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Galaxies without dark matter halos?



 
 
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  #1  
Old October 22nd 03, 10:25 PM
Ted Bunn
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Posts: n/a
Default Galaxies without dark matter halos?

From:
Newsgroups: sci.astro.research
Subject: Galaxies without dark matter halos?
References:

In article ,
Dag Oestvang wrote:

The attractiveness of the unified approach to spectral shifts is
that this approach clearly shows how spectral shifts are related
to the geometry of space-time.

In particular, spectral shifts due to curved space-time geometry
should never be thought of as ordinary Doppler shifts in flat space-time.
That is, it is not meaningful to approximate curved space-time with
flat space-time and at the same time keeping spectral shifts due
to curved space-time; such a scheme would be inconsistent.


This is precisely the position that I'm disagreeing with. Thank you
for stating it so lucidly.

I claim that it the following procedure is a perfectly meaningful,
consistent, and moreover extremely useful way to describe the
expanding Universe on small scales:

1. Decide to approximate spacetime as flat over a finite neighborhood.

2. Lay down a coordinate system (such as Riemann normal coordinates)
that "does the best job possible" of approximating curved spacetime as
flat spacetime over this neighborhood.

3. Calculate the redshift of nearby galaxies using the standard
Doppler-shift formula.

This procedure works: it gives the right answer, up to errors of order
(size of neighborhood) / (spacetime curvature scale). And in this
approximation, the galaxy's redshift is a Doppler shift.

A key principle of general relativity is that it reduces to special
relativity over small scales. This is just an example of that.

I honestly don't understand why this procedure is any different from
what we do all the time when we use special relativity to analyze
experiments in terrestrial labs. We know that spacetime in the
vicinity of the Earth isn't flat, but we also know that pretending it
is flat is an excellent approximation over small enough length and
time scales. In circumstances in which we're willing to ignore errors
of order (length scale of experiment)/(spacetime curvature scale), we
cheerfully pretend spacetime is flat, lay down appropriate
coordinates, and apply special relativity. We can do exactly the same
thing in smallish neighborhoods of an expanding
Friedmann-Robertson-Walker spacetime.

I suspect that some people who say that you can't do this believe
something like the following: if you follow the procedure I've
outlined above, you calculate redshifts of zero for the other
galaxies, because the redshifts themselves are of the same order
(neighborhood size / curvatur scale) as the errors. But that's
not so. If you lay down Riemann normal coordinates in a neighborhood
of an FRW spacetime, you find that other galaxies are moving away from
us in those coordinates in accordance with Hubble's law.

-Ted



--
[E-mail me at
, as opposed to .]
  #2  
Old October 27th 03, 12:18 PM
Tim S
external usenet poster
 
Posts: n/a
Default Galaxies without dark matter halos?

on 25/10/03 9:28 pm, at
wrote:

In article ,
Dag Oestvang wrote:
Ted Bunn wrote:


I claim that it the following procedure is a perfectly meaningful,
consistent, and moreover extremely useful way to describe the
expanding Universe on small scales:


Well, in my opinion you are mistaken, see below.


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.

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.

(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.)


snip further clarification

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.

By the way, spacetime is very weakly curved in the vicinity of
Earth's surface, so the assumption happens to be fairly unconstraining
in this case.

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.


I can't speak for Dag Oestvang, but your comparison seems to present the
following difficulties:

1) The geometry of spacetime in the immediate vicinity of the earth is not
even approximately FRW.

2) You and the baseball are (a fortiori) not even approximately comoving.

Crucially, the following more specific statements are true:

1') The geometry of spacetime in the immediate vicinity of the earth is (to
a very good approximation) static.

2') The baseball is following a geodesic, but you aren't: you have a proper
acceleration of 9.8 m s^-2 due to the upward force exerted by the tower.

The bulk of the baseball's redshift as measured by you is uncontroversially
due to the non-zero relative velocity of the ball wrt you. You mention an
additional 'gravitation redshift', but I maintain that this redshift isn't
gravitational in the strict sense, but is actually _accelerational_. That
is, it's the extra redshift due to acceleration that would also be measured
by an accelerating observer in flat Minkowski spacetime.

