Short question about diffeomorphism invariance

[nrqed]

But what is the same "form"? Can't we say that the Coriolis term is in fact present in a non-rotating frame, but it just happens to be zero? I can't provide a detailed example for the Newtonian theory of the top of my head, but if you take Maxwell's equations on flat Minkowski spacetime, write them in Lorentz covariant form with the metric explicitly in the equations, then turn all partial derivatives to covariant derivatives (comma to semicolon rule), wouldn't one get a generally covariant form of the equations? In an arbitrary coordinate system, the Christoffel symbols in the covariant derivatives will be non-zero, but in a Lorentz inertial coordinate system, the Christoffel symbols will be zero.

Ok, point well taken.


But then this implies that general covariance is a red herring, right?
So you are saying that Rovelli's discussion is completely incorrect? (I am not being provocative here, I am asking an honest question because I truly want to understand the hole argument and the ''no prior geometry'' argument) What about Einstein's hole argument? Is it a red herring too? Was Einstein wrong too about making a big deal out of general covariance, or does Rovelli misrepresent what happened historically? If the hole argument is not a red herring, what is the correct way to phrase it (not invoking general covariance)?

I guess that what I really want to understand is the argument for no prior geometry. Rovelli seems to rely on general covariance to reach that conclusion. I am willing to accept that general covariance should not play a role in the argument. But then, what *is* the argument?? Some reference cited by Atyy says that no prior geometry is simply a consequence of the metric being a dynamical field. But what is the definition of a dynamical field, and how does that imply that there is no prior geometry?

It seems as if people think of dynamical vs non-dynamical fields as being related to how field behave under an active diffeomorphisms. Is that correct? What is the exact definition?
What Carroll says about active diffeomorphisms does not clarify the issue because he does not talk about how fields transform, he just talks about the transformation of the points in the manifold and of the coordinates (at least in the link that Atyy provided).

I tried to make my questions as specific as possible. Thanks for the feedback.

 

[nrqed]

Let me ask a follow-up question:
The description of the hole argument made on Wikipedia (and I have seen this argument made in other places) says:

Einstein noticed that if the equations of gravity are generally covariant, then the metric cannot be determined uniquely by its sources as a function of the coordinates of spacetime. The argument is obvious: consider a gravitational source, such as the sun. Then there is some gravitational field described by a metric g(r). Now perform a coordinate transformation r r' where r' is the same as r for points which are inside the sun but r' is different from r outside the sun. The coordinate description of the interior of the sun is unaffected by the transformation, but the functional form of the metric for coordinate values outside the sun is changed.

I don't understand the point made here. I mean, of course the functional form of the metric will change since we changed the coordinate system! We could say the same thing about an electric field and a charge distribution, it seems to me. I mean:

"The argument is obvious: consider a charge distribution confined to a certain volume. Then there is some electric field described by E(r). Now perform a coordinate transformation r-> r' where r' is the same as r for points which are inside thcharge distribution but r' is different from r outside. The coordinate description of the interior of the volume containing the charge distribution is unaffected by the transformation, but the functional form of the electric field for coordinate values outside is changed"

So what? This does not imply anything special in E&M, why does it have a profound meaning in GR? (I know that I am missing something important!) Can someone elaborate?

 

[Finbar]

It seems as if people think of dynamical vs non-dynamical fields as being related to how field behave under an active diffeomorphisms. Is that correct? What is the exact definition?

The metric is a dynamical field in GR because you have to solve the Einstein solutions to find it (or rather the equivalence class of metrics). If it was non-dynamical then you would have to put in some metric by hand.


Maybe the point is that in other theories you can get away with writing equations in a non-generally covariant form. But in gravity you can't. Further more when you go to quantum theory symmetries become much more important than in classical theory. For example observables are the generators of symmetries i.e. momentum generates space translations etc. Quite possibly diffeomorphisms generate some kind of observables? Entropy? Also if one wants to quantize the metric, in a path integral say, you need to somehow integrate over only the class of gauge equivalent metrics. So maybe the general point is that when the metric is dynamical you need to worry more about general covariance since it plays an
"active" role. Where as in a theory with a fixed metric general covariance is more passive.

I think a related issue, that Fra is getting at (?), is to split diffeomorphisms into two classes
those which correspond to moving to co-ordinates for which correspond to the same observer's point of view and those for which relate to a different observer. For example in flat space we could move from a non-accelerating frame to an accelerating one and we would observe a different vacuum in each. Again in the quantum theory this distinction is much more meaningful than in classical theory where there is no concept of the vacuum( i.e. no creation or annihilation operators).

We could say the same thing about an electric field and a charge distribution, it seems to me.

Yes, I think the hole argument was just Einstein coming to terms with gauge invariance and gauge fixing. So its not special to gravity.

