Exotic Smoothness IV: Physical Models

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 \mathbb{R}^4

One of the most unexpected features of studying exotic \mathbb{R}^4 is that there are both “large” and “small” versions (I will refer to exotic \mathbb{R}^4 as R_{\theta}, which is a common convention). The large R_{\theta} are what we expect; manifolds homeomorphic to standard \mathbb{R}^4 but not diffeomorphic to it. Constructing models using large R_\theta seems a bit strange, since we can confirm through experiments that when we use standard \mathbb{R}^4, we can both model and predict the behavior of the universe. If we tried using a large R_\theta 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” R_\theta. These are exotic \mathbb{R}^4 which can be embedded in standard \mathbb{R}^4. 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 \mathbb{R}_{\theta} can be used to construct models of actual physical phenomena.

Dark Matter

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 \mathbb{R}_{\theta} 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.

Normal Matter

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 f:M\to M' 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.

Inflation

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 \mathbb{S}^3\times \mathbb{R} but has an exotic differential structure, denoted \mathbb{S}^3\times _\theta\mathbb{R}. 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 \Sigma at some point in the evolution. Inflation occurs because in order to transition between a trivial sphere and \Sigma, 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 \mathbb{S}^3\times \mathbb{R} (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.

Semiclassical Gravity

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

Z=\int [dg]\exp \left(\frac{i}{\hbar} S[g]\right)\rightarrow \sum_{(M,g)}\exp \left(\frac{i}{\hbar} S[g]\right)

where we replace the integral (which is generally poorly-defined) with a sum over all smooth, inequivalent (up to diffeomorphism) manifolds M and metric g. 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 M 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 \mathbb{C}P^2, 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 \hbar), 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.

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Exotic Smoothness III: Existence

So far, I have discussed what exotic smoothness is and have tried to motivate the basic reason why it might be important for physics. In this post I will talk about the existence of exotic smooth structures and how we see a hint of a deep connection to physics.

Existence

Fair warning; to really understand how we know exotic smooth structures exist, one needs quite a bit of difficult mathematics. It’s easy to see why this question would at least be hard to answer; to show two smooth manifolds M, N are exotic, we need to show that we can find a homeomorphism between them, but that there is no diffeomorphism between them. So the brute force method would be to check every function that could possibly exist – obviously too hard! The usual approach one takes to answer such questions is to look for well-defined characteristics that, for instance, all diffeomorphic manifolds have. These are generically called invariants, and usually come with famous names like Euler, Donaldson, Seiberg, and Witten. Proving that such invariants exist (and are actually useful!) is arguably one of the deepest rabbit holes in pure mathematics, so I will not attempt any kind of complete account here. I will give some important results, and try to provide simple arguments for why they work. It will turn out that the most interesting dimension for exotic smoothness is also the dimension we care the most about for physics.

Low Dimensions (<4)

In dimensions under 4, there are no exotic smooth structures. In dimension 1, there are only 2 non-diffeomorphic smooth manifolds, the 1-sphere and the interval [0,1], and they are not homeomorphic (since any map [0,1]\to \mathbb{S}^1 cannot be continuous). In dimension 2, you can decompose any (closed) smooth manifold into pieces which looks like spheres, tori, or projective planes, and again we find no exotic smoothness.

Dimension 3 is a little trickier, and in fact has only recently been solved by Perelman in 2003. Prior to that, we had the following conjecture:

Geometrization Conjecture (Thurston)Every smooth, closed 3-manifold can be decomposed into prime manifolds, and every prime, closed 3-manifold can be cut along tori so that the interiors are one of 8 geometries.

The geometries are essentially 3-dimensional generalizations of the spheres and tori from the analogous 2-dimensional result. If the geometrization conjecture is true, then the following classical conjecture of Poincare (1904) would also be true:

Poincare Conjecture: The only simply-connected closed 3-manifold is \mathbb{S}^3.

Perelman used Ricci flow to show the existence and uniqueness of these 8 geometric structures, and proved both the previous conjectures. The Ricci flow is worth a very brief aside (actually worth more then I’m giving it). It tells us that Ricci curvature on a manifold is given by the flow of the metric:

\partial_t g_{ij}=-2R_{ij}

Perelman showed that no matter what metric you start with, you flow to one of the 8 possible geometries. The Ricci flow is currently an area of active research, considering higher dimensional Ricci flows and how these flows tend towards singularities, for instance.

A significant result in math circles, Perelman was awarded a Fields medal (which he did not accept) and $1 million from the Clay Mathematics institute (also not accepted). For our purposes, this story ends with: there is no exotic smoothness in dimension 3!

