Short question about diffeomorphism invariance

[marcus]

There is a clear write up in Wuthrich's thesis, p49. http://philosophy.ucsd.edu/faculty/wuthrich/pub/WuthrichChristianPhD2006Final.pdf

Thanks for finding that source Atyy. I've been busy with other stuff and haven't looked at it.
I did find confirmation in Sean Carroll's thing of what you see in Rovelli. Rovelli's certainly not presenting anything unusual.
In Carroll "Intro to GR" (which seems pitched to undergraduates, style-wise) on page 138 it says:
==quote==
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_µν and matter fields ψ, and φ : M → M is a diffeomorphism, then the sets (M, g_µν , ψ) and
(M, φ* g_µν , φ* ψ) represent the same physical situation.
==endquote==

That is, you can totally moosh and morph and it is still the same physical reality. You aren't limited to an "isometric" diffeo.

In other words physical realities correspond to equivalence classes under diffeomorphism of metrics. They don't correspond to single metrics but to diffeomorphism classes of metrics. All those that you can make by morphing the one you started with.

So there is no rubber sheet. The equiv class carries the more abstract idea of a geometry-without-an-underlying-manifold.

Note that Carroll gives the mathematical truth, but then reassures his undergrads with some condescending pablum which basically says "don't worry your little heads about this". He gives the impression that diffeo is "just a highbrow change of coordinates" the basic message is "I gave you the equation, we aren't going to use it, so no need to think much about it."

Carroll is carrying on the "half century of confusion" that MTW complained about---diffeomorphism invariance going incognito, anonymous (as MTW put it) under the mask of "just like a change of coordinates".

Carroll is a master of comfortable communication---gift of the gab. The important thing is he gives the equation and says the two represent the same physical situation. His spin after that can be ignored.

Above just my humble view of course

“Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität.”

 

[atyy]

Above just my humble view of course

“Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität.”

Yes, I believe Einstein was confused.

 

[marcus]

...What I dislike about Rovelli's work is that he makes simple things so obscure so that he can say LQG is really addressing deep conceptual issues that other quantum gravity people don't.

Atyy try thinking it from the other direction! According to what you say here, you read Hawking Ellis, and Wald, and Carroll and you didn't get the idea! You thought Rovelli was saying something different! Only Rovelli made the idea of diffeo invariance clear enough to get through to you. So you thought he was saying different:

But since GR models physical reality by manifold and metric, only manifolds that are isometric describe the same physical reality (eg. Hawking and Ellis, p56). In the most common sense of diffeomorphism (Hawking and Ellis, Wald, Carroll, but not Rovelli's), two manifolds related by a diffeomorphism do not necessarily represent the same physical spacetime - only manifolds related by isometric diffeomorphisms .

The objective evidence would indicate that you got a wrong understanding from reading H&E and Wald and Carroll. You thought only isometries gave the same physical situation. That means that H&E Wald Carroll expository writing was obscure. It left you with a radically erroneous conception.

Actually Rovelli is a pretty good writer. I think the main reason some people (not necessarily you) find him difficult to read might be that they start with an attitude of disbelief and resentment. If you obstinately doubt everything you read it will make it more difficult to "get it."

For example people coming from just being overwhelmed by string mathematics, much of which depends on postulating a prior geometry---depends on a set geometric background---will naturally be reluctant to accept the idea that nature is not that way.

The disapprobrium seems in part like a classic case of punishing the messenger. The bringer of cognitive dissonance.

 

[MTd2]

Let me see from a spin foam point of view. Minimal size means the term locality is restricted the link be nodes, so a ruler can measure a minimum size between nodes. But this size can be arbitrarily small if inferred in other reference frame, so the set node + link + node is just another example of extended object. Right?

I would really want an answer about this

 

[atyy]

In other words physical realities correspond to equivalence classes under diffeomorphism of metrics. They don't correspond to single metrics but to diffeomorphism classes of metrics. All those that you can make by morphing the one you started with.

This is also true in special relativity.

 

[marcus]

But this size can be arbitrarily small if inferred in other reference frame,

I would really want an answer about this

What does it mean for a length (or an area) to be arbitrarily small if you cannot measure that small?

I must have cited the same 2003 Rovelli paper dozens and dozens of times here at this forum.

The title is something like "Reconciling discrete area spectrum with Lorentz invariance."

You have some physical object that defines an area and you observe the area from a stationary and a moving frame.
The expectation value of the area operator can be made arbitrarily small even though the spectrum (the possible results of any particular measurement) is discrete and has a smallest possible value.

so a ruler can measure a minimum size between nodes. But this size can be arbitrarily small if inferred in other reference frame

No this is not true.
Take the case of area. No matter how fast the other reference frame is going it cannot measure and get a positive area for an answer that is smaller than the minimum eigenvalue of the operator. Sometimes it will get the answer ZERO (so the average or expectation value can be quite small) but it will never measure a positive area smaller than the minimum.

This is the kind of thing you encounter in a quantum theory of geometry. A quantum theory is about measurement and observation. It is not about "what is there" at micro scale. It is about what we can measure and what we can SAY about the micro world. It is about the limitations on the information which we can get.

