14/05/05 20:50:01.44
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May 30, 2013 by cduston
Exotic Smoothness IV: Physical Models
So far, I have introduced some of the basic notions of smooth manifolds, what exotic smoothness is, and (very superficially!) how we know it exists.
In this post I will talk about how one can go about constructing a physical model which includes exotic smooth structures, and what kinds of behavior we can expect.
“What problems can exotic smoothness solve?” might be a summary for this post, but as we will see, there is more conjecture then problem solving.
Large and Small Exotic R^4
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Dark Matter
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The Brans Conjecture
Localized exotic smoothness can mimic an additional source for the gravitational field.
Of course, this conjecture is quite vague, but what Brans had in mind was exactly a solution to the dark matter problem.
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Normal Matter
I think it’s fair to say the Brans conjecture has not been proven yet ? specifically, there is not currently a model of dark matter which can be compared to (and thus verified by) observations.
However, there has certainly been work done which suggests that exotic smoothness can mimic mass in more limited ways. For instance,
Torsten Asselmeyer-Maluga (you will see his name come up frequently in connection with this topic ? he has been diligently working on getting very interesting results for over a decade now) has shown
that the intersection of some special surfaces in 4-manifolds (which represent points of which a homeomorphism f:M →M' fails to be a diffeomorphism) can create non-zero curvature terms (1997).
In other words, the failure of two 4-manifolds to be diffeomorphic at points can mimic mass terms.
This can be extended (see here and here), so that it appears that this result is quite general, and can be used to construct matter with a variety of internal symmetries.