Physics of Differentiable Manifolds: Part I

Often when trying to study general relativity, the most difficult aspect to understand rigorously is the underlying mathematics. Before anyone can do general relativity, the concepts of differentiable manifolds, smooth curves, vectors & co-vectors, tensors, tangent spaces and cotangent spaces, and many more have to be fully fleshed out. There are excellent texts out there that help achieve this goal, such as Sean Carroll’s Spacetime and Geometry. Here, however, we’ll provide a list of definitions with a few nuggets of information towards the end as a useful summary of the topics discussed in this post. Eventually, we will motivate how quantum field theory and general relativity can both be generated with concepts starting with a differentiable manifold with a metric.

Differentiable Manifold: An $n$-dimensional $\textit{differentiable manifold}$ is a set $\mathcal{M}$ together with a collection of subsets $\mathcal{O}_\alpha$ such that;

  1.   $\bigcup_\alpha \mathcal{O}_\alpha = \mathcal{M}$, that is the subsets $\mathcal{O}_\alpha$ $\textit{cover}$ $\mathcal{M}$.
  2.   For each $\alpha$ there is a one-to-one and onto map $\phi_\alpha : \mathcal{O}_\alpha\to\mathcal{U}_\alpha,$ where $\mathcal{U}_\alpha$ is an open subset of $\mathbb{R}^n.$ These are called $charts$ or $\textit{coordinate systems}$. The set $\{\phi_\alpha\}$ is called an ${atlas}$.
  3.   If $\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ overlap, i.e., $\mathcal{O}_\alpha\cap\mathcal{O}_\beta\ne \emptyset$, then $\phi_\beta\circ\phi_\alpha^{-1}$ maps from $\phi_\alpha(\mathcal{O}_\alpha\cup\mathcal{O}_\beta)\subset\mathcal{U}_\alpha\subset\mathbb{R}^n$ to $\phi_\beta(\mathcal{O}_\alpha\cup\mathcal{O}_\beta)\subset\mathcal{U}_\alpha\subset\mathbb{R}^n$. We require that this map be smooth (infinitely differentiable).

So, now that we have a manifold, we need functions mapping to and from the manifold that are smooth.

Smooth Scalar Function: A function $f:\mathcal{M}\to\mathbb{R}$ is smooth if, and only if, for any chart $\phi$, $f\circ\phi^{-1}:\mathcal{U}\to\mathbb{R}$ is a smooth function. $f$ is also known as a scalar field.

Smooth Curve: A smooth curve on a differentiable manifold $\mathcal{M}$ is a smooth function $\lambda : I \to \mathcal{M}$, where $I$ is an open interval in $\mathbb{R}$. By this we mean that $\phi_\alpha\circ\lambda$ is a smooth map from $I\to\mathbb{R}^n$ for all charts $\phi_\alpha.$

So the million dollar questions are “how does this all work” and “how do I remember this?” We’ll first start by discussing how to remember how these mappings all fit together with the following image:

This diagram shows the maps that we have discussed above. It is worth mentioning that $\mathcal{M}\subset \mathbb{R}^n$, and that $\mathcal{I}\subset \mathbb{R}.$ A few tricks to remember what is going on would be to see that the terms with $f$ are on the right side of the circle, and that both subsets of and $\mathbb{R}^n$ itself are located at $\pi, \&\ \pi/2$ respectively with $\mathbb{R}$ remaining to fill in spots. With this, you can recreate the circle in a pinch.

Of these mappings, $\lambda$ works in the following way. Suppose that $\lambda\equiv\lambda(t)$ for some $t\in \mathcal{I}\subset\mathbb{R},$ then $\lambda(t): t\mapsto (x^0(t),x^1(t),x^2(t),\cdots,x^n(t))$ where $\{x^\mu(t)\}$ are the coordinates of a point on the manifold. Then the scalar function $f$ takes this set of coordinates and maps it back to $\mathbb{R}$ in some way.

Example: Let $\lambda:\mathbb{R}\to\mathcal{S}^n$ be a smooth curve, such that $\lambda(t)|_{t=t_0} = p\in\mathcal{S}^n$ where $\mathcal{S}^n$ is an $n-$sphere of unity, meaning its radius is one. Then on this manifold, we can define a difference function, $d\equiv d(\{x^\mu(t)\})$ such that $$d(x^\mu(t)-x^\mu(t_0))=v|t-t_0|.$$ The difference function is an example of a smooth function as it maps points, or lengths in this case, on the sphere to a number in $\mathbb{R}.$ The astute observer might wonder what that $v\in\mathbb{R}$ is doing in the above equation, and to that, I say, all in due time … (it’s the velocity but you didn’t hear it from me).

However, the example is deeper than that. For those of you that have done any analysis, you will recognize this form as the mean value theorem if $d(x^\mu)$ is a linear function. Those of you with more geometry experience might see this as the arc length. In fact, and this may or may not blow your mind, but this equation is related to geodesics on a sphere and to how metric spaces operate.

So, this is great and all, but why is this useful?

The usefulness of a differentiable manifold is that on such a space, we can define a coordinate system with the charts, $\phi_a$. This coordinate system can then provide operations that we use in $\mathbb{R}^n$ space, specifically a derivative. To have a derivative, we need the following objects: smooth functions, smooth curves, a tangent vector, a coordinate basis, and finally a differential element. We explore all these objects because we wish to describe some physics on the manifold, and to discuss physics, we need equations of motion. These equations of motion, which stem from the action $$\mathcal{S} = \int_V d^4 x \mathcal{L}$$ where $V$ is the volume and $\mathcal{L}$ is the Lagrangian density, need the above objects defined rigorously before being able to do calculus on the manifold.

In fact, it could be argued that the manifold that we start with, equipped with some metric tensor, is the only real requirement to perform and discover new physics. All physics stems from calculus on a manifold + some meaningful restrictions imposed by reality.

Now that we have built the foundations of our space, next time we will move onto discussing derivatives on the manifold while continuing to develop this idea of geodesics.

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