#differential geometry

Notion of Position - we need to be able to discuss where things are in our spacetime

Notion of Continuity - moving particles don’t teleport, ie their paths through spacetime are continuous, so we need to be able to state this in the language of our spacetime structure

Locally similar to R⁴ - spacetime (or at least the one we’re familiar with) has three spacial dimensions and one time dimension, so spacetime should be 4-dimensional, and hence look like R⁴ on small scales

Notion of Vector Fields - vector fields have a habit of cropping up in physics (think Maxwell) so it’d probably be useful to have a way of talking about a vector field defined on our spacetime

Notion of Differentiability - we’d like to be able to take a parameterised path of a particle through our spacetime and work out its velocity, acceleration, jerk, etc at each point, hence we need a notion of differentiability and derivatives

Curvature - if you’ve read anything on general relativity, you’ll know that curvature of spacetime is a key concept; this is the notion that prevents us from defining spacetime as simply R⁴ with extra structure, since R⁴ is flat

Length of Paths - again, lengths of world lines (paths through spacetime) play a key roll in general relativity and, in particular, the observations of special relativity; therefore this is something we need to be able to discuss

Past and Future - particles don’t move backwards in time, so we need to be able to define which direction past and future are for each point of our spacetime

So then how are we to build such a structure?

Well, a natural starting point is of course the mathematician’s best friend: a set. Hence we’ll start with a naked set, and gradually add more and more structure to it until we fulfil our conditions for the properties of a spacetime.

A set S is sufficient for defining position, since we can simply assert that each element p ∈ S of our set corresponds to a unique position. But can we discuss continuity using this structure?

We need extra structure!

Accommodating Continuity

The most minimal structure capable of defining continuity is called a topology. A topology is a subset of the power set, O ⊆ 𝒫(S), where elements of O are called the open sets in S.

O must also be closed under finite intersection and arbitrary union, and contain ø and S. This means the union of any collection of open sets is open, and the intersection of any pair of open sets is open.

So what even is an open set meant to be and how does it define continuity? Well, roughly, open sets can be thought of as defining a loose notion of “closeness”. If two elements of S both belong to some open sets in O, they can be thought of as “close”, and smaller such shared open sets correspond to being “closer”.

In Rⁿ, open sets correspond to (unions/intersections of) solid n-dimensional disks with their boundary removed. In R, this is simply an open interval. As an exercise, try and think of what open sets might look like in 2- and 3-dimensional Euclidean space.

Now it might become believable that open sets, defining some notion of “closeness”, might be able to give us a definition for continuity, since continuity essentially means that nearby outputs come from nearby inputs.

Formalising this, we get the following definition of continuity:

“A map f: M -> N, where M has the topology S and N has the topology T, is called continuous iff for any open set of N, A ∈ T, its preimage under f is an open set of N, ie f⁻¹(A) ∈ S”

Here, the preimage of A under f simply means the set of all points in M that get mapped to an element of A by f. As an example, if A is the interval (0, 4) and f(x) = x², then the preimage of A under f is the interval (-2, 2), since this is the set of all points that are between 0 and 4 after squaring.

Having identified that topology is required for discussions of continuity, we will require that our spacetime be a topological space, ie a set S equipped with a topology O.

Locally Euclidean

Luckily, we don’t actually need any extra structure to describe the fact that our spacetime structure should look like R⁴. We do however need to restrict which topological spaces are allowed to be spacetimes to include only those that satisfy this property.

To do this we’ll need to formalise the idea of being “locally like R⁴”.

What we actually mean when we say a space is like R⁴ is that if we have some small subset of it, that subset should be essentially indistinguishable from some subset of R⁴ given the structure provided to the subset.

If you’ve studied any abstract algebra or topology before, this might bring to mind the idea of isomorphism (or homeomorphism as it’s referred to in topology) and this is exactly the right idea! The aforementioned subset should have some bijection with a subset of R⁴ that preserves topological structure.

In terms of what it means to preserve topological structure, we look to the property that topology is intended to study: the bijection should be continuous in both directions.

Formalising properly (and generalising to arbitrary Rⁿ), we get the following definition:

“A topological space (M, O) is locally like Rⁿ iff, for any point p ∈ M, there exists some open set U ∈ O containing p such that U is homeomorphic to (has a bijection, which is continuous in both directions, with) Rⁿ”

In the literature, such a space is referred to as an n-manifold, or a manifold of dimension n.

Notice that choosing a specific homeomorphism for some open set U is equivalent to drawing a continuous coordinate grid onto U.

A specific choice of homeomorphism is called a chart, and a collection of charts whose domains completely cover the space is called an atlas. These definitions will be important in later discussions. An example of a chart on a subset of 2D Euclidean space could be assigning a polar coordinate to each point. I’ll leave it as an exercise to the reader to check that this assignment is continuous in both directions.

So then, we simply must require that our spacetime is not just a topological space, but specifically a 4-manifold.

Are We Done?

It might be tempting, given we just found no addition structure beyond topology is needed to describe the locally Euclidean nature of our spacetime, to hypothesise that we already have enough structure to encompass all the required properties.

Let’s test this hypothesis out by trying to define a consistent notion of differentiability of curves through our spacetime.

Say we have some parameterised curve γ: [0, 1] -> M. Is it differentiable?

Well, let’s take advantage of the fact that our spacetime is a 4-manifold! We can choose some chart x: U -> R⁴ (supposing that γ lies entirely in some open set U; no generality is lost here since, if this is not the case, we can do a similar procedure piecewise over different open sets covering γ) and map γ to the curve (x o γ) which is a curve in R⁴.

Since we know how to tell if curves in R⁴ are differentiable, we could say that γ is differentiable if (x o γ) is differentiable.

This is the right idea, but there’s a problem: this is dependent on our choice of chart.

Given two charts x,y: U -> R⁴, there is no reason to expect that the differentiability of γ’s image under both matches. It could be that, for instance, (x o γ) is differentiable but (y o γ) is not!

Our charts could disagree on the differentiability of γ!

To see exactly why this is the case, we can draw the following commutative diagram:

Here, we see that, because the composition of two functions is differentiable exactly when both of the functions are themselves differentiable, the charts x and y will agree on the differentiability of γ if and only if (y o x⁻¹): R⁴ -> R⁴, often called the chart transition map from x to y, is differentiable.

In equational form:

y o γ = y o (x⁻¹ o x) o γ = (y o x⁻¹) o (x o γ)

There is no reason to expect that (y o x⁻¹) be differentiable because charts, and by extension chart transition maps, are only required to be continuous, thus there is no guarantee that our definition of differentiability is independent of our choice of chart!

We still need more structure after all…