enough yet to know. Either way, the description
following it is extremely misleading. Equation (4) certainly does not
justify the conclusion that special relativity doesn't reproduce the
cosmological time dilation to first order; it does reproduce it.

When you take an FRW spacetime and approximate it as flat (which you
can do to a good approximation at low redshift in any FRW spacetime,
or exactly at arbitrary redshift in the Milne model), the velocity of a
galaxy relative to you is not the peculiar velocity v_{pec}. It's (to
first order, anyway) the peculiar velocity plus the Hubble velocity.
So the special-relativistic prediction should not be

\gamma_{SR} = (1-v_{pec}^2/c^2)^{-1/2},

as they say, but

\gamma_{SR} = (1-v^2/c^2)^{-1/2}.

This is a good example of the usefulness of the Milne model in
error-checking. If you ever draw a conclusion that says that the
Milne model isn't consistent with special relativity, then it's 100%
guaranteed that you've made a mistake, because the Milne spacetime is
*precisely* the same as the Minkowski spacetime of special relativity.

-Ted

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

In article ,
Phillip Helbig---remove CLOTHES to reply wrote:

I've now finished reading the paper I mentioned a couple of posts back
in this thread, astro-ph/0310808. The authors mention that the
cosmological time dilation---(1+z), independent of the cosmological
parameters---does not agree with the special-relativistic prediction
EVEN TO FIRST ORDER (Eqs. 3--5). If so (and if not please point out
their error), then special-case cosmological models such as the Milne
model (i.e. lambda=0, Omega_matter=0; obviously a limiting case) cannot
be expressed in the language of SR and agree with observations.

Equation (4) is wrong. Either that, or the authors mean something
quite different by it than what it seems to mean; I haven't read the
paper carefully enough yet to know. Either way, the description
following it is extremely misleading. Equation (4) certainly does not
justify the conclusion that special relativity doesn't reproduce the
cosmological time dilation to first order; it does reproduce it.

When you take an FRW spacetime and approximate it as flat (which you
can do to a good approximation at low redshift in any FRW spacetime,
or exactly at arbitrary redshift in the Milne model), the velocity of a
galaxy relative to you is not the peculiar velocity v_{pec}. It's (to
first order, anyway) the peculiar velocity plus the Hubble velocity.
So the special-relativistic prediction should not be

\gamma_{SR} = (1-v_{pec}^2/c^2)^{-1/2},

as they say, but

\gamma_{SR} = (1-v^2/c^2)^{-1/2}.

This is a good example of the usefulness of the Milne model in
error-checking. If you ever draw a conclusion that says that the
Milne model isn't consistent with special relativity, then it's 100%
guaranteed that you've made a mistake, because the Milne spacetime is
*precisely* the same as the Minkowski spacetime of special relativity.

-Ted

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

In article ,
Dag Oestvang wrote:

But here we part company. That is, I believe that when Omega=1,
any sensible interpretation of cosmological spectral shifts as Doppler
shifts in flat space-time breaks down at all scales.

To justify this view it is useful to consider the geometry of the
Milne model in some detail. Since this is just the empty FRW model,
space-time is flat. In fact, space-time is a piece of Minkowski
space-time; namely the region inside the future light cone of
some point in Minkowski space-time. Moreover, the hypersurfaces
t=constant have hyperbolical geometry and the expansion of
the comoving observers is exactly described as a field of 4-vectors
in flat space-time.

Now consider a model where Omega is non-zero, but small (as in
example 3 above). In this case the hypersurfaces t=constant
still have hyperbolic geometry. This means that for every point
in space-time we can find a neighbourhood where the curvature of
space matches the curvature of space in the vincinity of an event in
the Milne model to the desired accuracy. Besides, in such an
neighbourhood space-time can be considered flat to the desired
accuracy. But this means that the expansion of the comoving
observers can be described exactly as in the Milne model to the
desired accuracy in the chosen neighbourhood; i.e. that V to a good
approximation comes from a field of 4-vectors in flat space-time.

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.
You seem to be saying this:

Given a point in an Omega = 1 FRW spacetime and a level of desired
accuracy epsilon, there is no finite-sized neighborhood of that
point in which spacetime can be approximated as flat to an accuracy
epsilon.

Is that what you're saying? If so, then that completely explains
our disagreement. This statement is wrong.

In fact, this statement is precisely equivalent to

An Omega = 1 FRW spacetime is not a Lorentzian manifold.

Or, if we define the word "spacetime" to mean "Lorentzian manifold"
(which is essentially what we do in general relativity), to

An Omega = 1 FRW spacetime is not a spacetime.

More or less by definition, a Lorentzian manifold is one that can be
approximated arbitrarily well on sufficiently small scales as
Minkowski spacetime. (Compare to Euclidean manifolds: a Euclidean
manifold, by definition, can be approximated arbitrarily well on
sufficiently small scales as flat.) In the epsilon-delta language
beloved of mathematicians, a Lorentzian manifold must satisfy the
condition that, at any point P, for any epsilon0, there is a delta0
such that spacetime deviates from Minkowski by less than epsilon in a
neighborhood of radius delta of P.

To put it more concretely, suppose you lived in an Omega = 1 FRW
Universe at a time 14 billion years after the big bang. You draw a
sphere of radius one millimeter. Do you really claim that spacetime
deviates from flat by more than 10% within that sphere? I swear to
you, on a stack of Misners, Thornes, and Wheelers, that it doesn't.
(If it did, how did anyone ever manage to do experiments confirming
special relativity?)

If your above statement doesn't mean what I wrote above, then
the only other meaning I can ascribe to it is this:

Given a point in an Omega = 1 FRW spacetime, the
constant-cosmic-time hypersurface through that point, and a level
of desired accuracy epsilon, there is no finite-sized neighborhood
of point P in which the hypersurface can be approximated as flat
(Euclidean) to an accuracy epsilon.

That's a slightly different statement, but it's still false. This
statement is equivalent to the statement that the constant-cosmic-time
hypersurface is not a Riemannian manifold, which it is.

Incidentally, when approximating an FRW spacetime as flat in the
neighborhood of a point, the surfaces of constant time do differ from
surfaces of constant cosmic time at second order. That is, if T
stands for the time coordinate in Riemann normal coordinates that best
approximate spacetime as flat, and t stands for cosmic time, then

T = t + O((r/R)^2),

where the small quantity r/R is the ratio of the distance to the
Hubble distance.

That is, when Omega=1
space is no longer hyperbolic but flat. This means that we cannot
find any event in the Milne model where the curvature of space
matches the curvature of space in a Omega=1 model.

Two points he

1. We can't exactly, but we *can* to any desired accuracy, in a small
neighborhood. (Again, this is the definition of a Riemannian
manifold.)

2. Worrying about whether we can approximate the constant-cosmic-time
hypersurface to arbitrary accuracy is a bit of a red herring anyway.
What matters is whether we can approximate *spacetime* to arbitrary
accuracy. (It doesn't really make much difference, though, since
we can do both.)

-Ted


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