# Short question about diffeomorphism invariance

#### yossell

Yossell, did you get over that worry yet?
Thanks Marcus - I'm going to spend some time thinking about Carroll's constructions and what you say. I know that diffeomorphisms *can* moosh things up, my worry is what happens when the metric field itself is also dragged around in the creation of the new model. I'll get back when I've got something new to say.

Best

#### atyy

One note of caution. Marcus is talking about diffeomorphisms on pseudo-Riemannian manifolds, where diffeomorphisms are not isomorphisms - only isometries are.

I am talking about diffeomorphisms on smooth manifolds, where they are isomorphisms.

An isometry is a mapping where you move the points on the manifold with a diffeomorphism, and then also move the metric with the pullback of the diffeomorphism.

#### marcus

Gold Member
Dearly Missed
... ("Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything." http://relativity.livingreviews.org/Articles/lrr-2008-5/ [Broken])
Atyy, where is that in the Rovelli livingreviews article? You are a lot quicker than I am at finding.

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#### Haelfix

At the beginning of the thread, I warned that this was going to become a semantic war. This particular question always does. Its happened to several generations of physicsists, and it happens to every single grad student i've ever known (including yours truly once upon a time). Not surprisingly either, consider the large amount of textbooks on the subject, each with different notation (in some cases sloppy) and different interpretations of the math. Obviously, it will be even more difficult when restricted to an internet forum.

The fundamental problem is that you can always obscure what is or is not dynamical/absolute or fixed in a theory, simply by performing a gauge transformation and/or field redefinitions with constraints (about which more later). Likewise, symmetries are not always manifest. You really need to perform a Hamiltonian analysis to disentangle what is what (and even then it can be tricky). The one thing that is true I think, is that under most definitions that I know off, general covariance in physics is an examples of a dynamic symmetry and not an invariance principles. The former is a form of redundancy of description, the latter constrains what terms can or cannot be written by the laws of physics under certain transformations.

Now, it has been said that GR is the only theory that respects active coordinate transformations (active diffeos for short) by Rovelli. Well, you can readily find a definition that makes the above statement true alternatively you can follow Carrol and find a definition that makes it false, however beware interpreting this too far one way or the other as the following illustration shows:

Consider fixing a coordinate system in GR. This explicitly breaks an infinite amount of mathematical diffeomorphisms and leaves only a small finite amount of them unbroken (these we call the isometries of the system). Incidentally the global symmetries of the theory are the diffeomorphisms that do not go to the identity at infinity but preserve some sort of asymptotic boundary data

But anyway, lets make the coordinate system the Schwarschild metric for ease. This description is still GR, and it is still evidently invariant under passive diffeomorphisms (coordinate changes from say Schwarschild coordinates to Kruskal coordinates) but the system is no longer acted on nontrivially by the full DIFF(M) group, instead by just a small subgroup thereof. This scenario is qualitatively similar to anything that might happen in say Newtonian physics or SR, where only a subgroup of the full Diff(M) group corresponds to active transformations (for instance: galilean translations in the case where the manifold is R^3 with +++ signature).

I wrote the above in a language for emphasis of the similarity with gauge transformations. Indeed, this is completely analogous to what happens in gauge fixing in Yang Mills theories. However there we do not say (as a matter of language) that QED, written in Lorentz gauge, is not gauge invariant. Instead we might say the U(1) symmetry is no longer manifest.

So the point is that while gravity, and only gravity has as its core dynamical symmetry (in some suitable formalism) the *full* diffeomorphism group (where it acts on all objects of the theory) you can always write it in a physical form which is qualitatively similar to any other theory. Likewise, you can make any other theory, look like GR by suitably geometrizing it (see Newton-Cartan gravity in MTW) except that you will discover that various d.o.f are actually only acted on nontrivially by a much smaller subgroup upon closer inspection.

Further, like gauge invariance, general covariance (at least in the sense of a infinitesimal pushforward operation alla Carrol) is still just merely a redundancy of description. The physical content is identical to the gauge fixed or coordinate fixed description.

#### Haelfix

As a different post, I thought i'd point out a simple example of how one can very easily confuse a theory with an absolute fixed object vs one that is free to vary.

The first thing you might try to do is to take the variation of every tensor or differential form in the theory (action), and see how it behaves. The reasoning being that only a dynamical object produces something nontrivial.

