What is spacetime, mathematically? (Part 2)
In the last part I introduced some motivations for our spacetime definition and we constructed a locally Euclidean structure capable of accommodating continuity, that being a topological 4-manifold.
We also attempted to define differentiability of curves using charts, but ran into inconsistencies dependent on our choice of chart.
In this post, I’ll describe the extra structure required to solve these inconsistencies, and introduce the notions of a tangent space, tangent bundle, and vector field.
Accommodating Differentiability
The big issue we had with accommodating differentiability is depicted in the following commutative diagram:
We saw that our chart-based definition of differentiability is only consistent if for any pair of charts x,y: U -> R⁴, the chart transition map (y o x⁻¹) is differentiable.
The issue here is we’re being too general with the set of possible charts we are picking from.
To maintain the property that our space is locally Euclidean, we need the set of possible charts to form an atlas (a collection of charts whose domains cover the whole space), but we’re currently using the largest possible atlas: the set of all charts with no restrictions.
If we require a choice of restricted atlas A, excluding certain charts such that chart transition maps are differentiable for any pair of charts, then we ensure our definition is consistent! Such an atlas is called a differentiable atlas.
Since in general there are multiple such atlases we could choose to restrict ourselves to, this choice of atlas is extra structure that we have to add to our spacetime construction.
In particular, our spacetime is now a triple (M, O, A) containing a set M, a topology on that set O, and a differentiable atlas A, such that (M, O) is a 4-manifold.
This structure is called a differentiable 4-manifold. It finally allows us to accommodate a notion of differentiability.
In fact, we can be more specific! A Cⁿ-atlas is an atlas A such that for any two charts x,y in A, the chart transition map from x to y has a continuous nth derivative. A C∞-atlas (I couldn’t find a superscript infinity character sorry!) is also known as a smooth atlas.
Thanks to a result proven by Hassler Whitney, we know that any Cⁿ-atlas, for n at least 1, has a smooth subatlas. As such, we will only discuss “smooth k-manifolds” for the rest of this series of posts, since any Cⁿ k-manifold can be given a smooth structure by this theorem.
Vector fields on a space are of utmost importance in physics (just look at Maxwell), and in particular smooth vector fields appear especially frequently.
So what is a vector field? Intuitively, a vector field on a manifold is an assignment of a tangent vector to each point on the manifold, and a vector field is smooth if this assignment is smooth.
But what even is a “tangent vector”? And maps require their domain and range to both be differentiable manifolds to even talk about them being smooth, so how can we talk about this assignment being smooth??
To properly define (smooth) vector fields on a differentiable manifold, we first need to discuss tangent spaces, and the tangent bundle.
(Smooth) Vector Fields Tangent Spaces
Given a smooth n-manifold (M, O, A) and a point p, the tangent space to M at p (which we usually write as TₚM) is an n-dimensional vector space.
The elements of this tangent space correspond to the velocities of curves passing through p.
For instance, given a curve γ: [0, 1] -> M for which γ(a) = p for some a ∈ [0, 1], the velocity of γ as it passes through p (which I’ll denote vᵧ,ₚ) is an element of TₚM.
What exactly is vᵧ,ₚ? Well, perhaps unexpectedly, it is a map vᵧ,ₚ: C∞(M) -> R from the set of smooth scalar fields on M to the real numbers.
Why? Well, rather than thinking of tangent vectors as arrows emanating from a point on the manifold, we tend to think of them as directional derivative operators. If we want to know the derivative of some smooth scalar field f: M -> R in the direction of the velocity of the curve γ at the point p, we compute vᵧ,ₚ(f).
Essentially vᵧ,ₚ specifies a direction in which to take a derivative of a smooth scalar field. It hence takes a smooth scalar field as its input and returns its directional derivative in the direction it represents.
Because I’m chronically lazy, it’s left as an exercise to the reader to prove that the set of tangent vectors forms a real vector space. Alternatively, you can trust me, because I’m trustworthy, and you should believe everything I say.
A note: Any linear algebraists reading will be screaming at me right now for my glaring omission of the cotangent space at p, Tₚ*M, so I’ll get that out of the way now.
