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Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemanniann 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?

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I once had a discussion with an expert in exotic structures on 4-manifolds who said that a group of physicists claimed that the reason for dark matter is exotic structures on $\mathbb{R}^4$. I don't know if anything ever came out of it, but it is an interesting idea. –  Jeremy Pecharich Sep 30 '11 at 16:39
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5 Answers 5

Regarding the first part of this question, in four spacetime dimensions there are no known generic violations of the cosmic censorship hypothesis while above four dimensions there is good evidence that cosmic censorship is violated without fine tuning of initial conditions. The best evidence that I know of for the latter statement comes from the analysis of the Gregory-Laflamme instability in http://arxiv.org/pdf/1006.5960 .

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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.

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Here is an argument whose conclusion is that it is unlikely that exotic 4d manifolds are physically important in general relativity. For physical reasons (mainly causality, stability, and determinism) the most important spacetimes are globally hyperbolic (or can be openly embedded in a globally hyperbolic spacetime). A well known result, showed by Geroch (JMP, 1970), is that a globally hyperbolic spacetime is homeomorphic to $S\times \mathbb{R}$, where $S$ is a 3d manifold. While it has been expected by many that this result could be strengthened to a diffeomorphism, a definitive proof seems to have only appeared rather recently (arXiv, 2003).

In other words, any exotic smooth structure would have to be a feature only of the 3-manifold $S$. But as you point out in your question, the exotic smooth structures you are interested in are a purely 4d phenomenon.

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There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejedudice. You may find this related blog discussion here useful: Local and global supersymmetry

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Not really an answer (because there really isn't one), but:

http://www.actaphys.uj.edu.pl/vol40/pdf/v40p3079.pdf

(in case this is not accessible from where you are:

EXOTIC SMOOTH 4-MANIFOLDS AND GERBES AS GEOMETRY FOR QUANTUM GRAVITY∗ Jerzy Król Institute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice iriking@wp.pl)

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