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This Week's Finds in Mathematical Physics (Week 206)



 
 
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  #12  
Old May 18th 04, 12:17 AM
Tobias Fritz
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Lee Smolin told me some neat stuff about MOND - that's "Modified
Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain
the strange behavior of galaxies without invoking dark matter. The basic
problem with galaxies is that the outer parts rotate faster than they
should given how much mass we actually see.

If you have a planet in a circular orbit about the Sun, Newton's laws
say its acceleration is proportional to 1/r^2, where r is its distance to
the Sun. Similarly, if almost all the mass in a galaxy were concentrated
right at the center, a star orbiting in a circle at distance r from the
center would have acceleration proportional to 1/r^2. Of course, not all
the mass is right at the center! So, the acceleration should drop off
more slowly than 1/r^2 as you go further out. And it does. But, the
observed acceleration drops off a lot more slowly than the acceleration
people calculate from the mass they see. It's not a small effect: it's a
HUGE effect!

One solution is to say there's a lot of mass we don't see: "dark matter"
of some sort. If you take this route, which most astronomers do, you're
forced to say that *most* of the mass of galaxies is in the form of dark
matter.

Milgrom's solution is to say that Newton's laws are messed up.

Of course this is a drastic, dangerous step: the last guy who tried this
was named Einstein, and we all know what happened to him. Milgrom's
theory isn't even based on deep reasoning and beautiful math like
Einstein's! Instead, it's just a blatant attempt to fit the experimental
data.
And it's not even elegant. In fact, it's downright ugly.

Here's what it says: the usual Newtonian formula for the acceleration
due to gravity is correct as long as the acceleration is bigger than

a = 2 x 10^{-10} m/sec^2

But, for accelerations less than this, you take the geometric mean
of the acceleration Newton would predict and this constant a.

In other words, there's a certain value of acceleration such that above
this value, the Newtonian law of gravity works as usual, while below this
value the law suddenly changes.

Any physicist worth his salt who hears this modification of Newton's law
should be overcome with a feeling of revulsion! There just *aren't* laws
of physics that split a situation in two cases and say "if this is bigger
than that, then do X, but if it's smaller, then do Y." Not in fundamental
physics, anyway! Sure, water is solid below 0 centigrade and fluid above
this, but that's not a fundamental law - it presumably follows from other
stuff. Not that anyone has derived the melting point of ice from first
principles, mind you. But we think we could if we were better at big
messy calculations.

Furthermore, you can't easily invent a Lagrangian for gravity that makes
it fall off more *slowly* than 1/r^2. It's easy to get it to fall off
*faster* - just give the graviton a mass, for example! But not more
slowly. It turns out you can do it - Bekenstein and Milgrom have a way -
but it's incredibly ugly.

So, MOND should instantly make any decent physicist cringe. Esthetics
alone would be enough to rule it out, except for one slight problem: it
seems to fit the data! In some cases it matches the observed rotation of
galaxies in an appallingly accurate way, fitting every wiggle in the graph
of stellar rotation velocity as a function of distance from the center.

So, even if MOND is wrong, there may need to be some reason why it *acts*
like it's right! Apparently even some proponents of dark matter agree
with this.


Doesn't it seem unresonable to discard a theory as successful as GR?
Or is it somehow possible to fit MOND into the framework of GR, like by
modifying the field equations, perhaps by including torsion?

Everybody is talking about "dark matter" or alternative theories, when it is
not even really clear what the predictions of GR a recently I heard a
talk about the "averaging problem" in GR; basically, the message was that
we do not know if it is valid to take an average energy-momentum-tensor,
put it into the field equations and see the result as an average metric. By
googling, I found the following paper:

http://arxiv.org/abs/gr-qc/9703016

which also has some references.

What do the experts think?
--
hang my head drown my fear
till you all just disappear

reverse my forename for mail! - saibot

  #13  
Old May 18th 04, 12:23 AM
Christine Dantas
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wrote in message ...

Personally, I can't get past my theorist's objections to MOND. It
doesn't play well at all with general relativity, and I just don't
believe that general relativity is completely on the wrong track.