Tₚ*M is the real vector space of real valued linear functions from TₚM to R. Elements of Tₚ*M, “tangent covectors”, correspond to gradient functions. This means that given some smooth scalar field f: M -> R, there is a tangent covector at p denoted df: TₚM -> R which takes some tangent vector v and returns the directional derivative of f in the direction of v.
The difference here is that the smooth scalar field is fixed rather than the direction in which the derivative is taken.
One final note is that a choice of chart x: U -> Rⁿ whose domain contains p induces a canonical basis on TₚM (and Tₚ*M):
For each point p and basis vector û in Rⁿ, we can form the curve γ(t) = ût + x(p). This then induces a curve on the manifold (x⁻¹ o γ), which passes through p (when t = 0). This curve has an associated tangent vector at p, which becomes a basis vector for TₚM.
Furthermore, let x¹, x², …, xⁿ be real scalar fields on U, where xᵏ(p) gives the kth component of x(p). Then dxᵏ for all k ∈ [1, n] ∩ N gives a basis of Tₚ*M.
Notice that both vector spaces necessarily have the same dimension as the manifold.
In general, a bundle is simply a triple (E, B, π), where E and B are smooth manifolds and π: E -> B is a smooth surjection.
E is called the total space, B the base space, and π the projection map.
A fibre over a point p ∈ B is the preimage of p under the projection map.
A section σ: B -> E of a bundle is a map from the base space to the total space such that (π o σ) is the identity on the base space. This essentially means that σ maps points of the base space to points of their fibres.
The best example here is to imagine an infinite vertical cylinder as the total space, and a single horizontal circular cross-section as the base space, with the projection map simply mapping a point on the cylinder to the point on the circle that it sits directly above (or below). The fibre over a point p is then a vertical line passing through p. A section is obtained by intersecting any nonvertical plane with the cylinder.
An example of a collection of fibres is depicted below:
Given an n-manifold M, its tangent bundle is (TM, M, π). Here, TM is the disjoint union of TₚM over all p ∈ M, and for v ∈ TM, π(v) gives the point whose tangent space v belongs to.
Note that this means that the fibre over any point p ∈ M is simply TₚM.
To make this a true bundle, we need to give TM the structure of a smooth manifold.
To construct a topology on TM, we simply assert that a subset of TM is open if and only if it is the preimage of some open set in M under the projection map. This is the coarsest topology such that the projection map is continuous.
It’s more difficult to construct a canonical smooth atlas on TM, but it is possible:
Given the smooth atlas A on M, for each chart x ∈ A, we can use the induced basis from x on TₚM for every p in the domain of x to produce an induced chart on TM. Given a point v ∈ TM, this induced chart will map v to the point in R²ⁿ whose first n coordinates are exactly the coordinates of (x o π)(v), and whose next n coordinates are the components of v when written in the induced basis on the tangent space that v belongs to.
I once again leave it as a (trust) exercise to the reader to show that this new map is a chart, and that the set of all such induced charts forms a smooth atlas on TM.
When the dust settles, we have a smooth 2n-manifold structure on TM, realising (TM, M, π) as a bundle.
For this bundle, a section can be interpreted as assigning to every point on the manifold a vector from its tangent space.
This is exactly the definition of a vector field!
For those just chewing at their nails, terrified that I will neglect the cotangent bundle, you are right to fear. The cotangent bundle is constructed similarly enough to the tangent bundle that I don’t deem it necessary to go through the construction in detail. It is, as expected, a bundle (T*M, M, π) where T*M is given a canonical smooth 2n-manifold structure induced by the structure on M.
(Smooth) Vector Spaces: Take 2
We can now aptly and formally define a vector space on M as a section of the tangent bundle on M.
Since the tangent bundle is endowed with a smooth structure, we can now also specify, in a well-defined manner, that a smooth vector field is a smooth section of the tangent bundle.
Similarly, a (smooth) covector field is defined as a (smooth) section of the cotangent bundle.
This concludes our discussion of vector fields!
In my next post(s), I’ll bravely approach the topic of tensor fields, connections, and more! :3
