Here are some thoughts about the (active) diffeomorphism invariance in GR. Some of my notations are slightly sloppy and there might be some inaccuracies, but I think I have clarified the issue for myself. I'll be glad to hear criticism of these notes. The above issue is somewhat controversial and obscure, but it originated with the famous Einstein's hole argument and lingered for a hundred years. On the other hand, diffeomorphism invariance is one of the key concept in String Theory and Quantum Loop Gravity, so it is very popular whenever GR is addressed. I am going to argue that in classical GR, diffeomorphism invariance is unimportant and somewhat misleading. Einstein's hole argument is also misleading. In fact from Category Theory point of view, while diffeomorphisms are isomorphisms in the category of smooth manifolds, the isomorphisms in the category of Riemann manifolds are the diffeomorphisms that leave the metric invariant. Here I have in mind the metric not just as a tensor product of the cotangent bundle over the manifold, but the equivalent requirement that all distances from a given point to any other point are preserved by the morphism. On the other hand, in the quest for quantum gravity, often the scene of the action is a smooth manifold, but the metric is something that has to emerge later as a the expectation value of a quantum object. The smooth manifolds without metric are naturally diffeomorphism invariant.
Let me introduce briefly the active diffeomorphisms. A smooth manifold \cal{M} is a topological space that is locally homeomorphic to ℝ^{n} . If U_i \subset \cal{M} is a collection of open sets, the images of the homeomorphisms h_i(U_i)=E_i \subsetℝ^n are called charts and whenever two charts overlap, h_j \circ h_i^{-1}: E_i\cap E_j\rightarrow E_i\cap E_jare smooth (infinitely differentiable)ℝ^{n}\rightarrow ℝ^{n} functions. They are called transition functions. (Active) Diffeomorphisms are defined in a similar way. If d: \cal{M}\rightarrow \cal{N}, it is a diffeomorphism iff h_j \circ d \circ h_i^{-1} is a smooth ℝ^{n}\rightarrow ℝ^{n} function. Here g_j (V_j) \rightarrow ℝ^{n} is a chart of \cal{N} . Sometimes the transition functions, which can be thought as coordinate changes are called passive diffeomorphisms. Whenever we have a Riemann manifold, i.e. a smooth manifold with metric, we can extend the diffeomorphism to the whole bundle and transform the metric at point P\in \cal{M}to the metric at the point d(P)\in \cal{N}. Note however, that the isomorphisms of the category of smooth manifolds are the one to one diffeomorphisms, while the isomorphisms in the category of Riemann manifolds are those one to one diffeomorphisms for which the metric at point P is the same at the one at d(P). Diffeomorphisms d: \cal{M}\rightarrow \cal{M} for which the above is not satisfied can be thought as simple relabeling of the points of the manifold.
A specific application of active diffeomorphisms in GR is Einstein's hole argument. Let's have a Riemann manifold, with a metric that is solution of Einstein's equations (with proper boundary and initial conditions): a Cauchy problem. Let's take an open set (the hole) in the region where we have found the solution and change the coordinates there. We have switched to another chart, and we can find the solution of the Einstein's equations in the new coordinates. As a function of the coordinates, the solution will be everywhere the same, except in the hole. Let's re-interpret the coordinate change as a new solution of Einstein's equations in the region of the hole. This is possible, because of the general covariant form of Einstein's equations. If we take the coordinates of point P in the old solution, we re-interpret the coordinate transformation as a map of the same chart onto itself, which produces an active diffeomorphism of the hole onto itself. In this transformation, also all matter fields are moved around (i.e. their values at the original points are moved to the images of these points). An important and confusing question is whether the new solution has the same metric, i.e. if it is the same Riemann manifold. It is not hard to see that the metric at the point P, labeled by the same coordinates in the old and in the new solution is different (as well as all material fields at that point). But does the solution correspond to a new Riemann manifold? Einstein was able to show that if we take physical signals that label the events (points), the distance between any pair of such physical events is the same in the new and in the old solution. Einstein claimed that the points of the spacetime by themselves do not have physical meaning, and only the intervals between the physical events are real. But one can alternatively say, that the active diffeomorphism in the hole was just relabeling of the space-time points, which was meaningless and unnecessary. The manifold remained the same, because the points labeled by physical events have the same metric in the two solutions. It is interesting that some philosophers take Einstein's remark as a confirmation of Aristotle's, Descartes' and Mach's viewpoint that space(time) is not real and it is just a tool to describe the ordering of the physical objects (events).