High Dimensions (>4)

It turns out that for high dimensional manifolds, there is only one story to tell: the h-cobordism theorem. The h stands for homotopy, which is the process of continuously (not smoothly!) deforming one object into another. The classic example is homotoping a coffee mug into a donut (gif from Wikipedia, of course…):

Mug_and_Torus_homotopy

The h-cobordism theorem essentially says that if you can do this between two manifold of dimension d>4, then the two manifolds are diffeomorphic. The technique this time is to decompose the manifold into pieces (called handles), and make sure these handles can be untwisted in such a way to make a homotopy. The reason they always can is something called The Whitney Trick, which essentially relies on having “enough space” to move the handles around. Now, this trick only works in dimensions 5 or greater, so the h-cobordism theorem fails to tell us anything about dimension 4. Seeing a trend yet?

One thing that’s different about the higher dimensional case is that there actually are exotic smooth structures. In fact, the first example of exotic smoothness (in any dimension) were the exotic 7-spheres found by John Milnor (1957). There are many other specific examples of exotic smoothness in higher dimensions, but since the universe is (apparently) dimension 4, we will turn there now.

Dimension 4

So the problem is that decomposition techniques generally fail in dimension 4, due to the added complexity but failure of the Whitney disk trick. Now, the topological version of the h-cobordism theorem works; meaning that two manifolds that are homotopic in dimension four are also homeomorphic. Of course, that doesn’t help us very much because we are at least in the category of continuity; want we want is the difference between continuous and smooth. Well, in this short introduction I won’t give any more details (perhaps in another post), but it turns out that in dimension four the thing you really need is the intersection form. This is (roughly) a matrix which describes how two-dimensional surfaces intersect in a four-dimensional manifold. By a complete classification of these forms (done by Freedman and Donaldson), you can do things like try and decompose the manifold while preserving the intersection form. This leads to some contradictions, the most interesting of which leads to the existence of exotic \mathbb{R}^4These would be smooth 4-manifolds which are homeomorphic to our usual \mathbb{R}^4, but which are not diffeomorphic to the usual \mathbb{R}^4. Things are even worse (or better!) – there are infinitely-many exotic \mathbb{R}^4!

So the situation is this;  in terms of exotic smoothness, dimension 4 is special. This presents a major motivation for studying exotic smoothness in the context of physics. We have already discussed that since exotic smooth structures are not smoothly equivalent, we would not expect any results which relied on calculus (like physics!) to be the same on both of them. Of course, this would not matter if we were studying the physics of space alone – since it is 3-dimensional, there is no exotic smoothness. But as soon as we move to the dimension in which all our fundamental theories are based, exotic smoothness suddenly becomes non-trivial.

This is either a very significant observation, or it is not! The next post will discuss how we might try to study exotic smoothness in physics, from both model-building and observational standpoints.

Exotic Smoothness II: What is so Exotic?

In the previous post I tried to give a brief overview of what a smooth manifold is, and why we need to use them for physics. During the construction of a manifold, some choices were made, and in this post I will discuss how these choices might not be unique, and why we should care. By way of review, a graphical representation of the process of constructing manifolds is given by the following illustration:

Hierarchy_v2

Equivalent Atlases

Recall that an atlas on a manifold is a collection of sets U_i and charts \phi_i:U_i\to U with some restrictions on the kind of functions we can have as overlaps \phi_{ij}=\phi_i\circ \phi_j^{-1}\to U_i\cup U_j. If the overlap functions are at least continuous, we have a topological manifold, and if they are smooth we have a smooth manifold. We can actually have anything in between (or other possibilities, like piecewise linear manifolds), but these two categories are sufficient for us to understand. The space U is a subset in some vector space upon which the manifold is modeled; in physics this is often \mathbb{R}^n or \mathbb{C}^n. Banach spaces can also be considered, as long as one keeps in mind that infinite-dimensional spaces have a few quirks that you must keep track of. A nice text which uses Banach spaces wherever possible is  Foundations of Differential Geometry by Lang.

So, how do we know that our choice of atlas \{(U_i,\phi_i)\} is unique?

Definition: Two atlases \{(U_i,\phi_i)\}, \{(V_i,\psi_i)\} on a smooth manifold are compatible if their union is again an atlas. This compatibility is an equivalence relation, and an equivalence class of an atlas is called a differentiable structure. 

