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When reading a bit on exotic spheres and exotic $\mathbb{R}^4$'s, I came across some papers of Carl H. Brans and Torsten Asselmeyer-Maluga:

  • "Exotic differentiable structures and general relativity" (1993),

http://arxiv.org/abs/gr-qc/9212003

  • "Exotic Smoothness and Physics" (1994)

http://arxiv.org/abs/gr-qc/9405010

  • "Exotic Smoothness on Spacetime" (1997)

http://arxiv.org/abs/gr-qc/9604048

  • "Smooth quantum gravity: Exotic smoothness and Quantum gravity" (2016)

http://arxiv.org/abs/1601.06436

Without having read the papers, I was wondering:

  • Are these ideas of considering exotic structures "commonly" thought of as useful to pursue?

  • Are there any "significant" results?

These are vague questions, I understand that, but I'm just trying to get an idea on whether bringing these kinds of "deeper" mathematics into the physics is (in this particular case) something worth doing. Of course, if someone knows more and is willing to share, I'm willing to read.

B. Pasternak
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1 Answers1

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I have communicated with both of these fellows. The mathematics is based on Donaldson's theorem that in four dimensions there exists an infinite number of atlases of charts on a manifold that are homeomorphic but not diffeomorphic. I am not able to go into the mathematics, for it is pretty deep. It centers around the moduli space of self-dual connections $SD/{\cal G}$ such that it is the union of all moduli for each metric $g$ $$ SD/{\cal G}~\simeq~\large\cup_{g}{\cal M}_g. $$ The set of possible metric or self-dual connections leads to a projectivization of this to the space of metrics, which turns out to be uncountable and most of its elements are not diffeomorphic.

The important thing I think is the moduli space, not so much that four-manifolds are arbitrarily smooth. If one thinks about this it is not empirically testable whether spacetime is arbitrarily smooth. We can't experimentally probe infinitesimals. As a result this is more of a consistent model system than something one should think about as physically relevant.

Lawrence B. Crowell
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