Hello all,

Concerning MOND x GR, the recent paper by Bekenstein seems to be
a relevant contribution to this issue (see below).

Regards,
Christine Dantas
INPE/Brazil


================================================== =======================
astro-ph/0403694
Relativistic gravitation theory for the MOND paradigm
Jacob D. Bekenstein

The modified newtonian dynamics (MOND) paradigm of Milgrom can boast
of a number of successful predictions regarding galactic dynamics;
these are made without the assumption that dark matter plays a
significant role. MOND requires gravitation to depart from Newtonian
theory in the extragalactic regime where dynamical accelerations are
small. So far relativistic gravitation theories proposed to underpin
MOND have either clashed with the post-Newtonian tests of general
relativity, or failed to provide significant gravitational lensing, or
violated hallowed principles by exhibiting superluminal scalar waves
or an a priori vector field. We develop a relativistic MOND inspired
theory which resolves these problems. In it gravitation is mediated by
metric, a scalar field and a 4-vector field, all three dynamical. For
a simple choice of its free function, the theory has a Newtonian limit
for nonrelativistic dynamics with significant acceleration, but a MOND
limit when accelerations are small. We calculate the beta and gamma
PPN coefficients showing them to agree with solar system
measurements. The gravitational light deflection by nonrelativistic
systems is governed by the same potential responsible for dynamics of
particles. Consequently, the new theory predicts gravitational lensing
by extragalactic structures that cannot be distinguished from that
predicted within the dark matter paradigm by general
relativity. Cosmological models based on the theory are quite similar
to those based on general relativity; they predict slow evolution of
the scalar field. For a range of initial conditions, this last result
makes it easy to rule out superluminal propagation of metric, scalar
and vector waves.
================================================== =======================

  #14  
Old May 18th 04, 01:37 AM
John Baez
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Default This Week's Finds in Mathematical Physics (Week 206)

In article ,
Torquemada wrote:

I checked out this article: http://www.astro.umd.edu/~ssm/mond/astronow.html
and there's an example of a graph of rotation velocity vs. radius showing
one of the wiggles JB mentions, with the note "even the kink observed in the
gas distribution is reflected in the rotation".

Forgive me for being a little sceptical [....]


We should all be VERY skeptical as far as both MOND and dark matter
are concerned! I'm not trying to get people to accept MOND, just to
talk about this stuff.

[...] but the Newtonian prediction has
exactly the same kink. In fact, the MOND curve is just the Newtonian curve
scaled up. Just about any reasonably well behaved modification of the
Newtonian formula that has a scaling effect that brings the Newtonian curve
roughly in alignment with measured results is going to have that kink.


I guess you're right, but aren't you basically buying MOND if you
posit a "reasonably well behaved modification of the Newtonian formula
that has a scaling effect that brings the Newtonian curve roughly in
alignment with measured results"?

After all, the real point of MOND is not the specific formula for
the gravitational force which I wrote down in "week206". I may not
have explained this well, but as Ted Bunn notes, all sorts of roughly
similar formulas would also fit the data about equally well. The
problem is, all these formulas require us to accept that gravity doesn't
work as expected at large distances! - or more precisely, at low
accelerations. Accepting any one would require us to toss general
relativity out the window. And all of them force us to dream up
theories of forces that die off more slowly than 1/r^2 - a difficult
task.

Or are you suggesting that dark matter could also explain these galaxy
rotation curves? For that, I guess the dark matter distribution would
have to closely mimic the visible matter distribution - see for example
the plot for the galaxy NGC 1580 in Figure 3 on page 39 he

http://xxx.lanl.gov/abs/astro-ph/0204521

Do proponents of dark matter claim this is how it works? I thought
otherwise. (But mind you, I'm no expert, so I could be ignoring all
sorts of important stuff.)

I thank Ethan Vishniac and Steve Carlip for telling me about some
things to read... but I haven't read 'em yet!






  #15  
Old May 18th 04, 01:46 AM
John Baez
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In article , Morris Carré
wrote:

wrote about Modified Newtonian Dynamics:


then the actual acceleration of an object is

a = a_N if a_N a_0
a = sqrt(a_N a_0) if a a_0

Here a_0 is some fundamental constant.


