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.

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One thought on “Exotic Smoothness I: Smooth Manifolds and Physics

  1. Pingback: Exotic Smoothness II: Why Exotic? | Christopher Duston's Home

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