Ok, so let’s say we have all the possible sets \mathcal{U}_i on the manifold, and all the possible charts on these sets \Phi_i. We start to group them into atlases

(\mathcal{U}_1,...,\mathcal{U}_j,\Phi_1,...,\Phi_j),~(\mathcal{U}_{j+1},...,\mathcal{U}_k,\Phi_{j+1},...,\Phi_{k}),...

with smooth overlap functions. Now we check to see if they are unique, which means taking a union of all pairs to see if they form another valid atlas. This gives us a bunch of equivalence classes of smooth structures, which one could denote them with a representative from each class

[\{U_i,\phi_i\}],[\{V_i,\psi_i\}],...,\quad U_i,V_i,...\in\mathcal{U}_i~\phi_i,\psi_i,...\in \Phi_i.

(If you know what I am talking about, don’t fret with the notation, I am just trying to be illustrative).

Definition: A topological manifold with inequivalent smooth structures is said to posses exotic smooth structures.

At this stage it’s not obvious why these things might be called “exotic” – from this constructionist point of view, they appear to be totally reasonable. To see why this “exotic” label might be appropriate, we need to know a few more things about manifolds.

Up to this point we have been studying manifolds intrinsicly, but looking at their points, sets, charts, and atlases. One can also compare manifolds by studying the maps between them. By considering the properties of maps

f:M\to N

between manifolds M and N we can learn something about the relative properties of the manifolds. For instance, if f is a continuous map with continuous inverse f^{-1}, we refer to M and N as homeomorphic. If f is smooth with a smooth inverse, M and N are diffeomorphic. A simple example of this can be found by looking at maps between different sized circles. Consider the explicit embedding of a 1-sphere of radius r in \mathbb{C}, that is

\mathbb{S}_r^1=\{(z\in\mathbb{C}||z|=r\}.

Now let’s define the map f:\mathbb{S}^1_a\to \mathbb{S}^1_b by

f(r,\theta)=(\frac{b}{a}r,\theta).

As long as a,b\neq 0 this map is continuous with an continuous inverse, so \mathbb{S}_a^1 and \mathbb{S}^1_b are homeomorphic. In fact, this map and it’s inverse are smooth as well so they are also diffeomorphic. In fact, since a and b are arbitrary we have just proved that all 1-spheres are diffeomorphic. Notice that there was no essential difference between determining if these manifolds are homeomorphic and determining if they are diffeomorphic. This might be what we expect; functions that are continuous but not smooth are not too far on the fringe of mathematical curiousity, but most functions we write down are smooth.

Ok, technically we only proved that “all 1-spheres represented as subsets of \mathbb{C} are diffeomorphic”; we didn’t use the chart construction outlined in the previous post. If we used the charts (which we generally want to do, since explicitly embeddings of manifolds are rather hard to come by), we would have a diagram like the following (for any i and j with U_i\cup V_j\neq 0:

Exotic_Overlap

Notice that in order for the diagram to be commutative the function f:M\to N must satisfy the same conditions as the overlap functions. Since the charts are only local, this would be a local condition like

f_{ij}:=\psi_i\circ f\circ \phi_j^{-1}:\phi_j(U_j)\to \psi_i(V_i)

for any atlas \{(U_i,\phi_i)\} for M and atlas \{(V_i,\psi_i)\} for N. However, since we already know that the two atlases we have chosen are not compatible, there is at least one overlap

\phi_i\circ \psi_j^{-1}:\psi_j(U_i\cup V_j)\to \phi_i (U_i\cup V_j)

that will not be smooth (if it was, this would be a valid overlap function in the smooth category and the atlases would be equivalent). Thus we have another definition for exotic smoothness:

Definition: Two manifolds which are homeomorphic but not diffeomorphic are exotic smooth structures (with respect to each other).

I hope it is now more clear why we called them “exotic” in the first place; for something to be such a structure, it must be impossible  to find a smooth function f:M\to N, while it is  possible to find such a continuous function. All the functions in all the world are checked, and not a single one works.

I will stop here; next time I will give some more details about the existance of exotic smooth structures and then why they are of particular importance in physics.

Exotic Smoothness I: Smooth Manifolds and Physics

Since this WordPress home is a new site, I thought I would try to add a little bit of new content, in a bloggish format. Since you can read my technical work here, I will try to review some of my more interesting results through (semi-)regular postings. I won’t necessarily shy away from technical things – rather, I will try to be as precise as possible without getting bogged down in details. The goal will be for a reader to gain a correct but superficial understanding whilte requiring minimal outside source material. This will be a more of an introduction and overview then you might get by reading a published paper.

Smooth Manifolds and Physics

In short, exotic smoothness is a mathematical feature of models of spacetime (the universe or parts of it). This feature has many implications, from cosmology and dark matter to quantum gravity, but has not been directly verified. Exotic smoothness is also of deep mathematical interest, but since my work focuses on how exotic smoothness could be used to construct physical models, that is what I will mostly focus on.