So I would say simply that you are trying to reason about the length of a link, the separation between two nodes. We don't do that. The question is not well posed.

 

[marcus]

This is also true in special relativity.

? I don't understand. Could you be joking? SR works in Minkowski space which is itself not invariant under diffeomorphism. Diffeos don't preserve flatness. You must have some obscure idea in mind. Please explain.

 

[atyy]

? I don't understand. Could you be joking? SR works in Minkowski space which is itself not invariant under diffeomorphism. Diffeos don't preserve flatness. You must have some obscure idea in mind. Please explain.

I am not joking. If you move the metric with a diffeomorphism, you will preserve flatness.

So SR is invariant under diffeomorphisms that move manifold, metric and matter. Diffeomorphisms that move the metric are isometries.

SR is not invariant under diffeomorphisms that move manifold and matter without moving the metric. The point is that GR is invariant under such diffeomorphisms, because the metric has become matter, and you move it automatically once you move matter ("dynamical fields").

 

[MTd2]

Sometimes it will get the answer ZERO (so the average or expectation value can be quite small) but it will never measure a positive area smaller than the minimum.

I agree with you. The minimum size can be made as small as possible due Lorentz contraction. I don't understand what you understood from me.

BTW, what is a well posed question?

 

[MTd2]

BTW, That thing about E8XE8 inside SO(3,1) doesn`t make sense because that E8 is the dynkin diagram of the group of symmetries of the equation of the ALE space whose the hypersurface correspond to the group of rotations of the icosahedron, which is a finite subgroup of SU(2). So, we would be talking about an internal space of a node, at best. So, forget about this idea.

 

[marcus]

The minimum size can be made as small as possible due Lorentz contraction.

No, that is precisely the point. The spectrum of the area observable cannot be changed by Lorentz contraction. "Minimum size" can only mean the smallest positive eigenvalue. This does not change.

It does not make any sense to refer to an expectation value as a minimum size. There is no minimum positive expectation value.

This is explained simply and clearly in the 2003 Rovelli paper. You would save yourself some time if you looked at it. It is short. I gave a paraphrase of the title earlier, but maybe you were unable to find it. Here is the link and abstract. (It was published 2003 but the arxiv abstract is from the previous year),

http://arxiv.org/abs/gr-qc/0205108
Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
Carlo Rovelli, Simone Speziale
12 pages, 3 figures Physical Review D67 (2003) 064019
(Submitted on 25 May 2002)
"A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area ..."

In LQG context, the minimal area is the minimal positive eigenvalue.
That does not change for the boosted observer.
The boosted observer sees the same minimal length (or minimal area).
What changes is the probability distribution (i.e. if you repeat the experiment many times, the average or expectation value.)

 

[marcus]

I agree with you. The minimum size can be made as small as possible due Lorentz contraction.

I think you did not understand and you did not agree with me when you wrote that.

It is false that (in Lqq context) the minimum size can be made as small as possible due to Lorentz contraction.

I don't understand what you understood from me.

That is correct. I believe I understand what you tried to say. It is what someone unfamiliar with LQG would expect---that boosting would cause the min length to contract. But actually it does not cause the min length to contract.

BTW, what is a well posed question?

I will try to think of one.

 

[marcus]

...
To really understand the invariance under diffeomorphisms (as you, and Rovelli, define it) I would need to see the proof. It is probably very short. Do you know a good reference where it is done clearly?
...

I'll try to sketch a proof. Recall that a diffeo φ: M --> M moves points in the manifold around. A coord change leaves the manifold unaffected and just moves points in R^d..

We already know that the Einstein equation is invariant under coord change. That can just be cranked out. Solutions remain solutions if you just remap the coordinates with a function k: R^d --> R^d.

OK now let's pick a point in the manifold and take a very modest diffeomorphism that slightly moves that point and its immediate neighbors, but doesn't take them out of a particular coordinate chart f: U --> R^d

(See wiki, definition of a manifold, atlas of coordinate charts, as a convenience we stay in one region U so we only need one coordinate function mapping U, into R^d)

Now we can define a "fake coordinate change" function k : R^d --> R^d

k = f(φ(f^-1(x)))

Start with x in R^d
Go with f^-1 up into the open set U in the manifold.
Then move stuff around with phi
Then come back down with f, and you are back in R^d.

Since this is a map from R^d to R^d, it can be treated as a coordinate change. And as usual it preserves solutions to Einstein equation. All coordinate changes do.
But see what the coordinate change does! If you look at how m gets mapped in the new coordinates k(f(m)) it give the same answer as f(φ(m))

kf = fφ

Keeping the points in the manifold the same and using the new coords kf gives the same result as doing the diffeomorphism φ (moving points in the manifold) and using the old coordinate function f.
Now the first of these (changing to new coords kf) preserves solutions, so therefore the diffeomorphism must also preserve solutions.

This is just the sketch of a proof. I think it is how a proof should go, if one were to write out all the greek letters and the arrows.

