In previous questions of mine here and here it was established that Special Relativity, as a special case of General Relativity, can be considered as the theory of a (smooth) Lorentz manifold $(M,g)$ where $M$ is globally diffeomorphic to (the standard diff. structure of) $\mathbb{R}^4$ and $g$ is a globally flat metric tensor field with signature $(+—)$. Note that I'm not mentioning time orientation here, which is probably a grave conceptual error on my part.
There is the notion of the general linear group $GL(T_pM)$ of each tangent space, and each of these has a subgroup $O_g(T_pM)$ of those linear maps, which preserve $G$. This Lie group is isomorphic to $O(1,3)$ of matrices preserving the minkowski metric. Similarly one obtains other $GL(T_pM)$ subgroups as analogues of (and isomorphic to) commonly considered subgroups of $O(1,3)$ such as its connected-component-subgroups. On the other hand, the group $$\text{Isom}(M)=\{\phi:M\to M|\ \phi \text{ is a diffeomorphism and } \phi_*g=g\}$$ of isometries of $M$ is isomorphic to the Poincaré group $\text{P}$ as a Lie group, as established in the second linked question above. There are no canonical choices for the isomorphisms of Lie groups mentioned here.
In particular, transformations $g\in\text{Isom}(M)$ as acting on points of spacetime $M$ can not be considered as generated by "linear maps and translations", because the addition of spacetime points is not defined. In fact I want to formulate Special Relativity in this unusual way is mainly to achieve this and distinguish clearly between transformations acting on tangent vectors and on events(=spacetime points).
Next I want to consider the applications of the above transformations to various things, and the meaning of "invariance, covariance and form-invariance" of said things under transformations. This is related to this question, which I don't fully understand the answer to. I'm thinking about making another question later concerning this terminology, understood within the above geometric framework (Feel free to comment on if this is a good idea. The questions has been asked a thousand times but I never fully understood the answers). For now, more pressing matters:
Consider a map $\phi:M\xrightarrow[]{C^{\infty}}\mathbb{R}$, which we think of as scalar field. Choosing two arbitrary sets of global coordinates $x,y:M\xrightarrow[]{C^{\infty}}\mathbb{R}^4$ we obtain coordinate representations $ \sideset{_x}{}{\phi}:=\phi\circ x^{-1} $ and $ \sideset{_y}{}{\phi}:=\phi\circ y^{-1} $. Clearly $\sideset{_x}{}{\phi}=\sideset{_y}{}{\phi}\circ J$ where $J:=(y\circ x^{-1}):\mathbb{R}^4\xrightarrow[]{C^{\infty}}\mathbb{R}^4$ is the coordinate transformation from $x$ to $y$ coordinates. Consider the special case where $J$ is constant and equals a Poincaré transformation as acting on tuples of numbers. Clearly in this case there is a unique $S\in\text{Isom}(M)$ such that
$$\sideset{_x}{}{\phi}=\sideset{_y}{}{(\phi\circ S)}:=(\phi\circ S)\circ y^{-1},$$
which is given by $S=x^{-1}\circ y$. What I'm trying to say is that there is a unique way to transform a field $\phi $ into a field $\phi'=\phi\circ S$, such that the coordinate representations match. I think this is called "active and passive" transformations. The correspondence between these could be expressed by defining a coordinate representation of a diffeomorphism $\sideset{_y}{}{ S}:=y\circ S\circ y^{-1}$ and noticing that $\sideset{_y}{}{ S}=J$. (I'm somewhat suprised by not getting an inverse somewhere…)
My first question is: Is there an important conceptual difference here? We established the symmetry groups transforming vectors and spacetime points, which are not a priori coordinate changes. The "indepencence" of physical reality from the choice of a coordinate system seems like a triviality to me and should be a given in ANY theory of physics that uses coordinates, or am I looking at this wrong? The spacetime symmetries on the other hand are non-trivial assumptions about the nature of space and time and for Poincaré symmetry characteristic of Relativity Theory (right?). Here the differences in transformation groups considered within different theories (Class. Mech=Galilei, SR=Poincaré, GR=(???)Diff) should have nothing to do with being too lazy to define coordinate-independent integrals and derivatives, right? I got the impression that all that talk about "Lorentz-tensors" (i.e. objects that "transform like tensors under Lorentz transformations but not under more general ones") in the special relativistic formulation of Classical Electrodynamics is mostly due to being too lazy to explain covariant derivatives?! Same motivations as the claim that "accelerated motion is not described/describable by Special Relativity" – to me a preposterous statement.
Assuming there is an important fundamental distinction between symmetries of spacetime and being able to choose an arbitrary coordinate system to describe physics, what is the motivation to define the corresponding transformation of a scalar field by $\phi'=\phi\circ S^{-1}$ where $S^{-1}$ is some space-time transformation or symmetry? Also how should one interpret the statement that the symmetry group of General Relativity is the full Diffeomorphism group? In particular I'm still not certain about whether or not it is correct to think of an "observer" as implying a choice of coordinates that discribe what this observer measures. This clashes with the statement "all observers are equivalent" (interpret as: "any coordinates you like can be used to describe the same physics") being used to mean things like "all observers see the same speed of light". Certainly if there was an ether one could still use any coordinates one liked.
Recently I've stumbled uppon this, where Terence Tao claims (if I understood correctly) that fixing coordinates is a special case of gauge fixing. Question 2: Does that mean that space-time symmetry like Poincaré symmetry is a special case of gauge symmetry? This is as opposed to (my impression concerning) the standard presentation where only symmetries under transformations of "internal" degrees of freedeom are called such. I think the questions are very closely linked. My hope is certainly that the answer to number 2 is "no" (while the claim by Tao is of course true). Thank you very much indeed for reading all this and in advance for any answers or comments.
Best Answer
As you say, that we can choose any coordinate system we like to do physics is a triviality, and not the point of general covariance. The point is that coordinate transformations/diffeomorphism of $M$ are symmetries of the Einstein-Hilbert action in the sense that the action is invariant under them.
The significance of the isometries of $M$ is that to each isometry there belongs a Killing vector field, and to each such Killing vector field there corresponds a conserved quantity.
Both of these statements are different from saying that the invariance under general coordinate transformations is a gauge symmetry. One may indeed view general relativity as a gauge theory whose gauge group is $\mathrm{GL}(4,\mathbb{R})$ and whose gauge field are the Christoffel symbols $\Gamma^\mu$ viewed as a $\mathrm{GL}(4,\mathbb{R})$-valued field. The unusual thing here is that the gauge transformations $M \to \mathrm{GL}(4,\mathbb{R})$ are induced as the Jacobian matrices of diffeomorphisms $ M \to M$. If one consequently views the connection given by $\Gamma$ as a dynamical variable in its own right, its equation of motion induced by the Einstein-Hilbert action are precisely the conditions for it to be Levi-Civita. This property is special to GR/the E-H action and not a feature of all generally covariant theories, although the E-H action is almost the unique action for a generally covariant theory in which the metric is dynamical