# Short question about diffeomorphism invariance

#### nrqed

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I am posting my question in this forum because it is about a basic conceptual aspect of LQG discussed in Rovelli's book Quantum Gravity.
He makes the following statement on page 67 (here, "e" refers to the vierbein):

Now, if $e$ is a solution of the equations of motion, and if the equations of motion are generally covariant, then $\tilde{e}$ is also a solution of the equations of motion. This is because of the relation between active diffeomorphisms and changes of coordinates: we can always find two different coordinate systems on M, say x and y, such that the function $e^I_{\mu} (x)$ that represents $e$ in the coordinate system x is the same function as the function $\tilde{e}^I{\mu}(y)$ in the coordinate system y. Since the equations of motion are in the same in the two coordinates, the fact that this function satisfies The Einstein equations implies that $e$ as well as $\tilde{e}$ are physical solutions.

I do not understand the part in boldface. First, he means that the *functional form* of $e$ and $\tilde{e}$ is the same, when he says that the two functions are equal, right? (which is different from saying $e(x) = \tilde{e}(y(x))$).

If that's the case, then I don't follow the logic of the argument. First, I don't see in what way the relation with active diffeomorphisms plays a role....is he assuming that the theory is invariant under active diffeomorphisms? It seems to me that one only needs to use the freedom to make changes of coordinates to obtain the result.

A second question is:if we had a scalar function f instead of a one-form like e, then it seemes to me that we could not make the argument that we can always find a different coordinate system such that f and f' can be made equal. Am I missing something?

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#### marcus

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A passive diffeo (or change of coords) is a map Rd --> Rd
(see top of page 64 for additional details, I won't give the full definitions)

An active diffeo is a map M --> M (other details at bottom of page 63)

It may seem strange to you that he should be so careful about explaining the fact that

invariance under change of coords implies invariance under active diffeo. But there is something to prove because they are different animals.

So you might get confused just because he is taking his time and going slow.

It is the invariance under (active) diffeo that eventually (on page 68) implies that points in spacetime have no meaning. Because active diffeos stir and smoosh the whole manifold M around and can take point A to point B. You have looked at the "Hole Argument" on page 68? There's that famous Einstein quote from 1916 that the principle of general covariance deprives time and space of the last remnant of physical reality.

So coordinate change (which doesn't move points of the manifold M) looks harmless.
But what Rovelli is explaining, where you asked about it, is that you cannot buy coordinate change invariance without also getting full diffeomorphism invariance.

Formally the two kinds of maps look different. One is M --> M and the other is Rd --> Rd. So there is something to prove, even though it may be intuitively obvious to you.

Should we go over the argument? Paraphrase it? Maybe some other poster will step in. Otherwise I will tomorrow (bed-time now). Or maybe it is clear already nrqed?

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#### nrqed

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A passive diffeo (or change of coords) is a map Rd --> Rd
(see top of page 64 for additional details, I won't give the full definitions)

An active diffeo is a map M --> M (other details at bottom of page 63)

It may seem strange to you that he should be so careful about explaining the fact that

invariance under change of coords implies invariance under active diffeo. But there is something to prove because they are different animals.

So you might get confused just because he is taking his time and going slow.

It is the invariance under (active) diffeo that eventually (on page 68) implies that points in spacetime have no meaning. Because active diffeos stir and smoosh the whole manifold M around and can take point A to point B. You have looked at the "Hole Argument" on page 68? There's that famous Einstein quote from 1916 that the principle of general covariance deprives time and space of the last remnant of physical reality.

So coordinate change (which doesn't move points of the manifold M) looks harmless.
But what Rovelli is explaining, where you asked about it, is that you cannot buy coordinate change invariance without also getting full diffeomorphism invariance.

Formally the two kinds of maps look different. One is M --> M and the other is Rd --> Rd. So there is something to prove, even though it may be intuitively obvious to you.

Should we go over the argument? Paraphrase it? Maybe some other poster will step in. Otherwise I will tomorrow (bed-time now). Or maybe it is clear already nrqed?
Thank you Marcus for your time. It is appreciated.