In short, gravity is completely irrelevant in example B. The whole thing
could be done without significant alteration in SR. Note the irrelevance of
tidal effects, which are the identifying mark of GR.

By contrast, cosmological redshift is essentially gravitational; it cannot
be reproduced in flat spacetime. The role of the relative motion of the
galaxies is to produce frame-dragging, a GR effect, and it is the
frame-dragging that is responsible for the 'expansion of space' and hence
the cosmological redshift. The approximation in which the curvature is
negligible is precisely the approximation in which the redshift is
negligible.

I suppose it is possible that the following idea might be made to work,
although it doesn't feel right to me:

If we assume that our FRW model is a spatially flat one, then the spatial
slice at any given moment of cosmological time is Euclidean, and hence has
an unambiguous notion of distance between us and the distant galaxy. We can
then calculate the rate of increase of distance between us and the galaxy as
a function of cosmological time (or our proper time, which hopefully isn't
too different), and get a relative velocity, to which we could attribute the
redshift. (Note that this description is coordinate-free.)

However, conceptually this feels dubious to me. With a genuine doppler
shift, the redshift is basically determined at the point that the light is
emitted (assuming the observer doesn't accelerate in the meantime). With the
gravitational redshift, the redshift depends on what happens to space during
the time of flight of the light. That, as it were, is 'when' the wavelength
gets 'stretched'.

I also haven't done the calculations to check that this would come out
numerically right; if it didn't, that would of course throw off the whole
idea anyway.

snip

Tim

[Mod. note: it does come out numerically right: d(proper
distance)/d(cosmological time) is equal to the naively calculated
recession speed in the limit z - 0. Personally, I think this isn't a
coincidence -- mjh]
  #3  
Old October 27th 03, 12:18 PM
Tim S
external usenet poster
 
Posts: n/a
Default Galaxies without dark matter halos?

on 25/10/03 9:28 pm, at
wrote:

In article ,
Dag Oestvang wrote:
Ted Bunn wrote:


I claim that it the following procedure is a perfectly meaningful,
consistent, and moreover extremely useful way to describe the
expanding Universe on small scales:


Well, in my opinion you are mistaken, see below.


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.

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.

(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.)


snip further clarification

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.

By the way, spacetime is very weakly curved in the vicinity of
Earth's surface, so the assumption happens to be fairly unconstraining
in this case.

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.


I can't speak for Dag Oestvang, but your comparison seems to present the
following difficulties:

1) The geometry of spacetime in the immediate vicinity of the earth is not
even approximately FRW.

2) You and the baseball are (a fortiori) not even approximately comoving.

Crucially, the following more specific statements are true:

1') The geometry of spacetime in the immediate vicinity of the earth is (to
a very good approximation) static.

2') The baseball is following a geodesic, but you aren't: you have a proper
acceleration of 9.8 m s^-2 due to the upward force exerted by the tower.

The bulk of the baseball's redshift as measured by you is uncontroversially
due to the non-zero relative velocity of the ball wrt you. You mention an
additional 'gravitation redshift', but I maintain that this redshift isn't
gravitational in the strict sense, but is actually _accelerational_. That
is, it's the extra redshift due to acceleration that would also be measured
by an accelerating observer in flat Minkowski spacetime.

In short, gravity is completely irrelevant in example B. The whole thing
could be done without significant alteration in SR. Note the irrelevance of
tidal effects, which are the identifying mark of GR.

By contrast, cosmological redshift is essentially gravitational; it cannot
be reproduced in flat spacetime. The role of the relative motion of the
galaxies is to produce frame-dragging, a GR effect, and it is the
frame-dragging that is responsible for the 'expansion of space' and hence
the cosmological redshift. The approximation in which the curvature is
negligible is precisely the approximation in which the redshift is
negligible.