Now consider a complicated action that also possessed a fixed field g. Define two new complex fields phi and phibar, such that g = (phi+phibar)/2 and rewrite your action in terms of these new fields.

You now have an intepretational problem now, since the new fields phi will have components that do not necessarily vanish under variation and you might mistakenly think the phantom complex components have now suddenly changed your one fixed object into two fully dynamical ones. Only solving for the eom will show that in fact you're degrees of freedom were not quite independant and that the constraint ends up eating the fictitious d.o.f.

Well, something qualitatively similar but more complicated happens with gravity in the Hamiltonian formulation. There you end up with the at first glance bizarre statement that the dynamics are identically zero, or alternatively that everything seems frozen. However as everyone knows the resolution is simple, the dynamics were not really zero after all, instead they were merely hiding in the constraints (called the Hamiltonian constraint).

The heurestic point I wish to make (without going into pages of explicit math showing loads of examples of this explicitly --many textbooks do this better), is that in theoretical physics over and over again, you will find situations where various symmetries or truisms about a set of equations are hidden or not manifest. However, the physical content or observables do not care which form you write it in, they dont care about humans interpretations, so long as an answer exists that can be compared with experiment thats all that really matters.

#### atyy

Atyy, where is that in the Rovelli livingreviews article? You are a lot quicker than I am at finding.
http://relativity.livingreviews.org/Articles/lrr-2008-5/ [Broken] , section 5.3.

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#### atyy

The one thing that is true I think, is that under most definitions that I know off, general covariance in physics is an examples of a dynamic symmetry and not an invariance principles.
Thanks for your long write ups in #79 & #80!

What is a dynamic symmetry? In Weinberg's gravity text, he has a "Principle of General Covariance", which he distinguishes from general covariance, says it's a dynamical symmetry, and is equivalent to the principle of equivalence. I do recognize his PGC to be what everyone else says is the EP, and which in my understanding is really the hypothesis of minimal coupling - same as using the so-called "gauge principle" to get minimal coupling between the electromagnetic and electron fields. Is a dynamical symmetry another name for minimal coupling, or is it a different principle?

#### Finbar

MTW's book "Gravitation" makes an excellent observation which amplifies the point Rovelli is making. Diffeomorphism invariance means something different from what people normally understand by general covariance.
Einstein and others saying "g.c." when they meant "d.i." led to a half century of confusion, say from 1915 to 1975.

When you say g.c. people think of coordinate change, which does not change the distances between points or anything essential about the geometry of the manifold itself.
Other physics besides GR can be formulated in a way that allows coordinate change.

A diffeomorphism φ: M --> N is just an invertible smooth map, which
can be between two different manifolds with completely different shape/curvature and it can completely change the distances. Points m m' can get mapped to points n n' with different distance from each other.

The case we are considering here is φ: M --> M, and the same thing is true.
Points m m' m" which are certain distances apart can get mapped to points
φ(m) φ(m') φ(m") which are completely different distances apart.

Whatever physical theory you want to talk about, Newton, Maxwell, QED, QCD it is not diffeomorphism invariant.

But GR is. And people seem to have been slow to realize this.

This is the "anonymity" that caused the "half century of confusion" that MTW talk about. Einstein did not come out and say "diffeomorphism invariance", but he in effect concealed the radicalism of the concept by calling it "covariance". So people kept thinking it was mere coordinate change stuff.

==MTW quote==
Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.
==endquote==

αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑

I'm sorry Marcus but what you say here is not true. Diffeomorphisms do not change the proper distances between events or curvature invariants. Newton, Maxwell, QED, QCD can all be formulated in a diffeomorphism invariant way.

Manifolds that are diffeomorphic are the same manifold. You seem to be claiming that there
exist diffeomorphisms which cannot be preformed by making a coordinate change.

The confusion that MTW talk about is your confusion. There's no physics in diffeomorphism invariance. Its the fact that the space-time geometry is dynamical that sets GR apart.

#### marcus

Gold Member
Dearly Missed
I'll think about it. I could be mistaken. In any case thanks for the comment!

What do you mean by the "proper distance between events". What is an event?

Maybe you can tell me a bit about it. We have a smooth manifold M. Say it has a metric. (We aren't doing GR necessarily, just diff. geom.)