I do understand the overall idea, which you summarized nicely.
As you said they are different animals so indeed there is something to prove. It is the proof that gives me trouble. The cornerstone of the proof reside in the step that I boldfaced in my OP. It is that sentence that I am trying to understand. I guess I have two questions:

By saying ''the function e and $\tilde{e}$ are the same", what does he mean? I think he means that the functional forms are the same (and not the value evaluated at the same point is the same)

How does one see that one can always have two different coordinate systems that give
$e = \tilde{e}$ ?

Thanks again

#### marcus

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Let's see if we can prove it. You can check me to see if my notation is understandable and whether I have it right or not. We may want to make an additional assumption like the active diffeo stays within the coordinate patch, just for convenience. Let the manifold be M and let systems of coordinates be written with letters like x or y

Here are two different systems of coordinates
x: M --> Rd
y: M --> Rd

We have a diffeo f: M --> M
and we want to find a change of coordinates, from x to y, that will undo the effect of the diffeo on what you call the functional form of e(m)

In x coords, the functional form of e(m) is e(x-1(s))
You take a point s in Rd and map it up to point m in the manifold and then find e(m)

The diffeo has this effect, e(f(m)). You moosh with the diffeo first and then do e.

What we want is a new coord system y such that the functional form of e(f(m)) is the same as the old functional form.

e(f(y-1(s)) should = e(x-1(s)) for any point s in some appropriate neighborhood in Rd

I think that means that given x and the diffeo f, we have to find coords y such that
f(y-1(s)) = x-1(s)

This is just half done. I have to go temporarily back later.

#### nrqed

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Let's see if we can prove it. You can check me to see if my notation is understandable and whether I have it right or not. We may want to make an additional assumption like the active diffeo stays within the coordinate patch, just for convenience. Let the manifold be M and let systems of coordinates be written with letters like x or y

Here are two different systems of coordinates
x: M --> Rd
y: M --> Rd

We have a diffeo f: M --> M
and we want to find a change of coordinates, from x to y, that will undo the effect of the diffeo on what you call the functional form of e(m)

In x coords, the functional form of e(m) is e(x-1(s))
You take a point s in Rd and map it up to point m in the manifold and then find e(m)
Hi Marcus,

Thanks for your help. I understand your approach, it fits well with my understanding of the problem. But there is one point that I need to clarify. It seems to me that when Rovelli is talking about the functional form of the function e(x), he is talking about something that maps a point in Rd to the space of one-forms. So in your notation, we should be talking about e(s) and not e(m). Do you see what I mean?

#### atyy

I don't understand Rovelli's general argument, but the particular part you mention is also in Wald, beginning on the bottom of p438.

#### nrqed

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I don't understand Rovelli's general argument, but the particular part you mention is also in Wald, beginning on the bottom of p438.
Thank you! I will read that.

#### nrqed

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Also Carroll's notes http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll5.html , the part beginning at "We are now in a position to explain the relationship between diffeomorphisms and coordinate transformations." (I haven't read this part carefully, but I quick glance seems to show it's the same argument as Wald's)
Thanks again! Sounds well explained.

I am still interested in understanding Rovelli's argument.

Thanks,

Patrick

#### marcus

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whew! Atyy, thanks so much for getting me off the hook! I had to be out for much of today and got distracted. Very glad you took over.
AFAICS Rovelli does not prove the relation between diffeo and coord transformation. That is more appropriate to a GR textbook. He motivates it with an example (French and English coordinates for a map of air temperature) and invokes it, leading up to the "Hole argument" on page 68.

The whole thing is motivational and introductory, giving perspective. If you buy that gravity theory is invariant under coord change then you buy that it is diffeo invariant. Then (what he really wants to show) if it is diffeo invariant then the points of spacetime have no physical meaning---no objective reality. Something Einstein pointed out in 1916, quotes from a paper and a letter. So then we need a new formulation of physics in which there is no underlying continuum for fields to be defined on.

