## Sunday, 4 August 2013

### 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.

1. From my reading of Wald he seem to be using the word 'diffeomorphic' for your (LE), not 'isomorphic'. In what sense are you using the word isomorphic here?

If you use it synonymously with diffeomorphism I would still agree with you that Wald's statement is stronger than Carroll's, since if my understanding is correct two manifolds M and N can be diffeomorphic without having the same differential structure determined by their atlas, and thus not equal. (That is if one take the differential structure as being a criterion for determining manifold equality). Carrol seem to define M and N to be equal if they are diffeomorphic which might explain the apparent differences of what it means to be diffeomorphism invariant.

Furthermore do you take the topology of M to be determined by the charts of it's maximal atlas?

2. Thanks for the comment - by structures $M_1$ and $M_2$ being isomorphic I mean there's a bijection between the carrier sets which preserves all structure (relations, distinguished subsets, etc.). I use a modification of the definition of a what a manifold is from Robbin and Salamon,

http://www.math.ethz.ch/~salamon/PREPRINTS/diffgeo.pdf

Following them (roughly), I define a "manifold presentation" to be a pair $(A,C)$ where C is an atlas; and I define a manifold to be a pair $(A,C^{+})$, with $C^{+}$ the maximal atlas determined by $C$. So, distinct manifold "presentations" can give the same manifold, because you can get the same maximal atlas. (Yes, the topology is determined uniquely by the atlas $C$. The converse sometimes doesn't hold though for some weird manifolds.)

Carroll's formulation -- which is fairly standard in a lot of GR literature -- is that if $\phi : M \to M$ is a diffeomorphism, then $(M,g,T)$ and $(M,\phi[g], \phi[T])$ represents the same physical world. I agree that this is correct - but is the condition "if $\phi : M \to M$ is a diffeomorphism" necessary? Do we have to move the points around in a smooth way? Or can we move them around in a crazy way?

Wald's formulation implies something much stronger. Exposing the carrier set and the (maximal) atlas, spacetime models have the form $(A,C,g, T)$ where A is carrier set and C is a maximal atlas on A, g the metric and T maybe the energy tensor. Let $\phi : A \to A$ be any permutation of the carrier set whatsoever (may be as topologically crazy as you like), then $(A,C,g,T)$ and $(A, \phi[C],\phi[g],\phi[T))$ are isomorphic (in the sense meant in model theory). On Wald's formulation, they represent the same physical world. So, my conclusion is that the condition "$\phi : M \to M$ is a diffeomorphism" is unnecessary.

Wald's formulation just amounts to "isomorphic spacetime models represent the same physical world".

The mathematical point is kind of silly/obvious. If you have a structure $M = (A, R_1, ...)$ and take any bijection $f : A \to A$, then $(A, f[R_1], ...)$ is automatically isomorphic to the original. E.g., suppose $(\mathbb{R},<)$ is the ordered reals and $f : \mathbb{R} \to \mathbb{R}$ any bijection. Then $(\mathbb{R},f[<])$ is isomorphic to $(\mathbb{R},<)$.

My note on this is here,

But the responses I get to it in talks are "no, Jeff, that's wrong; it has to be a diffeomorphism of M to itself" or "yes, that's right, but obvious".

Jeff

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4. Hi Jeff! Thanks so much for you reply.
I do not think your conclusion is neither obvious or it's subtlety appreciated by many physicists. Being a physicist myself I became interested in the subject of diffeomorphism invariance after reading Northon, Stachel and Rovelli. Also because I'm a physicists, I was a little uncomfortable with the proofs in your note. E.g. in Lemma 2 where you appealed to diffeomorphisms being isomorphisms between differentiable manifolds. It also seemed to me that you did not establish the equivalence of your version of LE (above in the blogpost) and Wald's. Would you not need the converse of Lemma 2 and 3 for that?

Btw: Do you have any recommendations on literature for an interested physicist to read up on isomorphism theory, and the structuralism of Russel? I have some knowledge of topology, smooth manifold theory and Riemannian geometry.

1. Mikael, is the part that isn't clear this part:

"Let $\mathcal{M}_1 = (M,g,...)$ and $\mathcal{M}_2 = (N,h,...)$ be spacetime models, with carrier sets A and B respectively. Let $\pi : A \to B$ be a bijection such that

(i) $N = \pi_{\ast}M$

(ii) $h = \pi_{\ast}g$

(iii) and $\dots$"

Is this the bit that isn't clear? I.e., that this is equivalent to saying that $\mathcal{M}_1$ and $\mathcal{M}_2$ are isomorphic?

Jeff

2. No. In your blogpost you write "Leibniz Equivalence (LE)
If spacetimes (M,gμν,ψ) and (N,hμν,θ) are isomorphic, then they represent the same physical situation. This is equivalent to Wald's wording, which can be read here."

Wald states that models (M, g, T) and (N, (push*g), (push*T)) are physically equivalent iff there exists a diffeomorphism F: M -> N. However he does not state that N has to be equal to (push*M). Thus it seems that for your version and Wald's to be equivalent you would have to show that M and N are diffeomorphic iff N = (push*N), which again would mean that they are isomorphic.

Mikael

3. Mikael, thanks - I see. So, yes:

$\phi : M \to N$ is a diffeomorphism if and only if $N = \phi_{\ast}M$, if and only if $\phi: M \to N$ is an isomorphism.

(A diffeomorphism is an isomorphism of differentiable manifolds. That is, a bijection between carrier sets which "preserves" the differential structure.)

Cheers,

Jeff