TITLE: Is there a relation between 4-dimensional general relativity and exotic smooth structures on $\mathbb{R}^4$?
QUESTION [17 upvotes]: Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemannian manifolds of signature $(n-1,1)$.
I've heard more than once people say that the reason why there are many non-diffeomorphic smooth structures on $\mathbb{R}^n$ only for $n=4$ would be somehow "related to the fact that the spacetime in which we live is $4$-dimensional". Or, on the other hand, that there are exotic smooth structures only on $\mathbb{R}^4$ because "that's the dimension of spacetime".

So, first of all

1) Are there properties or phenomena of general relativity that are "specific" to $\mathbb{R}^{3,1}$, that is, the equivalent version on $\mathbb{R}^{n-1,1}$ does not hold for $n\neq 4$, or more generally on any other Lorentzian manifold homeomorphic to $\mathbb{R}^4$?

I know string theorists proved that if you cross $\mathbb{R}^{n-1,1}$ by a Calabi-Yau manifold then $n=4$ would result in a "critical dimension" for some properties (supersymmetry?) of string theory on the whole $\mathbb{R}^{3,1}\times\mathrm{CY}$, but I'm wondering if something special happens already at the level of GR.

And

2) Has anybody seriously considered the relationship between "the spacetime being $4$-dimensional" (i.e. $n=4$ being a "critical dimension" for GR/ string theory on $\mathbb{R}^{n-1,1}$) and the existence of exotic $\mathbb{R}^n$'s only for $n=4$? To which conclusions did they arrive? Which explanations did they provide?

REPLY [5 votes]: I assume by general relativity you mean classical general relativity as opposed to any quantum version of general relativity. With that in mind I will attempt to answer your questions:

Are there properties or phenomena of general relativity that are "specific" to $\mathbb{R}^{3,1}$, that is, the equivalent version on $\mathbb{R}^{n-1,1}$ does not hold for $n \ne 4$.

There are many such properties. The most obvious of which is the count of local degrees of freedom. (For a reference as to what "local degrees of freedom" in this context means refer to the the end of section 10.2 in Wald's General Relativity.) For $n < 4$ general relativity has no local degrees of freedom. For $n = 4$ general relativity has 2 local degrees of freedom per space-time point. For $n > 4$ general relativity has more than 2 local degrees of freedom per space-time point.

...or more generally on any other Lorentzian manifold homeomorphic to $\mathbb{R}^4$

Here I assume when you say homeomorphic you mean homeomorphic and not diffeomorphic. In other words your question is what properties of general relativity change when one changes the smooth structure on $\mathbb{R}^4$?
Beyond the obvious answer, a previously smooth metric is no longer smooth, I think this question is not answered in the current literature. 
The closest one comes to an answer is in section III of Witten's Global Gravitational Anomalies. There he argues that in more than $4$ dimensions taking the connected sum of a manifold $M$ with an exotic sphere $S$ is equivalent to placing a gravitational instanton on $M$. In four dimensions he is silent, waiting on the resolution of the smooth 4-dimensional Poincare conjecture. However, instantons are only really relavent for quantum general relativity.

Has anybody seriously considered the relationship between "the spacetime being 4-dimensional" and the existence of exotic $\mathbb{R}^n$'s only for $n=4$?

I'd say no one has seriously considered this relationship. There has been lots of speculation. For example a series of articles by Carl Brans an collaborators seem to raise a lot of questions in this direction but not reach any solid conclusions.
My guess is that one would have to prove, in a physicist's sense of the word, that the path-integral is only well-defined, whatever that might mean, due to the inclusion of gravitational instantons if $n=4$. I don't think we are anywhere near being able to do that yet.