I suppose it is possible that the following idea might be made to work,
although it doesn't feel right to me:

If we assume that our FRW model is a spatially flat one, then the spatial
slice at any given moment of cosmological time is Euclidean, and hence has
an unambiguous notion of distance between us and the distant galaxy. We can
then calculate the rate of increase of distance between us and the galaxy as
a function of cosmological time (or our proper time, which hopefully isn't
too different), and get a relative velocity, to which we could attribute the
redshift. (Note that this description is coordinate-free.)

However, conceptually this feels dubious to me. With a genuine doppler
shift, the redshift is basically determined at the point that the light is
emitted (assuming the observer doesn't accelerate in the meantime). With the
gravitational redshift, the redshift depends on what happens to space during
the time of flight of the light. That, as it were, is 'when' the wavelength
gets 'stretched'.

I also haven't done the calculations to check that this would come out
numerically right; if it didn't, that would of course throw off the whole
idea anyway.

snip

Tim

[Mod. note: it does come out numerically right: d(proper
distance)/d(cosmological time) is equal to the naively calculated
recession speed in the limit z - 0. Personally, I think this isn't a
coincidence -- mjh]
  #4  
Old October 29th 03, 09:55 AM
[email protected]
external usenet poster
 
Posts: n/a
Default Galaxies without dark matter halos?

In article ,
Tim S wrote:

1) The geometry of spacetime in the immediate vicinity of the earth is not
even approximately FRW.


That's certainly true, but it's not relevant to the point I was trying
to make. My argument was based on the fact that the geometry of spacetime
is approximately *flat* (i.e., Minkowski), not FRW.

To be a bit more precise, I'm basing my argument on the utterly banal
observation that everything is linear to first order. If you want to
study phenomena on length scales much smaller than the curvature
scale, then you can view spacetime as approximately Minkowski, with
small perturbations of order

epsilon = (length scale of observations) / (curvature scale of spacetime).

When you observe a frequency shift near the Earth's surface,
and you interpret it as a Doppler shift, that's precisely what you're
doing: you're pretending that spacetime is flat and applying special
relativity. You know that you're making errors because spacetime
is not perfectly flat, but you also know that those errors are
small (because epsilon is small under the circumstances). In short,

As long as you're willing to ignore errors of order epsilon, you're
allowed to pretend that spacetime is flat and interpret spectral
shifts as Doppler shifts.

I claim that interpreting the redshift of a nearby galaxy in an FRW
Universe as a Doppler shift relies on exactly the same assumptions,
and should be regarded as exactly as valid, as using a radar gun to
measure the speed of a Pedro Martinez fastball.

[OK, I confess I'm exaggerating a bit there. The quantity epsilon is
much smaller in the latter case than in the former, so you're making
less of an error. But the two cases are exactly the same in
principle: the act of interpreting the observed spectral shift as a
Doppler shift (as opposed to a gravitational shift) rests on
neglecting corrections due to spacetime curvature. If those errors
are small, and you're willing to ignore them, you can ignore them.]

1') The geometry of spacetime in the immediate vicinity of the earth is (to
a very good approximation) static.


Maybe I'm missing something, but I don't see the relevance of this
observation.

2') The baseball is following a geodesic, but you aren't: you have a proper
acceleration of 9.8 m s^-2 due to the upward force exerted by the tower.


Ditto.

I'll change my example to eliminate this feature, if you like. Put
yourself on a satellite in an elliptical orbit. Use a radar gun to
measure the speed of another satellite in an elliptical orbit. Again,
to a kick-ass approximation, the number that you get out can be
interpreted as a Doppler shift (as long as conditions are such that
epsilon is small, which is an easy condition to satisfy in these
circumstances). Again, the reason that's true is that you can
approximate spacetime as flat + perturbations of order epsilon to a
kick-ass approximation. Again, the above statements apply equally
well to a small neighborhood of an FRW spacetime.

The bulk of the baseball's redshift as measured by you is uncontroversially
due to the non-zero relative velocity of the ball wrt you.


The reason this is "uncontroversial" is precisely because spacetime
is flat to an excellent approximation. If that weren't true, then
you couldn't even unambiguously defined "relative velocity."

