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#1
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Physical theory which matches MOND results
I'm on a couple of weeks leave from my computing job at the
moment and I've been using the time to do some physics again. I've found that one of my old ideas gives the same result as the successful but purely empirical MOND theory which explains galaxy rotation curves, but derives it from a simple physical hypothesis. I've currently posted about it to the moderated newsgroup sci.physics.research and also e-mailed a copy to Prof. Milgrom, the author of the MOND theory, but it would be nice to discuss it with someone before I run out of leave (even if only to shoot it down so I can start thinking about something else). Perhaps to do it properly I should submit a letter to some journal, but if this is already known about that would be a waste of time, especially as I don't know where to start. Here are the details of the idea (which I hope is sufficiently specific and testable to call a "theory"): In local GR calculations, one normally assumes that space is flat at a sufficient distance from the local mass or masses which are being studied. However, if the local mass m constitutes a significant fraction m/M of the total mass of the universe, then this assumption seems questionable. If the universe is spatially finite at a given moment in time, then a more plausible assumption would be that the region containing a fraction m/M of the total mass would also loosely speaking contain a related fraction of the 3D "angle" needed to close the universe, so the limit would be "conical" rather than "flat". More specifically, a starting hypothesis might be that a sphere enclosing a fraction m/M of the total mass of the universe would effectively be slightly "conical", missing that proportion of its surface area at any given radius, in the same way that a 2D cone made out of paper has a fixed proportion of its circumference missing at a given radius. For the 3D case, this would mean that the effective radius of the sphere was reduced by a factor sqrt(1-m/M). This factor is then the cosine of the angle by which the "cone" diverges from flat space, so the sine of the angle is sqrt(m/M) and this also then gives the angle of deviation from flatness in radians. The conical space-time at distance r from the source is then curved by 1/r sqrt(m/M) so bodies moving through that space-time would appear to be accelerated by an extra gravitational acceleration c**2/r sqrt(m/M). In MOND, the extra acceleration is sqrt(G m a0)/r where a0 is an arbitrary constant set to be approximately 1.2e-10 ms**-2 to fit the experimental results. For our hypothesis to match MOND exactly, we only require that M is equal to (c**4 / G a0), which gives a value of approximately 10**54 kg. Although trivial methods of estimating the mass of the universe give a result a little lower, around 3 * 10**52 kg, this seems to me to be an interestingly close fit. This suggests that the existing MOND acceleration term could perhaps be written instead as c**2/r sqrt(m/M), where M is around 10**54 kg and may well correspond in some sense to the total mass of the universe, and that it might be possible to find a way to relate this physically to a "conical" limit instead of a "flat" limit in gravitational calculations. I must admit to some use of "analogies" rather than formal calculations in the way that I get from the cone angle being sqrt(m/M) to the space-time curvature being 1/r times that and hence the acceleration being c**2 times the curvature. I'm also not a professional physicist but as it is possible for readers to check out everything I've said for themselves that should not have any bearing on the value of the results. The "conical" model might also help how to analyze cases where more than one mass is involved. In particular, it seems that as the conical model effectively creates the large-scale curvature of the universe out of conical distortions surrounding individual masses, then these conical effects may not necessarily be spherically symmetrical, which might give rise to possible testable differences from the original MOND theory. Jonathan Scott |
#3
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I think that 10**54 kg is close enough given that M is a
"total" mass (or mass-energy) in some sense, rather than simply the mass of the observable universe. I'm not sure what the current estimates are, and they probably assume some dark matter which isn't needed with this type of theory, but I'm fairly sure that a value a couple of orders of magnitude on the high side is not completely ruled out. It is also possible that I've missed a factor of 1/(4 pi) or 1/(2 pi) or some other constant in the way that I derived the curvature based on an analogy with lower dimensions. If for example it were 1/(2 pi) then that would give M = 2.5 * 10**52 kg. However, I'd rather not add any such extra fudge factors without a good mathematical reason. A few years ago when I was studying GR I'm sure I could have checked that bit in minutes, but I'm a bit rusty now so unless I get inspired within the next day or two I'd prefer to just leave it for others to check for any missing constants. Jonathan Scott |
#4
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On rereading my GR text books, I now realize that the
assumption of flatness at infinity in the Schwarzschild solution is something that apparently results purely from spherical symmetry combined with Einstein's vacuum field equations. That means either that my suggestion conflicts with GR or that it conflicts with at least one of the assumptions behind the usual spherical solution. I'm continuing to investigate whether perhaps there is some assumption about the physical coordinates which would still allow the 3-D equivalent of a conical limit. However, given the success of MOND as opposed to GR in predicting galaxy rotation curves, I'd be more inclined to suggest that it's GR that needs fixing. I've also discovered that the standard GR solution for the shape of space around an infinite "cosmic string" involves a missing angle, so that planes perpendicular to the string have the geometry of a 2-D cone. On that basis I'd have expected to be able to get a 3-D conical effect for a sufficiently massive sphere. |
#5
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Looks like it does fit with GR after all. As far as I
can see, if one takes the Schwarzschild solution and multiplies the dr^2 component of the metric by any constant (creating the 3-D conical effect that I was looking for) it appears that the new equation is still a solution of the vacuum field equations. This is apparently normally ignored because it can be hidden mathematically by changing the scale of the time coordinate (which seems questionable to me because I think that adjusting the previously chosen background coordinates has physical implications). I think that I can therefore add the 3-D conical limit to the Schwarzschild solution merely by inserting a constant factor in the dr^2 term of the metric. Whether this has any effect on the geodesics (and hence reproduces the MOND effect) is yet to be seen. (Where are all the GR experts when you need them?) Jonathan Scott |
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