Short question about diffeomorphism invariance

[atyy]

My worry is that t these constructions really do little more than change the identity of the points of the manifold, and that the identity of points in M NEVER mattered, in GR, SR or Newtonian theory.

Yes, I understand similarly. It's not that the manifold doesn't exist, it's that without putting non-dynamical or dynamical fields on it, the points are experimentally identical. Just like electrons - different electrons are identical, but electron number is still something that we can measure experimentally.

And yes, the manifold is needed to define Newtonian physics, SR and GR.

Newtonian physics and SR both have a non-dynamical metric field - this corresponds to matter which does not interact with the dynamical fields of the theory. In Newtonian gravity, the non-dynamical metric could correspond to light rays. In Maxwell's equations on flat spacetime, the dynamical fields would be electromagnetic, while the non-dynamical metric is represented by measuring rods (although we know these ultimately interact with electromagnetic fields, at the everyday level, these are inert, since the charges have all clumped together and neutralized each other at large distance scales). The distinction of GR is that we deal with a field whose coupling is universal, and so it must be dynamical.

 

[marcus]

Oh - then I must have misunderstood the quote from Carroll. When we start with φ, a mapping from M to M, and then we compare (M, gµν , ψ) and (M, φ*gµν , φ*ψ), what is the new metric φ*gµν of this second model but a copying of g to other points of the manifold?

I think Carroll would explain. Don't take my word for it.

I'll tell you how I think of it, which isn't necessarily the way Carroll does, or the way that would be right for you.

I was taught that the differential structure gives you smooth real valued functions f(m) defined on the manifold. And the tangent vectors X at point m are "derivations" defined on the functions, that satisfy a few simple conditions (linearity...). This makes the tangent vector essentially be the operation of taking directional derivative in some direction.

That makes the operation of "pushing forward" very easy to define.
If φ(m) = n and X is a tangent vector at m, then one gets a new tangent vector at n by taking a function f defined around n, and pulling it back by φ and operating by X on it.

(φ*X)f = X(f.φ)

That's a quick way to see how φ maps tangent vectors.

Now a particular tensor might be a bilinear function of two tangent vectors, or it might (to take an even simpler example) be just a linear functional defined on tangent vectors.
You "pull back" an object like that by the diffeomorphism, by pushing forward some tangent vectors for it to eat (in its old location).

It's really not as complicated as it sounds and in some sense all the words I'm saying just obscure the central message, which is that tensors transform as they are pushed around by diffeomorphisms. It is not a simple copying operation.

Someone else may wish to correct me on this or describe this in some other way which they find preferable. I'm an old guy, my math courses were several decades ago. Happy to be exposed to anyone's alternative account.

 

[marcus]

Yossell, if the diffeo is going in the wrong direction for what you want to do, then of course use φ^-1.
And people have different notational conventions. I would be happy to write out, if you would like to see it, how I think you ship a package from one point to another, where the package is a linear functional defined on the tangent space. For example.
But you may have already figured that out, or you may like some different approach to defining the tangent space/bundle. So I'll just wait and see if there are questions.

 

[atyy]

Marcus, btw, I didn't say I disagreed with Rovelli, I just said his explanation is obscure, and I don't agree that he casts any light on the conceptual foundations of GR, and that the passge is poor motivation for LQG (ie. I do find LQG interesting, but not for Rovelli's reasons in fact Rovelli does give good reasons - but they are motivations for Aysmptotic Safety, and maybe string theory - not for LQG - so maybe I like Rovelli's argument after all, since I do like Asymptotic Safety and string theory - the former for its clarity of motivation, the latter for its visionary extension of GR!)

Also, the equation you quoted from Carroll is indeed an Rovelli agrees with - but it is Rovelli's definition of a passive diffeomorphism - which is Carroll's definition of an active coordinate transformation - so I agree with both Rovelli and Carroll that that is not what distinguishes GR from SR. ("Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything." http://relativity.livingreviews.org/Articles/lrr-2008-5/ )

 

[marcus]

My worry is that these constructions really do little more than change the identity of the points of the manifold,..

Yossell, did you get over that worry yet? Diffeomorphisms do radically more than just change the identity of points, since by mooshing the manifold around they also change the metric and transform all the stuff based on the tangent spaces to the manifold at the various points. At least that is how I see it. Do you still see things differently?

BTW I might mention that at least in the Riemannian case you can realize the manifold in a "coordinate free" way as a set M with a distance function d_g(m,n) defined for any two points in M. In pointset topology that would be called a "metric space". You take the Riemannian metric g and use it to find the shortest path distance between any two points, and you record all that d(m,n) information and then throw away the Riemannian metric g.
It is an intuitive way to think about a metric on a manifold. No need to imagine a bundle of tangent spaces---just picture the bare manifold and imagine that you know the distance d(m,n) between any two points.