Am I dreaming, or isn't there something "planckscalish" to the idea ?


Sort of:

The reason why Smolin was interested in MOND is that this constant

a_0 = 2 x 10^{-10} m/sec^2

is supposedly about equal the acceleration of the expansion of
the universe due to the cosmological constant, for objects that are...
one Hubble away? Or something like that... I'm too lazy to check,
I could be horribly far off, and Smolin shouldn't be blamed if I'm
misremembering and saying something really stupid!

Does someone have the energy to check?

[Moderator's note: The order of magnitude is about right, anyway. -TB]

Anyway, Smolin was wondering if the MOND acceleration scale was somehow
related to the cosmological constant - he said he's spent lots of nights
lying in bed staring at the ceiling trying to figure something out about
this, so far with no success.




  #16  
Old May 18th 04, 09:45 PM
John Baez
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Default This Week's Finds in Mathematical Physics (Week 206)

In article , John Baez wrote:

The reason why Smolin was interested in MOND is that this constant

a_0 = 2 x 10^{-10} m/sec^2

is supposedly about equal the acceleration of the expansion of
the universe due to the cosmological constant, for objects that are...
one Hubble away?


Does someone have the energy to check?


[Moderator's note: The order of magnitude is about right, anyway. -TB]


Okay, thanks. That's good enough: it's just one of those rough
numerical coincidences that makes you - or in this case, Smolin -
lie in bed staring at the ceiling wondering if there could be
something funny going on involving gravity and this particular
acceleration scale.

  #18  
Old May 20th 04, 09:48 PM
John Baez
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Default This Week's Finds in Mathematical Physics (Week 206)

In article ,
Tobias Fritz wrote:

Lee Smolin told me some neat stuff about MOND - that's "Modified
Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain
the strange behavior of galaxies without invoking dark matter. The basic
problem with galaxies is that the outer parts rotate faster than they
should given how much mass we actually see.


Doesn't it seem unreasonable to discard a theory as successful as GR?


Of course! I don't think anyone wants to discard GR because of the
dark matter problems. But, it does make sense to have some people
play around with other ideas.

Except for a few partisans, nobody will take MOND seriously until
it's extended to a full-fledged theory that matches the successes of
GR and isn't horribly ugly - or until it makes some prediction that's
almost impossible to match using conventional means (e.g. fine-tuned
dark matter).

But, it's still good to look at the rotation curves in the paper I
referred to, and wonder what's really going on!

Or is it somehow possible to fit MOND into the framework of GR, like by
modifying the field equations, perhaps by including torsion?


People are trying very hard to fit MOND into GR in all possible ways,
and also to design dark matter that mimics the predictions of MOND.
Bekenstein's new paper:

Jacob D. Bekenstein
Relativistic gravitation theory for the MOND paradigm
http://www.arXiv.org/abs/astro-ph/0403694

seems like the best attempt so far to make MOND into a respectable
theory. It's still not elegant.

Everybody is talking about "dark matter" or alternative theories, when it is
not even really clear what the predictions of GR a recently I heard a
talk about the "averaging problem" in GR; basically, the message was that
we do not know if it is valid to take an average energy-momentum-tensor,
put it into the field equations and see the result as an average metric.


It's a nonlinear equation, so of course this is only approximately right
at best. The question is: is the approximation good enough for practical
purposes?

Unless there's strong evidence that the approximation is *not* good enough,
I think it's a bit over-sensational to say "it's not even really clear what
the predictions of GR are". In every application of fundamental theories
of physics to real-world problems, people make approximations. Trying to
rigorously justify these approximations leads to difficult and interesting
problems in mathematical physics. But, we rarely claim that it's not clear
what the theory actually predicts until we have made everything rigorous!
So, claiming this here might fool nonexperts into thinking there's a big
problem with general relativity, when it's actually just "life as usual".

By googling, I found the following paper:

http://arxiv.org/abs/gr-qc/9703016

which also has some references.

What do the experts think?


I think someone, e.g. the author of this paper, should do some
back-of-the-envelope calculations to guess how much error is introduced
into astrophysical or cosmological calculations by means of this
averaging approximation. If it's a lot, this is a subject of real
importance in astronomy. If it's a little, this subject will mainly
be interesting to mathematical physicists.

