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
One of the most unexpected features of studying exotic is that there are both “large” and “small” versions (I will refer to exotic as , which is a common convention). The large are what we expect; manifolds homeomorphic to standard but not diffeomorphic to it. Constructing models using large seems a bit strange, since we can confirm through experiments that when we use standard , we can both model and predict the behavior of the universe. If we tried using a large to do the same thing, we had better get the same results. So we are lead to consider that maybe exotic smoothness doesn’t even matter; or indeed, if this was the only possibility it cannot matter – at least at this level of consideration.
However, there are also the “small” . These are exotic which can be embedded in standard . There are several different ways this can happen; I will not provide any details here (maybe in a later post). Right now, I’d like to skip how we know they exist and talk about how can be used to construct models of actual physical phenomena.
Dark matter is literally matter which does not shine. In other words it does not interact electromagnetically. The story of the discovery of dark matter is well-known; in 1933 Fritz Zwicky noticed that the velocity at which stars orbited spiral galaxies were significantly different than what was predicted by Newtonian mechanics. He found that you could correct this by adding extra matter which we could not see. This effect is very pervasive amoung spiral galaxies, and dark matter is also important for galaxy formation models and early universe cosmology. The only “direct detection” of dark matter has been done using lensing studies (the bullet cluster) – although I have never seen an analysis that has actually convinced me these detections are as “direct” as is claimed…but in any case, most physicsts and astronomers strongly believe that dark matter exists, despite the fact we have never detected it as particles (such as at LHC or other colliders). It has only been detected through its gravitational interaction with the surrounding environment.
Recall what gravitational lensing is; the path of the light is bent due to a large gravitational potential. In the case of the bullet cluster, the claim is that the light is bent more then what we can predict using just the luminious matter that we see. There is a conjecture that light crossing the boundary of a small would experience the same effect:
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. Light crossing through an exotic region of space might deflect exactly as if it had encountered a concentration of dark matter. If this conjecture was shown to be true, it would mean that what we thought was dark matter (and thus exotic when compared the well-established models of particle physics) would not be due to exotic physics but rather, exotic mathematics. In fact, this is not really modifying any of the well-tested theories on the books already, since the mathematics we are talking about here has always existed; we’ve just always ignored it.
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 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.
A recent article by Asselmeyer-Maluga and collaborator Jerzy Krol (here) has shown that one can generate inflation with exotic smooth structures. This is done by using a manifold which is topologically but has an exotic differential structure, denoted . By gluing these structures together one can construct a spacetime which has a 3-sphere in the causal past and future, but there is a topology change to the spatial section at some point in the evolution. Inflation occurs because in order to transition between a trivial sphere and , one must add more and more Casson handles (handles similar to those I mentioned before). This causes an exponential change in the spatial curvature, driving the inflation. As a kind of “price to pay” for this, one must give up the notion of global hyperbolicity, since any globally hyperbolic manifold is isometric to (with standard smoothness structure). In order for a globally hyperbolic manifold to not violate causality, there must be naked singularities. As one might expect this approach is quite detailed, but is one of the few examples of a model, built from exotic smooth structures, which reproduces the general features of observed phenomena.
Although a bit less sexy than using exotic smooth structures to describe dark matter or inflation, by studying them in the context of semiclassical gravity we can more directly see that a calculation is different with or without them. The basic idea is to start with a path integral
where we replace the integral (which is generally poorly-defined) with a sum over all smooth, inequivalent (up to diffeomorphism) manifolds and metric . Now, we may still not be able to write down this sum explicitly (since the full classification problem in 4 dimensions has not been solved), but if we restrict the manifolds and metrics to only those that solve the Einstein equations, we have some confidence that the sum contains the “most important terms” in the full path integral. This restriction to classical solutions is where this approach gets its name.
Now comes the exotic smoothness; say we wanted to do some calculation – a cross-section or expectation value a la statistical mechanics – and we wanted to “include all the spheres” in the sum. Well, since it’s not known if the sphere has exotic smooth structures, this is not possible since each different should be non-diffeomorphic. And, if it was possible, we would need to know if it even mattered. If we included all the exotic structures, would we even get something different? This is really a key question, since up to this point we (the royal ‘we’) have essentially been ignoring the exotic smooth structures. Was that the wrong thing to do?
The answer, at least in a restricted sense, is yes. By picking a specific example (which happens to be iterated branched covers of , see here), one can see that there is a difference between including a single space in that sum and including several members of an exotic family. Essentially, the exotic family has manifolds which are not isometric – that is, have different volumes. By forming an expectation value of volume (and expanding to first order in ), it can be shown that the answer you get depends on the specific members of the exotic family as well as the conformal scale factor. The behavior mimics a phase transition between the largest and smallest volumes. Thus, in semiclassical gravity there are at least some instances when the Bran Conjecture is certainly true.
Well, this was long post but I wanted to give the current state of model-building based on exotic smooth structures. I think I will stop here; much of my other work is related to this topic, but this is enough to know in terms of exotic smooth structure.