Sean Carroll on Leibniz Equivalence

Below is a formulation of Leibniz Equivalence from Sean Carroll's (online) Lecture Notes on General Relativity (Sc. 5, "More Geometry"):

Let's put some of these ideas into the context of general relativity. You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with metric $g_{\mu \nu}$ and matter fields $\psi$, and $\phi : M \to M$ is a diffeomorphism, then the sets $(M, g_{\mu\nu},\psi)$ and $(M, \phi_{\ast}g_{\mu\nu},\phi_{\ast}\psi)$ represent the same physical situation.

This principle might be called Weak Leibniz Equivalence, a version of the frequent claim that the "gauge freedom" of GR is $\text{Diff}(M)$. This version is fairly standard. For example, it is the same as John Norton's in his SEP article "The Hole Argument" (from note 5):

For any spacetime model $(M, O_1, \dots, O_n)$ and any diffeomorphism $h$ on $M$, Leibniz equivalence asserts that the two models

$(M, O_1, \dots, O_n)$ and $(hM, h^{\ast}O_1, \dots, h^{\ast}O_n)$

represent the same physical system.

And here is the formulation by Urs Schreiber in n-Lab:

In physics, the term general covariance is meant to indicate the property of a physical system or model (in theoretical physics) whose configurations, action functional and equations of motion are all equivariant under the action of the diffeomorphism group on the smooth manifold underlying the spacetime or the worldvolume of the system.

These formulations are weaker than Robert Wald's formulation (from General Relativity 1984: 438):

Leibniz Equivalence (LE)
If spacetimes $(M, g_{\mu\nu},\psi)$ and $(N, h_{\mu\nu},\theta)$ are isomorphic, then they represent the same physical situation.

(This is equivalent to Wald's wording, which can be read here.) Wald's formulation refers to pairs of spacetime models, rather than a single spacetime model.

(LE) implies the weaker principle. For suppose that $(M, g,\psi)$ is our spacetime model, and $\phi : M \to M$ is a diffeomorphism. (I ignore tensor indices.) Then, given definitions of a "pushforward" $\phi_{\ast}$, the models $(M, g,\psi)$ and $(M, \phi_{\ast}g, \phi_{\ast}\psi)$ are isomorphic. Then, by (LE), they represent the same physical situation.

However, (LE) is stronger than the weaker principle. Let $(M, g,\psi)$ be our spacetime model and let us identify the internal structure of the manifold $M = (X, \mathcal{C})$, where $X$ is the carrier set and $\mathcal{C}$ is a maximal atlas on $X$ ensuring that $M$ is a differentiable manifold. Now let

$\pi : X \to X$

be an arbitrary permutation of the carrier set $X$. We may pushforward the charts in $\mathcal{C}$ to $\pi_{\ast}\mathcal{C}$. The resulting space $(X, \pi_{\ast}\mathcal{C})$ is, by construction, isomorphic to the original. Now let $N = (X, \pi_{\ast}\mathcal{C})$. It follows that

$\pi : M \to N$

is a diffeomorphism.

Apply the diffeomorphism $\pi$ to the metric and matter field(s). The result $(N, \pi_{\ast}g, \pi_{\ast}\psi)$ is again isomorphic, by construction, to $(M, g, \psi)$. So, by (LE), $(M, g, \psi)$ and $(N, \pi_{\ast}g, \pi_{\ast}\psi)$ represent the same physical situation.

So, (LE) is much stronger than the claim that GR is "diffeomorphism invariant". (LE) says that General Relativity is "permutation invariant": one can permute the base set $X$ of a spacetime model $(M, g, \psi)$ any way one likes, so long as one pushes forward everything else along the permutation. It tells us that the identity of the points in the base set $X$ does not matter. Only the "overall structural pattern"---including the pattern involving the topology (i.e., the collection of open sets), the metric and matter fields---matters.

Carroll goes on to give an informal intuitive explanation.

Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard. When people say that GR is diffeomorphism invariant, more likely than not they have one of two (closely related) concepts in mind: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime. The first of these stems from the fact that the metric is a dynamical variable, and along with it the connection and volume element and so forth. Nothing is given to us ahead of time, unlike in classical mechanics or SR. As a consequence, there is no way to simplify life by sticking to a specific coordinate system adapted to some absolute elements of the geometry. This state of affairs forces us to be very careful; it is possible that two purportedly distinct configurations (of matter and metric) in GR are actually "the same", related by a diffeomorphism. In a path integral approach to quantum gravity, where we would like to sum over all possible configurations, special care must be taken not to overcount by allowing physically indistinguishable configurations to contribute more than once. In SR or Newtonian mechanics, meanwhile, the existence of a preferred set of coordinates saves us from such ambiguities. The fact that GR has no preferred coordinate system is often garbled into the statement that it is coordinate invariant (or "generally covariant"); both things are true, but one has more content than the other.

If the argument given above is correct, one should add to this that purportedly distinct configurations (of topology, matter and metric) for spacetime models $\mathcal{M}_1$ and $\mathcal{M}_2$ may be related by a permutation of the carrier set, even a completely wild permutation (e.g., one that simply transposes two distinct points). If so, they represent the same physical situation.