For instance, suppose you moved the experiment to a region just barely
outside the Schwarzschild radius of a black hole, such that the
quantity epsilon was not small. Then the question of whether an
observed spectral shift was Doppler or gravitational would not be
"uncontroversial": different people might analyze the situation using
different coordinates (one using Schwarzschild coordinates, one using
Kruskal coordinates, one using Eddington-Finkelstein coordinates).
They'd all agree on the observations, of course, but they'd disagree
on the interpretation.

The reason this doesn't happen for experiments near the Earth's
surface is that the quantity epsilon is small, which means that
there's an obvious coordinate system in which to analyze things
(the coordinate system that makes spacetime look flat).

In short, gravity is completely irrelevant in example B. The whole thing
could be done without significant alteration in SR. Note the irrelevance of
tidal effects, which are the identifying mark of GR.


I don't really understand why that's relevant, but if it's bothering
you, then let's switch from the dropped baseball example to my new
satellite example above. Let the two satellites by in very different
orbits (so that their separation is of order the orbital radii). It's
still much smaller than the curvature scale, so epsilon is still
small, but now tidal effects are important.

By contrast, cosmological redshift is essentially gravitational; it cannot
be reproduced in flat spacetime. The role of the relative motion of the
galaxies is to produce frame-dragging, a GR effect, and it is the
frame-dragging that is responsible for the 'expansion of space' and hence
the cosmological redshift.


This sort of statement is precisely the reason I want to harp on this
point. I think it's a very misleading way to think about the
low-redshift FRW Universe. One way to understand why is to think
of the Omega - 0 limit of the FRW spacetime (i.e., the Milne model).
When Omega = 0, spacetime becomes exactly Minkowski. It would surely
be absurd to deny that the observed spectral shift of test particles
in that spacetime was a Doppler shift. Yet as soon as Omega
becomes nonzero (even if it's arbitrarily small), all of a sudden
those observed redshifts are 100% due to "frame dragging" by the
(aribtrarily close to massless) stuff flying through the Universe?
I can't imagine seriously believing that.

The approximation in which the curvature is
negligible is precisely the approximation in which the redshift is
negligible.


That last statement is just factually false. I promise. Do the
calculation. Figure out the coordinate system that best approximates
a neighborhood of FRW spacetime as flat. Those coordinates will not
be comoving coordinates. They'll be coordinates in which the galaxies
are flying away from the origin at speeds given by Hubble's law.

-Ted


--
[E-mail me at , as opposed to .]
  #5  
Old October 29th 03, 09:55 AM
[email protected]
external usenet poster
 
Posts: n/a
Default Galaxies without dark matter halos?

In article ,
Tim S wrote:

1) The geometry of spacetime in the immediate vicinity of the earth is not
even approximately FRW.


That's certainly true, but it's not relevant to the point I was trying
to make. My argument was based on the fact that the geometry of spacetime
is approximately *flat* (i.e., Minkowski), not FRW.

To be a bit more precise, I'm basing my argument on the utterly banal
observation that everything is linear to first order. If you want to
study phenomena on length scales much smaller than the curvature
scale, then you can view spacetime as approximately Minkowski, with
small perturbations of order

epsilon = (length scale of observations) / (curvature scale of spacetime).

When you observe a frequency shift near the Earth's surface,
and you interpret it as a Doppler shift, that's precisely what you're
doing: you're pretending that spacetime is flat and applying special
relativity. You know that you're making errors because spacetime
is not perfectly flat, but you also know that those errors are
small (because epsilon is small under the circumstances). In short,

As long as you're willing to ignore errors of order epsilon, you're
allowed to pretend that spacetime is flat and interpret spectral
shifts as Doppler shifts.

I claim that interpreting the redshift of a nearby galaxy in an FRW
Universe as a Doppler shift relies on exactly the same assumptions,
and should be regarded as exactly as valid, as using a radar gun to
measure the speed of a Pedro Martinez fastball.