The essential thing that a diffeomorphism does, in that picture, is that it maps any, say, triple of points into some other triple separated by completely different distances. It completely changes the distances amongst any bunch of points.

 

[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]

... ("Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything." http://relativity.livingreviews.org/Articles/lrr-2008-5/ )

Atyy, where is that in the Rovelli livingreviews article? You are a lot quicker than I am at finding.

 

[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, let's make the coordinate system the Schwarzschild metric for ease. This description is still GR, and it is still evidently invariant under passive diffeomorphisms (coordinate changes from say Schwarzschild 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 don't care about humans interpretations, so long as an answer exists that can be compared with experiment that's 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/ , section 5.3.

 

[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?

 

[Haelfix]

The word "dynamic symmetry" utilized in this way is admittedly oldfashioned, and due to Wigner I believe. Nowdays people just say 'gauge symmetry' where it is understood that general relativity has similar properties.

http://geomsymm.cnsm.csulb.edu/courses/303/reading/wigner.pdf

 

[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]

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.

 

[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]

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?

 

[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.

 

[atyy]

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


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

His point is that people should do Asymptotic Safety - unfortunately not even Rovelli understood his point!

 

[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 possesses 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.

 

[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?

 

[marcus]

...
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.

JD, I'm interested in how you characterize the unique role played by the gravitational field in all of this.
The intuitive idea that the other fields rely on the metric tensor for their diffeomorphism invariance. What you say sounds like it might be a way of expressing what I have in mind.

Diffeomorphism invariance follows when the theory is devoid of background geometrical data...

I agree that if we include the gravitational field and transform everything together then the other fields typically acquire diffeomorphism invariance. And because then the geometry is represented by metric tensor the situation seems, as you put it, "devoid of background geometrical data." But suppose we consider the invariance question for some of these fields without putting g_μν into the picture. That may seem a strange, unmotivated thing to do---leave out the gravitational field. It seems to me that typically (perhaps with some exceptions which you can point out) the other fields fail to have a satisfactory formulation.

Maybe this is even obvious. Imagine using a fixed background metric, one that does not transform under diffeomorphisms, or no metric at all. How to put the other fields into covariant form?

It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields. That in order to transform properly the other fields must "ride" on the gravitational field. Does this make any sense to you?

 

[jdstokes]

Maybe this is even obvious. Imagine using a fixed background metric, one that does not transform under diffeomorphisms, or no metric at all. How to put the other fields into covariant form?

The answer is that you simply can't make the theory generally covariant. The theory will at most be invariant under the symmetries of your (fixed) background metric. Note that this is exactly the situation in Poincare-invariant QFTs, which are studied by particle physicists all the time.

To a particle physicist $\eta_{\mu\nu}$ is considered as a fixed matrix of numbers, not as a tensor field which transforms. The Noether currents associated with Poincare invariance are derived using this fact.

It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields.

I agree with this. Demanding general covariance forces one to couple matter to the metric (this is the definition of general relativity). Einstein's theory of gravity is obtained from general relativity by endowing the metric with its own (diffeomorphism invariant) kinetic term.

 

[marcus]

The gravitational field is the one diff-invariant one--morally :-D

...It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields. That in order to transform properly the other fields must "ride" on the gravitational field. Does this make any sense to you?

...I agree with this. Demanding general covariance forces one to couple matter to the metric (this is the definition of general relativity). Einstein's theory of gravity is obtained from general relativity by endowing the metric with its own (diffeomorphism invariant) kinetic term.

One way to put this--would you agree?--is to say that the other fields acquire their diff-invariance from the gravitational field.

Morally, the metric tensor is the source of diff-invariance, which the other fields get by "riding" on it or being formulated using it. This is just interpretive language, but I think it is in line with the idea of "no prior geometry" (phrase used in the MTW text), independence of geometric background, and what you mean by "devoid of background geometric data."

That may be why many of us, including myself, think of the gravitational field (the spacetime geometry) as playing a unique role.

 

[Finbar]

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.



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.



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?

I think your right. But once you have a manifold M with a metric defined on it you can push this metric forward onto a diffeomorphic manifold N and then the proper distances are preserved. You could of coarse define another metric on the N if you liked.

 

[yossell]

I think your right. But once you have a manifold M with a metric defined on it you can push this metric forward onto a diffeomorphic manifold N and then the proper distances are preserved. You could of coarse define another metric on the N if you liked.

Thanks, Finbar - I see what you mean. As atyy pointed out, there appear to be a lot of different ideas about how exactly the hole construction is to be done, and this may be what's behind so much apparent disagreement and confusion.