Another controversial issue is the treatment of active diffeomorphisms as gauge transformations. Gauge transformations of a dynamic variable (field) correspond to the same physical description (state), both in classical and quantum mechanics. The dynamic variables are redundant, the real physical object is the equivalence class defined by the gauge transformation. The reason that we use the particular redundant dynamic variables is that they are described by a nice linear objects, while the equivalence classes are hard to work with (they are not linear objects). Both in quantum and in classical mechanics, when we need to remove the redundancy, we choose a particular representative from each equivalence class (called also a gauge group orbit), by imposing a gauge-fixing condition. It is a surface that intersects each gauge orbit in a single point. In General Relativity, when described by the metric, there is no redundancy and no need to perform diffeomorphisms, which either bring us to a different point (with different metric) on the same Riemann manifold or are trivial isomorphisms of the Riemann manifold. There are two subtleties: 1. If the Riemann manifold possesses a symmetry, described by a Killing field, there are non-trivial isomorphisms, but they correspond to true symmetries, which map one physical state to another physical state and not to redundancies (gauge symmetries). 2. Sometimes in GR the description uses not the metric tensor but another quantity, e.g. the tetrads. They provide a redundant description of the Riemann manifold, but it makes manifest the local Poincare symmetry, which is a gauge symmetry for the tetrads. However it is unrelated to the diffeomorphisms.
One case where people use gauge fixing is the description of gravitational waves and in general any solution of the Einstein's equations, which is written with respect to some background metric. Now we have two different Riemann manifolds, both satisfying Einstein's equations. Let's treat the second manifold as a produced by infinitesimal diffeomorphism of the first one. Let's write the metric of the second manifold as g_{αβ}+h_{αβ} , where g_{αβ} is the metric of the background manifold and h_{αβ} is an infinitesimal change. Now we treat h_{αβ} as a tensor field on the old manifold. This is not possible in general and only the infinitesimal nature of h_{αβ} allows us to do this. It is a function of 20 arbitrary parameters in 4D: an infinitesimal 4x4 matrix (16 parameters), that transforms gij and the 4 infinitesimal coordinate shifts that move the point P to its new position. Requiring that the new metric is torsion free, imposes 6 conditions and that it is solution of Einstein's equations imposes another 10 conditions on the parameters. We are left with 4 parametric redundant description of h_{αβ} , but this is the price to pay for treating it as a tensor field on the old manifold. To remove the redundancy, we have to impose a 4 parametric gauge-fixing condition.
Finally most of the attempts to build quantum gravity desire to have a background-free description of the theory. This means that you don't start with a particular Riemann manifold, the metric has to emerge as the expectation value of some quantum operator in some physical state. But even in these descriptions of quantum gravity there is a “scene” at which the action happens: a smooth manifold. It is a hope only, that in an even more fundamental approach, the space-time itself will emerge from some other mathematical object, like spin-foam, twistor space or something else.
The “scene” of the current quantum gravity attempts is diffeomorphism invariant; two smooth manifolds, related by one to one diffeomorphism and without any additional structure are indistinguishable . That is why, diffeomorphism invariance is so essential in String Theory and Loop Quantum Gravity. However as we saw, it is not important to classical GR and also it is not clear that the true quantum gravity theory, necessarily will have to start from or go at an intermediate stage through the the diffeomorphism invariant smooth manifold without metric.
