Representing Spacetime as a Branched Covering Space

“We could lick gravity, but sometimes the paperwork is overwhelming”

-Wernher von Braun

Dr. von Braun (yes, that Dr. von Braun) was not just being a smartass German; this is a major stumbling block. General Relativity is a beautiful theory which is incredibly hard to use gracefully. There are several reasons, and I will just highlight two of them.

Spacetime is four-dimensional. The lengths between any two points in a four-dimensional space is given by a (usually symmetric) 4×4 matrix called the metric. A symmetric 4×4 matrix has 10 independent components. The Einstein field equations tells us how each of these 10 components evolve – via a set of 10 nonlinear partial differential equations. To find solutions to these equations (and thus determine the geometry of spacetime), we often exploit symmetries of the problem. For instance, in cosmology we assume spacetime is homogenous (the same at every point in space) and isotropic (the same in every direction), which reduces the number of unknown parameters of the metric from 10 to 1 (the scale factor). There is no general solution for the full Einstein equations.

General Relativity is background independent. I could talk all day about this, but this is the feature which separates GR from all other physical theories (electromagnetism for instance, and all quantum everything). This means the theory is completely self-contained – matter generates spacetime curvature and spacetime curvature acts like matter. Take electrodynamics; photons and electrons travel in a spacetime background, but do not change the spacetime background. That GR has this feature certainly suggests that it is “more fundamental” than other classical theories which still require humans to put in some features of the theory by hand. This makes the 10 components of the metric very dependent on each other (highly coupled), but also it means there is no natural way to do a perturbative expansion. Which is not to say linearized gravity (small changes to the gravitational field which occur on a fixed background) is not useful – that’s where most of what we know about gravitational waves comes from – but when you linearize gravity you break its background independence. This changes the fundamental structure of the theory and makes it behave like the others – which it doesn’t. There is no known way to study small changes to the gravitational field and preserve its background independence.

That’s all to say that General Relativity is hard because geometry in four dimensions is hard. One of the major topics of my work is to try and find some simplifications of the geometry to learn things about the physics. A key mathematical tool is:

Definition A covering space is a space $M$ together with a surjective map  $\pi: M \rightarrow B$ such that for every $x\in B$ there is a neighborhood $U$ of $x$ such

$\pi^{-1}(U)=V_1\cup V_2...\cup V_n$

where each $V_i$ is an open set in $M$. For every covering map $n$ is a constant integer, and we say that $M$ is an $n$-sheeted cover.

Wait not that. This:

Definition A branched covering space is a space $M$ together with a surjective map $\pi: M \rightarrow B$ such that outside of a finite set $R\subset B$, $M$ is a covering space.

The set $R$ is called the branch locus, and over the branch locus some of the sheets “collide”, so that there are now $m inverse images of the covering map. An illustration may be more helpful:

In this figure, (a) and (b) are examples of unbranched covering spaces, whereas (c) is a branched covering space of the circle with branch locus the point $L$. The picture makes it seem like the covers are “crossing” each other, but it’s more accurate to say they are “glued together” in some specific manner. In fact, every singular point is described locally via coordinates $z^n$ for an $n$-fold cover which is why this is very hard to draw in 3 dimensions. The details of why that produces a smooth manifold (as opposed to a figure 8 space, like what you’re probably thinking) are a bit too much for this post.

The reason all this belongs on my website is the following classical result:

Alexander’s Theorem:  Any compact smooth n-manifold ($n > 2$) can be represented as a covering space of the n-sphere branched over an $(n-2)$-complex.

That’s a lot of manifolds! What is most interesting is that since branched covering spaces are just copies of the base space intersecting in some funny ways (like the figure), this means we can represent arbitrary compact, smooth n-manifolds as spheres which have been glued together. This greatly reduces the possibility complexity, particularly when you start thinking about dimensions 3 and 4. The reason this works can be found in the original proof: you start with a triangulation of an arbitrary 3-manifold, and a map from the triangulation to the triangulation of a sphere. This triangulation consists of two 3-complexes, one which contains the point at infinity and one which does not. By just making sure the orientations all work out, one can map each complex in the manifold to one of the two in the sphere, with a map that is p to 1 everywhere but at the vertices and edges of the complex. This argument can be easily extended to higher dimensions.

In fact, this is the first of a whole set of similar results. For instance, you can realize any closed 3-manifold over one specific knot, and you can realize 4-manifolds not just over 2-complexes but over smooth embedded surfaces (provided the number of covers is at least 4). I won’t go into  where these specific results come from, but suffice to say it affords lots of opportunities for interesting model-building in physics – essentially by reparametrizing the degrees of freedom of the model in terms of this geometric and topological information.

In the next post I will present one example of an application which I have worked on – a reparametrization of the gravitational field which I make needlessly complicated and find some interesting connections to spinors and cosmic strings…