Why speaking about diffeomorphisms for change of coordinates?

Question

We often use diffeomorphisms to change coordinates on a smooth manifold $(M,A)$. But, from what I've seen, "changing coordinate" is simply a function $\psi\circ\phi^{-1}$, where $\phi$ and $\psi$ are charts.

It is clear that a diffeomorphism induces a change of coordinates but is the inverse also true? I don't see how to make a global transformation from all the $\psi_j\circ\phi^{-1}_i$'s wich don't necessarily agree on intersections.

This question is related to the fact that I don't really understand why GR is a gauge theory with gauge group Diff($M$). For me, invariance under coordinates transformations is more general than invariance under Diff($M$).

This question is the follow up of this question I posted in the math network but I didn't really get any answers.

Answer 1

This is a question I've also often gotten confused over. (For this answer I don't see the need to speak about charts and maps.) The way I see it is that often when Physicists talk about diffeomorphisms they really just mean a coordinate transformation. However as far as I'm aware, when considering diffeomorphisms you're looking at how tensors change under a pushforward and a coordinate transformation. E.g. for a tensor, we apply a pushforward to the coordinates of the tensor, evaluate the tensor at the point, and then make a transformation back to the original coordinates. I'm unsure if this is equivalent to what people mean by an active coordinate transformation (i.e. the physical spatial points change but the coordinates do not).

The key distinction between a coordinate transformation, in my opinion, is that for a diffeo the coordinates don't change, therefore the volume element $d^4x$ doesn't change, but $\sqrt{-g} d^4x$ would. Conversely, we know for a coordinate transformation it's the other way round.

This is the source I used, but I'm not sure how consistent this is with other text. E.g. Carroll says that diffeo's are an 'active' view and standard coord transformations are a passive view, but they're practically the same: we move the points on a manifold under the diffeo and then evaluate the coords of the new points. Wald instead talks about both active and passive diffeos, and says they're philosophically different but totally equivalent.

Answer 2

I'm not sure either, so I will try to explicate some terms frequently used. The transforms, in the sense you described them, we will call passive transforms. More generally, a passive transform can be any transform involving open subsets of one or two charts in the way you described, with the use-case, that you described, of their being chart-to-chart conversions. That's the epitome of a coordinate transformation.

In contrast, the active transforms are those which move the underlying points over all of $M$. This has to be reflected chart-wise as a family of transforms on $M$ indexed by two charts, and by the transforms on the charts, reflected by them.

Use-cases of interest (to me) are diffeomorphisms on $M$ that only affect a compact subset $U ⊆ M$ non-trivially. I don't know enough about diffeomorphisms or local diffeomorphisms to know if it is even possible. I assume it is. But it is the gist of my question Existence of a diffeomorphism matching a finite point cloud transform.

So, let $φ: M → M$ be a diffeomorphism on an $n$-dimensional differentiable manifold $M$, for some $n = 1, 2, 3, \ldots$ and $A$ is an open covering of $M$ containing a set of chart-maps $m_U: U → ℝ^n$, for $U ∈ A$, and passive chart-to-chart one-to-one conversion maps $$T_{UV}: ℝ^n ⊇ m_V(U ∩ V) ↔ m_U(U ∩ V) ⊆ ℝ^n,$$ such that the usual properties hold, e.g. $T_{UV}(m_V(x)) = m_U(x)$ for $x ∈ U ∩ V$ and $T_{UV} = {T_{VU}}^{-1}$. Then the active transform is indexed by a family of one-to-one maps $φ_{UV}: V_U ↔ U^V$ on $M$, where $V_U = φ^{-1}(U) ∩ V$ and $U^V = U ∩ φ(V)$, that is chart-to-chart compatible.

The chart-to-chart compatibility has to be on both the $U$ and $V$ ends. The transforms $φ_{UV}$ are reflected on the charts as $$m_{φUV}: ℝ^n ⊇ m_V\left(V_U\right) ↔ m_U\left(U^V\right) ⊆ ℝ^n,$$ such that $m_U ∘ φ_{UV} = m_{φUV} ∘ m_V$. (Sniff, sniff - I'm picking up commutative diagrams, somewhere. Bark. Bark.) On the $V$ side, chart-to-chart compatibility means $m_{φUV'} ∘ T_{V'V} = m_{φUV}$ when restricted to the domain $(V∩V')_U$. On the $U$ side, chart-to-chart compatibility means $m_{φUV} = T_{UU'} ∘ m_{φU'V}$, when restricted to the domain $V^{U∩U'}$.

Diffeomorphisms and active transforms - by the explicated definition I've used - have to be defined globally over all of $M$, while passive transforms - by the same explication - are only defined on an open subset of a chart in $M$, rather than over all of $M$.

Answer 3

Physicists often talk locally whilst mathematicians often talk globally.

For a mathematician, a diffeomorphism is a smooth bijection between two manifolds. It is defined locally, as manifolds are defined locally but it is a global notion. Since the domains and codomains of a chart are open they are also manifolds thus the physicists way of speaking of diffeomorphisms is a specialisation of that of mathematicians.

A gauge theory is essentially a vector bundle which is untied to the base spacetime manifold. We think of it as an internal space, say color, that is placed at each point of spacetime in a smooth manner. GR is not a gauge theory in this sense as when written as a vector bundle this has an isomorphism with the tangent bundle of the base spacetime manifold.