[OK, I confess I'm exaggerating a bit there. The quantity epsilon is
much smaller in the latter case than in the former, so you're making
less of an error. But the two cases are exactly the same in
principle: the act of interpreting the observed spectral shift as a
Doppler shift (as opposed to a gravitational shift) rests on
neglecting corrections due to spacetime curvature. If those errors
are small, and you're willing to ignore them, you can ignore them.]

1') The geometry of spacetime in the immediate vicinity of the earth is (to
a very good approximation) static.


Maybe I'm missing something, but I don't see the relevance of this
observation.

2') The baseball is following a geodesic, but you aren't: you have a proper
acceleration of 9.8 m s^-2 due to the upward force exerted by the tower.


Ditto.

I'll change my example to eliminate this feature, if you like. Put
yourself on a satellite in an elliptical orbit. Use a radar gun to
measure the speed of another satellite in an elliptical orbit. Again,
to a kick-ass approximation, the number that you get out can be
interpreted as a Doppler shift (as long as conditions are such that
epsilon is small, which is an easy condition to satisfy in these
circumstances). Again, the reason that's true is that you can
approximate spacetime as flat + perturbations of order epsilon to a
kick-ass approximation. Again, the above statements apply equally
well to a small neighborhood of an FRW spacetime.

The bulk of the baseball's redshift as measured by you is uncontroversially
due to the non-zero relative velocity of the ball wrt you.


The reason this is "uncontroversial" is precisely because spacetime
is flat to an excellent approximation. If that weren't true, then
you couldn't even unambiguously defined "relative velocity."

For instance, suppose you moved the experiment to a region just barely
outside the Schwarzschild radius of a black hole, such that the
quantity epsilon was not small. Then the question of whether an
observed spectral shift was Doppler or gravitational would not be
"uncontroversial": different people might analyze the situation using
different coordinates (one using Schwarzschild coordinates, one using
Kruskal coordinates, one using Eddington-Finkelstein coordinates).
They'd all agree on the observations, of course, but they'd disagree
on the interpretation.

The reason this doesn't happen for experiments near the Earth's
surface is that the quantity epsilon is small, which means that
there's an obvious coordinate system in which to analyze things
(the coordinate system that makes spacetime look flat).

In short, gravity is completely irrelevant in example B. The whole thing
could be done without significant alteration in SR. Note the irrelevance of
tidal effects, which are the identifying mark of GR.


I don't really understand why that's relevant, but if it's bothering
you, then let's switch from the dropped baseball example to my new
satellite example above. Let the two satellites by in very different
orbits (so that their separation is of order the orbital radii). It's
still much smaller than the curvature scale, so epsilon is still
small, but now tidal effects are important.

By contrast, cosmological redshift is essentially gravitational; it cannot
be reproduced in flat spacetime. The role of the relative motion of the
galaxies is to produce frame-dragging, a GR effect, and it is the
frame-dragging that is responsible for the 'expansion of space' and hence
the cosmological redshift.


This sort of statement is precisely the reason I want to harp on this
point. I think it's a very misleading way to think about the
low-redshift FRW Universe. One way to understand why is to think
of the Omega - 0 limit of the FRW spacetime (i.e., the Milne model).
When Omega = 0, spacetime becomes exactly Minkowski. It would surely
be absurd to deny that the observed spectral shift of test particles
in that spacetime was a Doppler shift. Yet as soon as Omega
becomes nonzero (even if it's arbitrarily small), all of a sudden
those observed redshifts are 100% due to "frame dragging" by the
(aribtrarily close to massless) stuff flying through the Universe?
I can't imagine seriously believing that.

The approximation in which the curvature is
negligible is precisely the approximation in which the redshift is
negligible.


That last statement is just factually false. I promise. Do the
calculation. Figure out the coordinate system that best approximates
a neighborhood of FRW spacetime as flat. Those coordinates will not
be comoving coordinates. They'll be coordinates in which the galaxies
are flying away from the origin at speeds given by Hubble's law.

-Ted


--
[E-mail me at , as opposed to .]
 




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