I think Atyy already found an online source where the proof is presumably written out, not just a sketch.
αβγδεζηθικλμνξοπρσςτυφχψωΓΔΘΛΞΠΣΦΨΩ∏∑

 

[MTd2]

Marcus, I wanted an argument using spin foams. How about this:
matter fields is something that lives on a spin foam like electical fields on a electrical grid structure. Any possible experiment done by whoever lives on that grid will be like obtaing an eigenvalue of a surface operator. But what moves is the matter field, so any measurement of an area will be irrespective of its energy-momentum.

Right?

 

[marcus]

If you look back you see Nrqed started this thread not about QG but about GR. It has always been about GR.

The difference between (1) diffeomorphism and (2)coordinate change leaving the manifold unaffected.
The difference between (1) diffeo invariance and (2) invariance under change of coords.
The fact that in GR a geometry is an equivalence class under diffeomorphism. Not bound up with any particular manifold or any particular metric on that manifold.
The ontological consequence of that:
In GR spacetime does not exist, it has no objective or physical existence. What exists in GR are the relationships among events. The geometry itself--like the smile on the face of the cat after the cat is gone. A web of geometric relationships is information but it is not a thing. No rubber sheet, in other words.

Nrqed was also asking how one can use (2) to prove (1). How coord change invariance can be used to finesse diffeo invariance. I sketched a proof and Atyy may have found an online source. I think it's a fairly trivial thing to show.

Out of respect for the topic, I believe we should not start chatting about QG stuff like spin foams in this thread. If you have an idea about spinfoam models, why not start a separate thread about it?

 

[marcus]

...I wanted an argument using spin foams. How about this:
matter fields is something that lives on a spin foam like electical fields on a electrical grid structure. Any possible experiment done by whoever lives on that grid will be like obtaining an eigenvalue of a surface operator. But what moves is the matter field, so any measurement of an area will be irrespective of its energy-momentum...

Nothing can move on a spin foam. A spin foam depicts a possible course of evolution.
In that very limited sense, is like a trajectory, or a world-line. It IS the motion. So nothing moves on it.

I can see you are pursuing some analogy. But the analogy is not clear yet. It might work better if you were talking about spin networks rather than spinfoam.
Spin networks describe geometry.
Spin foams are somewhat like Feynman diagrams, or the hypothetical paths in a path integral. They are alternative possible histories of geometry, so to speak, not geometry themselves.
They depict various ways that some change in a spin network might have happened.

But I still think that if you want to discuss your idea you should start a separate thread, since it doesn't fit in here (as far as I can tell.)

 

[yossell]

I was interested in the ongoing debate between you and Atyy - did it get resolved?

In Carroll "Intro to GR" (which seems pitched to undergraduates, style-wise) on page 138 it says:
==quote==
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 gµν and matter fields ψ, and φ : M → M is a diffeomorphism, then the sets (M, g_µν , ψ) and
(M, φ* g_µν , φ* ψ) represent the same physical situation.
==endquote==

That is, you can totally moosh and morph and it is still the same physical reality. You aren't limited to an "isometric" diffeo.

In other words physical realities correspond to equivalence classes under diffeomorphism of metrics. They don't correspond to single metrics but to diffeomorphism classes of metrics. All those that you can make by morphing the one you started with.

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. On the qualitative questions that physically matter, whether a manifold is flat, curved, whether the path an object travels is inertial, the constructions make no difference. Although φ is not an isometry, it doesn't change the fact that the two models constructed above are isometric - that there is some function φ' that is an isometry.

 

[marcus]

“Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität.”

Yes, I believe Einstein was confused.

I was interested in the ongoing debate between you and Atyy - did it get resolved?...

It is kind of you to ask. But for me the wonderful thing about talking with Atyy is the stimulating unresolution. The brilliant chimaera. The changeling aspect. We never quite agree but he forces me to think.

The direct answer to your question is "no". I'm happy with that and hope to hear more.

 

[marcus]

...
I did find confirmation in Sean Carroll's thing of what you see in Rovelli. Rovelli's certainly not presenting anything unusual.
In Carroll "Intro to GR" (which seems pitched to undergraduates, style-wise) on page 138 it says:
==quote==
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_µν and matter fields ψ, and φ : M → M is a diffeomorphism, then the sets (M, g_µν , ψ) and
(M, φ* g_µν , φ* ψ) represent the same physical situation.
==endquote==

That is, you can totally moosh and morph and it is still the same physical reality. You aren't limited to an "isometric" diffeo.

In other words physical realities correspond to equivalence classes under diffeomorphism of metrics. They don't correspond to single metrics but to diffeomorphism classes of metrics. All those that you can make by morphing the one you started with...

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. On the qualitative questions that physically matter, whether a manifold is flat, curved, whether the path an object travels is inertial, the constructions make no difference. Although φ is not an isometry, it doesn't change the fact that the two models constructed above are isometric - that there is some function φ' that is an isometry.

I think φ* has a substantial effect on the metric tensor. It does not simply pick g_µν up and copy it to another point of the manifold. I think we probably agree on that, so let me think about your function φ'.

I don't see how I would construct φ', Yossell.

 

[yossell]

I think φ* has a substantial effect on the metric tensor. It does not simply pick g_µν up and copy it to another point of the manifold.

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?

 

[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?