You apply a diffeomorphism just to the manifold and not to the metric, does it change "proper distances between events?"

I'd like to look at a few examples with you, vary the assumptions, and understand better.

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#### Finbar

$$ds^2 = g_{\mu \nu} dx^{\mu}dx^{\nu}$$

is invariant under diffeomorphisms

$$P_c=\int_c ds$$

over some curve c is a proper distance.

By an event I mean the set of points, one in each diffeomorphic manifold, that are mapped to each over.

I really think your confused if you think that diffeomorphisms really change the underlying space-time manifold.

From Carrol p. 429
If $$\phi$$ is invertible (and both $$\phi$$ and $$\phi^{-1}$$ are smooth, which we always implicitly assume), then it defines a diffeomorphism between
M and N. This can only be the case if M and N are actually the same abstract manifold; indeed the existence of a diffeomorphism is the definition of two manifolds being the same.

#### Finbar

I don't understand how you can apply a diffeomorphism on the manifold but not the metric?

If i have a diffeomorphism then this defines a pullback for the metric as well.

I use this metric and then proper lengths are invariant.

#### marcus

Gold Member
Dearly Missed
I don't understand how you can apply a diffeomorphism on the manifold but not the metric?

If i have a diffeomorphism then this defines a pullback for the metric as well.

I use this metric and then proper lengths are invariant.
I agree that you can define a pullback for the metric for the metric as well!

One can leave the same metric in place, and stir the points around with the diffeomorphism.
(In which case distances change. That may not seem motivated to you but I believe it is an option mathematically. )

OR one can map the points to new location AND transform the metric---do the pullback.

After your explanation I can convince myself that proper distances as defined here are unchanged in that case. Thanks for discussing this!

It seems to me that I have now agreed with you that GR is diffeo invariant. If you transform the metric (and move the matter accordingly of course) then nothing changes. GR is a theory of the metric. I'm convinced that it behaves right.

I still cannot convince myself (what a number of people have been saying) that all the other physical theories (Newton Maxwell QED...) are ALSO diffeo invariant in the same sense---on their own, so to speak, without the help of GR. Is this true. Does it make sense?

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#### Finbar

I think that the whole confusion comes from gauge fixing before you solve the Einstein equations. I've been reading

http://arxiv.org/pdf/gr-qc/9910079v2

I understand what he's saying but I don't really see the point.

The idea is that you can have two metrics g(x) and g'(x) that both solve the einstein equations. Then say that ds^2(x) is now different if you use either metric. But this is because actually the coordinate x refers to a different point on the same manifold depending on which metric you use. But if you find the diffeomorphism that relates g to g' you can then find the coordinate transform x --> y such that you can relate the points. At which point you see that ds^2 is the same.

In the end though the difference between active and passive diffeomorphisms is just down to interpretation. I think I now see that this interpretation only makes sense when you have to solve equations of motion to find the metric. But this is just pointing out in a rather confusing way how important diffeomorphisms are in GR.

What is wrong is to say that GR is the only diffeomorphism invariant theory. Diffeomorphism has a strict mathematical definition.

Im happier with using "background independent" instead of trying to twist "general covariance" or "diffeomorphism invariance" so that they mean something they do not.

#### Finbar

I still cannot convince myself (what a number of people have been saying) that all the other physical theories (Newton Maxwell QED...) are ALSO diffeo invariant in the same sense---on their own, so to speak, without the help of GR. Is this true. Does it make sense?
Well I can have QED live on a curved manifold with a fixed metric in some coordinate system.
I can write this theory in a generally covariant. Then I can preform a diffeomorphism and show that the action is invariant under these transformations. The gauge fields would then have to transform as vectors. I see no reason that the logic is any different. I can still find two different solutions to maxwells equations A'(x) and A(x) related by a diffeomorphism.

#### Finbar

His point is that people should do Asymptotic Safety - unfortunately not even Rovelli understood his point!
Why? I don't see the connection to Asymptotic Safety.

#### atyy

One can leave the same metric in place, and stir the points around with the diffeomorphism.
(In which case distances change. That may not seem motivated to you but I believe it is an option mathematically. )

OR one can map the points to new location AND transform the metric---do the pullback.
This is exactly the distinction I was making between a pure diffeomorphism and a diffeomorphism plus a pullback which is an isometry.

