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This Week's Finds in Mathematical Physics (Week 206)
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#12
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This Week's Finds in Mathematical Physics (Week 206)
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
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This Week's Finds in Mathematical Physics (Week 206)
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#14
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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! |
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This Week's Finds in Mathematical Physics (Week 206)
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
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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. |
#17
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This Week's Finds in Mathematical Physics (Week 206)
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#18
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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... |
#20
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This Week's Finds in Mathematical Physics (Week 206)
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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