The discussion is driving towards the two short paragraphs on page 75 that conclude that section of the book.
If you think of what is talked about on pages 65-69 as leading up to that, it gets clearer I think.

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#### atyy

I am still interested in understanding Rovelli's argument.
With respect to Rovelli's argument (which I don't understand), there are some other sources which deal with the issue, which I have found helpful.

One is MTW's statement that although general covariance was a founding principle of GR, it is in fact not, and the founding principle is "no prior geometry".

The distinction between general covariance and "no prior geometry" is found in Mattingly's http://relativity.livingreviews.org/Articles/lrr-2005-5/ [Broken]: "There are three general principles in general relativity relevant to Lorentz violation: general covariance (which implies both passive and active diffeomorphism invariance ), the equivalence principle, and lack of prior geometry. As we saw in Section 2, general covariance is automatically a property of an appropriately formulated Lorentz violating theory, even in flat space. The fate of the equivalence principle we deal with below in Section 2.5. The last principle, lack of prior geometry, is simply a statement that the metric is a dynamical object on the same level as any other field."

An extensive discussion is given in Giulini's http://arxiv.org/abs/gr-qc/0603087.

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#### nrqed

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With respect to Rovelli's argument (which I don't understand), there are some other sources which deal with the issue, which I have found helpful.

One is MTW's statement that although general covariance was a founding principle of GR, it is in fact not, and the founding principle is "no prior geometry".

The distinction between general covariance and "no prior geometry" is found in Mattingly's http://relativity.livingreviews.org/Articles/lrr-2005-5/ [Broken]: "There are three general principles in general relativity relevant to Lorentz violation: general covariance (which implies both passive and active diffeomorphism invariance ), the equivalence principle, and lack of prior geometry. As we saw in Section 2, general covariance is automatically a property of an appropriately formulated Lorentz violating theory, even in flat space. The fate of the equivalence principle we deal with below in Section 2.5. The last principle, lack of prior geometry, is simply a statement that the metric is a dynamical object on the same level as any other field."

An extensive discussion is given in Giulini's http://arxiv.org/abs/gr-qc/0603087.
Thanks again Atyy for the very interesting references!

Thanks to you and Marcus for the exchange. I figured out what was bugging me.
It was not the relation between coordinate changes and active diffeomorphisms, it was the meaning of general covariance. Now it's clear (I think!)

Thanks!

Patrick

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#### nrqed

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EDIT: For some reason, when I put a tilde on the new function T, it did not show up. SO I used T' instead.

Unfortunately, I now realize that there is still one point that is unclear to me.
However, I have understood enough to make my question well-defined, so hopefully it will be much easier to answer it.

My confusion comes from the following point. A coordinate system is a map from M to Rd. A scalar function T is a map from M to R. So they are not, essentially, different objects. My problem comes from the fact that they are treated differently under an active diffeomorphism.

What I mean is this: Rovelli states clearly that under an active diffeomorphism, the function T(P) changes (where P is a point in the manifold). Therefore, if we see an active diffeomorphism as dragging the points of the manifold to new positions, the scalar function does not "follow", it stays where it initially was, so that we get a new function $T'(P)$. (of course, in his example of winds and temperature, he does not drag the manifold, he drags the temperature field instead so it is the inverse to what I just described but the key point is the same: the points in the manifodl get moved relative to the scalar function).

Now I can state the source of my confusion. Consider now including a coordinate system on the manifold, that is a map X: $$M \rightarrow R^d$$, before applying the active diffeomorphism. We can now introduce a temperature function t(x) that maps Rd to R.