I can't imagine this "averaging problem" is big enough to explain the
effects that made people resort to dark matter and MOND, for example!

It might be relevant to understanding the details of hypernovae,
though: you've got a lot of dense matter moving around at relativistic
speeds, maybe turbulent, getting ready to collapse into a black hole...

  #19  
Old May 21st 04, 08:46 AM
Phillip Helbig---remove CLOTHES to reply
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Default This Week's Finds in Mathematical Physics (Week 206)


In article , (John Baez)
writes:

In article ,
Tobias Fritz wrote:

Lee Smolin told me some neat stuff about MOND - that's "Modified
Newtonian Dynamics", which is Mordehai Milgrom's way of trying to explain
the strange behavior of galaxies without invoking dark matter. The basic
problem with galaxies is that the outer parts rotate faster than they
should given how much mass we actually see.


Doesn't it seem unreasonable to discard a theory as successful as GR?


Of course! I don't think anyone wants to discard GR because of the
dark matter problems. But, it does make sense to have some people
play around with other ideas.

Except for a few partisans, nobody will take MOND seriously until
it's extended to a full-fledged theory that matches the successes of
GR and isn't horribly ugly - or until it makes some prediction that's
almost impossible to match using conventional means (e.g. fine-tuned
dark matter).

But, it's still good to look at the rotation curves in the paper I
referred to, and wonder what's really going on!


Perhaps MOND is now at the stage of the Old Quantum Theory. There were
lots of reasons not to accept it as the final theory, and some things
about it are in some sense just plain wrong. Nevertheless, it was
somehow on the right track, and led to full quantum mechanics later on.

Or is it somehow possible to fit MOND into the framework of GR, like by
modifying the field equations, perhaps by including torsion?


People are trying very hard to fit MOND into GR in all possible ways,
and also to design dark matter that mimics the predictions of MOND.
Bekenstein's new paper:

Jacob D. Bekenstein
Relativistic gravitation theory for the MOND paradigm
http://www.arXiv.org/abs/astro-ph/0403694

seems like the best attempt so far to make MOND into a respectable
theory. It's still not elegant.

Everybody is talking about "dark matter" or alternative theories, when it is
not even really clear what the predictions of GR a recently I heard a
talk about the "averaging problem" in GR; basically, the message was that
we do not know if it is valid to take an average energy-momentum-tensor,
put it into the field equations and see the result as an average metric.


It's a nonlinear equation, so of course this is only approximately right
at best. The question is: is the approximation good enough for practical
purposes?

Unless there's strong evidence that the approximation is *not* good enough,
I think it's a bit over-sensational to say "it's not even really clear what
the predictions of GR are". In every application of fundamental theories
of physics to real-world problems, people make approximations. Trying to
rigorously justify these approximations leads to difficult and interesting
problems in mathematical physics. But, we rarely claim that it's not clear
what the theory actually predicts until we have made everything rigorous!
So, claiming this here might fool nonexperts into thinking there's a big
problem with general relativity, when it's actually just "life as usual".

By googling, I found the following paper:

http://arxiv.org/abs/gr-qc/9703016

which also has some references.

What do the experts think?


I think someone, e.g. the author of this paper, should do some
back-of-the-envelope calculations to guess how much error is introduced
into astrophysical or cosmological calculations by means of this
averaging approximation. If it's a lot, this is a subject of real
importance in astronomy. If it's a little, this subject will mainly
be interesting to mathematical physicists.

I can't imagine this "averaging problem" is big enough to explain the
effects that made people resort to dark matter and MOND, for example!


Hasn't there been some progress here since 1997? I seem to recall that
some of the usual suspects---Ehlers, G.F.R. Ellis, Buchert---had made
some progress in the averaging problem in the last few years. It might
be worth a search of the archives for these three names to see if they
have published anything on this.
  #20  
Old May 22nd 04, 10:49 AM
alistair
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Can MOND make any useful predictions about electromagnetic phenomena?
It's a modification of Newton's laws and they can be used in gravity
and electromagnetism.

 




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