#### atyy

Why? I don't see the connection to Asymptotic Safety.
Ha, ha - just half kidding. I had in mind that all theories are invariant if you use a diffeomorphism to move everything about. GR is distinguished by being invariant if you use a diffeomorphism to move only the fields which are varied in the action (and the assumption of 4D). If we consider all theories in this class, we get the most general generally covariant Lagrangian, which is the starting point of AS.

#### yossell

I'd like to check my understanding in general, and in particular, whether I understand the terms being used in the same way as others.

Diffeomorphisms do not change the proper distances between events or curvature invariants.
In general, this is false. All that's required from a diffeomorphism is that it be smooth. There's nothing in the definition that requires that the proper distance between two points be preserved.

Manifolds that are diffeomorphic are the same manifold. You seem to be claiming that there
exist diffeomorphisms which cannot be preformed by making a coordinate change.
It's true that manifolds that are diffeomorphic are (essentially) the same manifold. But that's because a manifold has no metric structure defined on it. So understood, a manifold isn't yet anything like a space-time with a `shape'.

Is this Right? Wrong? Not even wrong?

#### atyy

I'd like to check my understanding in general, and in particular, whether I understand the terms being used in the same way as others.
You are right, but so is Finbar.

Map between smooth manifolds only
maths: diffeomorphism

Map between smooth manifold plus pullback of fields including metric
maths, Hawking and Ellis: isometry
Carroll, Wald: diffeomorphism or active coordinate transformation
Rovelli: passive diffeomorphism
Giulini: diffeomorphism covariance

Map between Riemannian manifolds plus pullback of dynamical fields excluding metric, unless the metric is a dynamical field
Carroll, Wald, MTW: no prior geometry
Rovelli: active diffeomorphism
Giulini: diffeomorphism invariance

#### jdstokes

Whatever physical theory you want to talk about, Newton, Maxwell, QED, QCD it is not diffeomorphism invariant.
This is wrong.

The first point to understand is that any physical theory can be written in a Lorentz invariant form. This includes all of the theories you mention above. I should point out that Newton is slightly different from the other theories in that it is not manifestly Lorentz invariant, This problem can be overcome, however, simply by introducing a preferred timelike direction. The underlying theory is still Lorentz invariant.

As is well known, any Lorentz-invariant theory can be given a coordiante-invariant formulation using the minimal substitution prescription; that is, replace the fixed matrices $\eta_{\mu\nu}$ by the metric tensor field $g_{\mu\nu} = g_{\mu\nu}(x)$, partial derivatives by covariant derivatives etc...

Diffeomorphism invariance is a property of a coordinate-invariant theory which does not possess any background geometrical data. This is simply objects which do not obey field equations of motion (such as the preferred timelike direction in Newton). The metric does not fall into this category, however.

Therefore, Maxwell, QED, QCD are perfectly good diffeomorphism-invariant field theories, by virture of their Lorentz invariance, and absence of background geometrical data.

This whole topic of coordinate invariance versus diffeomorphism invariance is notoriously poorly explained in the literature. In fact, I am not aware of a single reference which explains it to my satisfaction.

#### yossell

Wow - thanks atyy, that's a very helpful and comprehensive list which will be my desktop background for a few days. But it does show that we've all got to be careful before we disagree with each other and we may simply be talking past each other, operating with different definitions.

And I was told this kind of thing only happened in philowsowphicawl circles

#### jdstokes

I'm sorry Marcus but what you say here is not true. Newton, Maxwell, QED, QCD can all be formulated in a diffeomorphism invariant way.
Diffeomorphism invariance follows whenthe theory is devoid of background geometrical data. For this reason Newton is not diffeomorphism invariant.

What you mean to say is that all of the above theories can be formulated in generally covariant fashion. This is indeed correct. In the case of Newton, however, this covariantization comes at the expense of introducing a preferred timelike unit vector. Since this vector does not satisfy the field equations, and has to be inserted ad hoc, the theory is not diffeomorphism invariant.

All of this can be proven rigorously using the action formulation.

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#### atyy

And I was told this kind of thing only happened in philowsowphicawl circles
Now, should we discuss whether the Higss boson is due to spontaneous gauge symmetry breaking? :rofl:

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