Ok, now we apply the active diffeomorphism. The scalar function T(P) moves relative to the manifold, as we said before. The question is: what happens to the coordinate system X(P)?

a) At first, it seems as if it should behave the same way as the scalar function T(P).
However, if this was the case, the induced function t(x) would not change of functional form!!

b) The coordinate system is dragged along the manifold. In other words, each point P remains
asisgned to the same coordinate $x^\mu$. In that case, the scalar function
t(x) will change of functional form since a given coordinate is no longer assign to the same
point in the function. To be more mathematical, what can do here is to assign a coordinate system
to th emanifold *after* the active diffeomorphism and then we use the pullback of this mapping to
the original manifold (before the active diffeomorphism) to get a coordinate system on the original
manifold. In this way any given point is asisgned the same coordinates before and after the diffeomorphism.

It is clear that to get equivalence between active diffeomorphisms and changes of coordinates, we must adopt
the second point of view. But then my question is: why do we treat differently the coordinate system X, which is just
a mapping from the manifold to Rd and the saclar function T, which is a mapping from the manifold
to R? Why the different rules for these two mappings?

I hope my question is clearer now.

Thanks!

Patrick

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#### Finbar

I fail to see the physical significance of general covariance or diffeomorphisms. Newtonian physics can be expressed as a diffeomorphic invariant theory. What is physically interesting about GR is that there is no a priori geometry; that the geometry is dynamical.

I'm some what confused by the whole active/passive diffeomorphism thing. What exactly is the difference?

#### atyy

I fail to see the physical significance of general covariance or diffeomorphisms. Newtonian physics can be expressed as a diffeomorphic invariant theory. What is physically interesting about GR is that there is no a priori geometry; that the geometry is dynamical.

I'm some what confused by the whole active/passive diffeomorphism thing. What exactly is the difference?
How about as applied to the field in the action (rather than the equations of motion written in generally covariant form) - so Maxwell's equations would not come from a genrally covariant action, whereas the Einstein equations do?

#### atyy

I hope my question is clearer now.
I fail to see the physical significance of general covariance or diffeomorphisms. Newtonian physics can be expressed as a diffeomorphic invariant theory. What is physically interesting about GR is that there is no a priori geometry; that the geometry is dynamical.

I'm some what confused by the whole active/passive diffeomorphism thing. What exactly is the difference?
Is Rovelli's point really that active diff = passive diff? I think he actually wants to say active diff is *not* equivalent to passive diff, and GR is invariant under active and passive diff, whereas other theories are invariant only under passive diff.

I believe he is using the term "active diff" *differently* from Wald and Caroll - whose point is simply that you can use a diffeomorphism to change coordinates. Rovelli's point is really "no prior geometry", and his definition of an active diff is a diff on the "dynamical fields" only, whereas Wald and Carroll's active diff is a diff on everything that is geometrical including the metric, even if it is not dynamical. In SR, a Rovelli "active diff" will move everything except the metric, since the metric is not dynamical, so the physics will be changed, since the metric is an essential part of the theory, just that it's not dynamical. In GR, a Rovelli "active diff" will move everything including the metric, since the metric is dynamical, and so the physics will not change. (In short, I agree with Finbar.)

I think Rovelli said it much more clearly here (maybe he read MTW or Finbar between 2003 and 2008 ): "Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields. (Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything.)" http://relativity.livingreviews.org/Articles/lrr-2008-5/ [Broken]

So why is Rovelli using such confusing terms (apart from being confused - he really wants to do Asymptotic Safety, not LQG). I think it may be that although general covariance of the dynamical equations is not a principle solely of GR (since all theories can be written in generally covariant form, even SR and Newton) - the general covariance of the action is a distinguishing feature of GR (this is one the key ingredients in the definition of Asymptotic Safety).

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#### nrqed

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Is Rovelli's point really that active diff = passive diff? I think he actually wants to say active diff is *not* equivalent to passive diff, and GR is invariant under active and passive diff, whereas other theories are invariant only under passive diff.
My interpretation of what he is saying in his book is that active diffeomorphism *is* equivalent to passive diff. That, in itself, does not say anything special about GR or physics.
*Then*, the fact that Einstein's equations are generally covariant (that's the physical input)
implies that they are invariant under passive diffeomorphisms. Hence they are invariant under active diffeomorphisms, hence the points in the manifold have no physical meaning. That's my understanding of his explanations.