Exotic smoothness arises when we set down the foundations for general relativity; specifically, when we construct a smooth manifold. Let’s begin by turning general relativity on it’s head; assume spacetime (a model consisting of 3 space and 1 time dimension) is curved, and we would like to study objects moving around in it. This might be the opposite approach that Einstein took, but for this discussion it makes more sense. By saying spacetime is curved, I mean that the distance function (the metric) does not obey the Pythagorean identity. For instance, the distance function on a 2 dimensional curved surface might be

ds=\sqrt{dx^2+f(x,y)dy^2}

For some function f(x,y). Intuitively, this means that moving in the x-direction changes distances you measure along the y-direction.

We are immediately met with a problem; how are we going to do physics on a curved spacetime? Physics requires calculus (for the equations of motion, minimizing the action, etc), but if space is curved then derivatives will clearly be different depending on where we are in space. This problem is solved by a specific mathematical object called a manifold.

The first thing we need to do is start with the only thing we can define unambigiously; the set of all points \{p_i\} in a spacetime M. This is unambigious in the sense that we just take everyone point in space and time, label them by some i, and put them in a big basket. By grouping these points into sets U_j, we can define some notion of “points being near each other”. If we make sure each point p_i is in at least one set, we can define a topology on spacetime:

Definition: A system of subsets T=\{U_i\} for U_i\subset M defines a topology for M if

  • \emptyset and M are both in T.
  • The union of the elements of any subset of T is in T.
  • The intersection of the elements of any finite subset of T is in T.

M is now referred to as a topological space.

Now that we know which points are near other points, we would like to have some coordinate systems on each set. This will allow us to determine how close each point is to another, and also to perform local calculations.

Definition: A chart (U_i,\phi_i) of a topological space M is an open set U_i \subset X with a homeomorphism

\phi_i:U_i\to V,

where V is an open set in \mathbb{R} ^n. The image \phi_i(x)=(x_1,...,x_n)\in V are the local coordinates of M, and M is now an n-dimensional topological manifold.

Essentially this gives the qualitative definition of a manifold as “a space of dimension n which locally looks like a subset of \mathbb{R}^n“. These coordinate charts also tell us exactly how we can move in between each set:

Definition: The set of charts \{(U_i,\phi_i)\} is called an atlas on M and the homeomorphisms \phi_i must satisfy
\phi_{ij}:=\phi_{i}\circ\phi_j^{-1}:\phi_j(U_i\cap U_j)\to \phi_i(U_i\cap U_j)
on the overlap U_i\cap U_j \neq \emptyset.

Now, it is the nature of these transition functions \phi_{ij} which will concern us. If we want to be able to do calculus, we have to be able to move in between open sets, and therefore these functions must be differentiable. If they are infinitely differentiable (called C^\infty functions, meaning we can take as many derivatives as we want), we call M a smooth manifold. We can look at other kinds of manifolds too; topological manifolds have transition functions which are continuous, for example. But since we are interested in physics, the smooth category will be our primary focus.

Now, at this stage we might be ready to perform some calculations – we have coordinate systems and transition functions so any results which we derive locally in one coordinate system can be smoothly extended to any other coordinate system (note: this is normally done with tangent spaces and connections, but for the moment we will just consider “moving around the manifold” to require a complete specification of the \phi_{ij}). However, being careful investigators, we would want to know if there were any steps we have taken which were not unique – for instance, was there another choice of the complete set of transition functions \{\phi_{ij}\} that was a valid choice for our open sets?

It may seem like this is not a worthwhile question to consider; who cares if there is another way to do something, as long as we have at least one way that works? Well, mathematics aside, the issue is really with how we confirm our physical models to be correct. Say we construct our manifold, and perform a theoretical calculation. For instance, we determine how much a light ray deflects when it passes very close to a massive galaxy (gravitational lensing), and we confirm through careful observation that our calculation was correct, provided the mass of the galaxy has some value. How can we be sure the result is reproducible, if there was more than one way to do the theoretical calculation? Even worse, what if we find a second way of constructing the smooth manifold that gives us a different answer when we do the theoretical prediction? We might be lead to an incorrect value for the mass in the galaxy, which could have an effect on models of galaxy formation, cosmology, or even figuring out the total mass of the universe!

That’s getting ahead of ourselves, but this is the essence of exotic smoothness; a specific step in the construction of a manifold is not unique, leading to the possibility that any physical calculation we perform is incorrect.

Next time, I will carefully define what it means for two atlases to be the same, and give an overview of some existence results for exotic smoothness in various dimensions.