## General Relativity on Exotic R4s — Amenable Groups — Angels on the Head of a Pin? [closed]

I don't know if this is even a legitimate question. The Banach-Tarski Paradox has been answered in terms of Amenable groups. Consider G-sets (pseudo-groups) as actions of groups on sets. Now consider the Poincare Group (general relativity) action on 4-space (Pseudo-Riemann manifold which is a Lie group.) Now compare this action to the action of a Poincare group on an exotic R4 (also a Lie group). Does there exist a paradoxical decomposition for the pseudo-groups in question? In other words, can one instantiate the Banach-Tarski paradox for the Poincare action on an exotic R4s? For that matter, does there exist paradoxical decompositions for the Poincare action on a Pseudo-Riemann manifold as well?

Like I said, I don't even know if even makes any sense to pose the question this way. Certainly, one must be more restrictive with regard to considerations of compactness and connectedness with regard to the Poincare group to begin with.

Perhaps, some clarity in terms of issues of surjectivity and injectivity of homomorphisms from the Poincare group to the group of all automorphisms of the Lie group in question(i.e. pseudo-Riemann manifold vs exotic R4) would be helpful in fiber bundle language or co-homological descriptions. I'm assuming here that the Lie groups are not only manifolds but smooth manifolds (Hilbert's 5th problem answered in the affirmative.)

I guess what I'm driving at is whether the whole issue of exotic smoothness can have anything to do with general relativity. Certainly 4 manifolds support a large number of exotic forms of smoothness because there is no generalized Poincare Conjecture that applies to diffeomorphisms of R4s. The simple fact is that Einstein's notion of general covariance may apply to every dimension of manifold except the very one (4 dimensions) that we need.

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I'm not sure I fully understand your question, but it seems to me that you are mixing up several different kinds of structure: smooth structure (relevant for the notion of an exotic $\mathbb{R}^4$), (pseudo) Riemannian structure (relevant to general relativity), and measure theory (relevant to the Banach-Tarski paradox). I think it is the case that all exotic $\mathbb{R}^4$'s are measure theoretically equivalent (certainly this is true for all of the small exotic $\mathbb{R}^4$'s), so the Banach-Tarski paradox works in precisely the same way. – Paul Siegel Sep 9 at 15:37
If you want to know whether or not there is a counterpart of the Banach-Tarski paradox where $SO(n)$ is replaced by the Poincare group, it all comes down to whether or not the Poincare group is amenable (I think the answer is no in dimension 4, and hence there is a version of the Banach-Tarski paradox). But your final question - whether or not exotic structures are related to general relativity - seems to have nothing to do with the Banach-Tarski paradox. Still, given that excotic structures are constructed using ideas from gauge theory, it wouldn't surprise me if there is something there. – Paul Siegel Sep 9 at 15:44
I think the OP should reformulate the question to make it clearer (and to convince the reader that he knows the meaning of the words he is using). It could be a very interesting question... – Qfwfq Sep 9 at 16:01
You say "what I'm driving at is whether the whole issue of exotic smoothness can have anything to do with general relativity," but it is not clear how that pertains at all to the actual question, which is whether a certain group action admits a paradoxical decomposition. Also, could you please explain your reference to angels on the head of a pin? – Trevor Wilson Sep 9 at 16:42
This potentially may be an interesting question, so I will not vote to close, but I do encourage the author to revise the question. So far the posting looks like a collection of randomly generated phrases. – algori Sep 9 at 19:21
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