So why is Rovelli using such confusing terms (apart from being confused - he really wants to do Asymptotic Safety, not LQG). I think it may be that although general covariance of the dynamical equations is not a principle solely of GR (since all theories can be written in generally covariant form, even SR and Newton)
Really? If we write Newton's equations in a given coordinate system and then do a change of coordinates, the functional form will no longer be the same. I mean, that's why we have things like the Coriolis force in a rotating frame. Maybe I am missing something!

#### Fra

I find this discussion closely related to https://www.physicsforums.com/showthread.php?t=418467.

Loosely speaking, to me the conceptual meaning of ANY so called "passive transformation" be it poincare, diff or anything else, is in the context of a given observer simply a "relabelling" of events. So if you acknowledge that that the labes(coordinates, event index etc) are just for bookkeeping and that the choice of label has no physical significance, since type of invariance is somehow obvious.

What I associate with "active transformations" rather means, conceptually that you are transforming the observer (or the event manifold (and not just the lables).

The connection is that mathematically the same transformations defines the set of coordiante systems, as defines the set of observer.

IMO, the "physics" lies in the statement that the set of possible observers are generated exactly by the same mathematical transformation that relabels the events (the passive ones).

I think this statement can be discussed and there are issues with it, because from the inference perspective, one certainly wonder where is the information about these transformations stored, and what physical process allows it's inference?

I think this question is ultimately related to what's measurable and not, and who is measuring it. This is less problemativ in classic GR, but I think it's nontrivial when you take measurements and information also about transformations more serious than Einstein did in the "classical world".

I definitely think that one should make a clear distinction between passive and active transformations, beucase it's not clear to me that the coincidental mathematical similarity is quite correct and will survive the transition to QG.

Edit: the point one is tempted to make is that the "choice of observer" is also just a relabelling in the extended sense. But there is a problem with that, in the measurement perspective, since the observer is central. The vision of an observer independent transformation that fomr a birds perspective generates all possible observers, is not unproblematic. So the choice of observer in a seriously constructed intrinsic measurement theory, can IMHO not quite be put on par with the "choice of event LABELS".

This disitincion wasn't there for Einstein as he was looking for a deterministic classical model.

/Fredrik

#### Fra

hence the points in the manifold have no physical meaning.
I don't like how this is usually put. It's like saying that the observer has no physical meaning. While that makes sense in GR; please explain how it makes sense for a measurement theory.

I personally think it's more correct to say the point on the manifold have no objective(observer-independent) meaning.

The problem with Rovelli is that he ASSUMES that there is some objective meaning to the set of possible observers, and the generating transformations. Except for the mathematics I don't see how this is a inferrable statement, therefor I think it shold be reformulated.

/Fredrk

#### Finbar

So if I begin with one set of coordinates relating to some observer a passive diffeomorphism
would simply re-label the events as seen by this observer. On the other hand an active diffeomorphism is like moving to a coordinate system associated to a different observer?

This seems to make sense since under both diffeomorphisms the objective physics is still invariant. However for the active case we seem to have changed the order of events or possibly the gravitational field(metric).

The other idea is that diffeomorphisms are active because the metric is a dynamical field. This makes sense also but I'm not sure if this need be a prerequisite to define an active diffeomorphism.

With respect to the maxwell equations you can write them in a generally covariant form. For example if you were on a fixed curved space-time.

#### atyy

Really? If we write Newton's equations in a given coordinate system and then do a change of coordinates, the functional form will no longer be the same. I mean, that's why we have things like the Coriolis force in a rotating frame. Maybe I am missing something!
This is done in section 12.4 of MTW.

Also helpful, I think, is section 3.2 of Malament's http://arxiv.org/abs/gr-qc/0506065.

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#### nrqed

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This is done in section 12.4 of MTW.

Also helpful, I think, is section 3.2 of Malament's http://arxiv.org/abs/gr-qc/0506065.

Thanks again for the interesting references.

But then I don't understand Rovelli's point. Why is he stressing the general covariance of GR and what point is he making, exactly? Or are you saying that Rovelli's arguments are baseless?

Thanks

#### atyy

But then I don't understand Rovelli's point. Why is he stressing the general covariance of GR and what point is he making, exactly? Or are you saying that Rovelli's arguments are baseless?
I've never understood his argument there - perhaps it's right and poorly presented, or perhaps it is wrong, I don't know which - the only thing I understood from his book, is that there is an issue, but as to which the right statements and arguments are I only gathered from other books - I really think he said it much better in his later review, which is exactly what all modern textbooks say:

"Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields. (Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything.)" http://relativity.livingreviews.org/...es/lrr-2008-5/ [Broken]

So the statement "Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields" is equivalent to MTW's "no prior geometry".

And the statement "Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything" is what Wald and Carroll mean when they say you can use a diffeomorphism to change coordinates.

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#### nrqed

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Is Rovelli's point really that active diff = passive diff? I think he actually wants to say active diff is *not* equivalent to passive diff, and GR is invariant under active and passive diff, whereas other theories are invariant only under passive diff.

I believe he is using the term "active diff" *differently* from Wald and Caroll - whose point is simply that you can use a diffeomorphism to change coordinates. Rovelli's point is really "no prior geometry", and his definition of an active diff is a diff on the "dynamical fields" only, whereas Wald and Carroll's active diff is a diff on everything that is geometrical including the metric, even if it is not dynamical. In SR, a Rovelli "active diff" will move everything except the metric, since the metric is not dynamical, so the physics will be changed, since the metric is an essential part of the theory, just that it's not dynamical. In GR, a Rovelli "active diff" will move everything including the metric, since the metric is dynamical, and so the physics will not change. (In short, I agree with Finbar.)
That might be the key but could you make this more precise? Rovelli defines an active diffeomorphism as a map from M to M. How would you define mathematically a "dynamical" field and a "nondynamical" field?

I think Rovelli said it much more clearly here (maybe he read MTW or Finbar between 2003 and 2008 ): "Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields. (Every field theory, suitably formulated, is trivially invariant under a diffeomorphism acting on everything.)" http://relativity.livingreviews.org/Articles/lrr-2008-5/ [Broken]

This may be the answer to the question I posted earlier, when I asked why the coordinate system is dragged with the points when Rovelli is performing an active diffeomorphism while the field T is not (again, he actually drags the field, leaving the manifold unmoved but that's completely equivalent to dragging the manifold and leaving the field there). So I guess that, using your terminology, a "dynamical field" would be something which is *not* dragged together with the manifold under an active diffeomorphism. In that case, a dynamical field gets assigned to a different point in the manifold under such (nontrivial) active diffeomorphism. Of course, if the dyamical fields are dragged with the points in the manifold, nothing has changed and we get a "trivial active diffeomorphism".

I am surprised that the whole terminology is still unclear given how old Einstein's hole argument is!

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#### atyy

That might be the key but could you make this more precise? Rovelli defines an active diffeomorphism as a map from M to M. How would you define mathematically a "dynamical" field and a "nondynamical" field?
I think I better not discuss Rovelli's 2003 terminology since I've never understood it. Yes, it's not possible to make more than a heuristic distinction between dynamical fields and nondynamical fields - that's what the Giulini article I linked to says.

So GR really cannot be "no prior objects" - it it really is "no prior 4D spacetime geometry".

Should we generalise or not? I think the correct thinking is really not with LQG (as far as motivation goes, even though the formalism and especially the link to group field theory, non-commutative geometry, and condensed matter may be very fruitful). The correct conceptual thinking is stated by Polchinski "There are two possible resolutions. The first is ........... a nontrivial UV fixed point [asymptotic safety]. The second is that ... the extrapolation to arbitrarily high energies is incorrect [emergent gravity]" http://books.google.com/books?id=k4ZQ04viGWIC&dq=polchinski+string&source=gbs_navlinks_s

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