 

[atyy]

But then this implies that general covariance is a red herring, right?
So you are saying that Rovelli's discussion is completely incorrect? (I am not being provocative here, I am asking an honest question because I truly want to understand the hole argument and the ''no prior geometry'' argument) What about Einstein's hole argument? Is it a red herring too? Was Einstein wrong too about making a big deal out of general covariance, or does Rovelli misrepresent what happened historically? If the hole argument is not a red herring, what is the correct way to phrase it (not invoking general covariance)?

I don't know whether Rovelli's arguemnt is wrong, but I am pretty sure that general covariance (defined here as covariance under an arbitary change of coordinates) is a red herring - all modern textbooks agree on this point.
-MTW: The "no prior geometry" demand actually fathered general relativity, but ... disguised as "general covariance", it also fathered half a century of confusion."
-Weinberg: "It should be stressed that general covariance by itself is empty of physical content" [Weinberg however does define a "Principle of General Covariance" which is meaningful, but it is not general covariance and corresponds to what other people call the "Principle of Equivalence" or "minimal coupling".]
-Carroll: "Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard."

I guess that what I really want to understand is the argument for no prior geometry. Rovelli seems to rely on general covariance to reach that conclusion. I am willing to accept that general covariance should not play a role in the argument. But then, what *is* the argument?? Some reference cited by Atyy says that no prior geometry is simply a consequence of the metric being a dynamical field. But what is the definition of a dynamical field, and how does that imply that there is no prior geometry?

That the metric is a dynamical field is the definition of no prior geometry. I don't know if a general definition of a dynamical field exists, but if we restrict to theories derivable from a Lagrangian, then I would say a dynamical field is defined by a Lagrangian and its symmetries, with GR's principle being that the Lagrangian contains scalars made from the 4D spacetime curvature tensor. In general, there will be an infinite number of such possible terms, but taking an effective theory point of view, the Einstein-Hilbert action which contains only the lowest order term is good enough at low energies. (Asymptotic Safety is the hypothesis that a Lagrangian containing additional terms made from the 4D spacetime curvature tensor will have a non-trivial fixed point as energy increases; emergent gravity such as string theory is the hypothesis that Lagrangian terms made from the 4D curvtaure tensor are insufficient at high energies, and new degrees of freedom must enter.)

It seems as if people think of dynamical vs non-dynamical fields as being related to how field behave under an active diffeomorphisms. Is that correct? What is the exact definition?
What Carroll says about active diffeomorphisms does not clarify the issue because he does not talk about how fields transform, he just talks about the transformation of the points in the manifold and of the coordinates (at least in the link that Atyy provided).

The most common definition of "active diffeomorphism" is Carroll's and Wald, which is just a "diffeomorphism" and has nothing to do with "no prior geometry". Rovelli has used a differrent definition of "active diffeomorphism" by which he means "no prior geometry" (eg. last paragraph of section 4.1 of http://arxiv.org/abs/gr-qc/9910079). Anyone is allowed to make up their own terminology, no matter how confusing. In his 2003 book, he claims to be using the Carroll and Wald definition of "active diffeomorphism", but the passage makes no sense to me unless he is not using it. I personally think it best to avoid Rovelli's definition.

 

[Finbar]

Ok the following is a personal view.


I don't think that the *best* way to think about gravity is that it is the curvature of space-time. Although this interpretation is the most straight forward mathematically i think it some what misses the real meaning of general relativity. Instead if you think back to Einstein's original motivation, before he realized that Riemannian geometry was the language was elegantly expressed in, he was trying to generalise the idea that all motion is relative. So that all that is physical is the interaction between matter and forces. In this way gravity is just a force interacting with other forces and matter fields. What is special about gravity is that it actually "gauges away" space-time; it removes space-time as being physical in any sense. There is no space-time. I find it much more intuitive to think of gravitational interactions this way; thinking of the force of gravity acting directly on matter. What is physical then is just the relation of the matter to the gravitational force both of which are dynamical and not to any idea of space-time.


Viewing it this way then explains why the idea of space-time being emergent seems to be imbedded, already, in classical general relativity e.g. its relation to thermodynamics. The idea that there is no space-time is already there in the original motivations of Einstein it was just lost in the language of Riemannian geometry.

 

[haushofer]

I also opened a topic about this a while ago:

https://www.physicsforums.com/showthread.php?t=280232

Also, the paper of Norton is mentioned; he wrote a lot about the interpretation of passive and active coordinate transfo's and the history of it.

I still don't understand it completely, and I don't think I've ever met someone who does.

 

[nrqed]

I don't know whether Rovelli's arguemnt is wrong, but I am pretty sure that general covariance (defined here as covariance under an arbitary change of coordinates) is a red herring - all modern textbooks agree on this point.
-MTW: The "no prior geometry" demand actually fathered general relativity, but ... disguised as "general covariance", it also fathered half a century of confusion."
-Weinberg: "It should be stressed that general covariance by itself is empty of physical content" [Weinberg however does define a "Principle of General Covariance" which is meaningful, but it is not general covariance and corresponds to what other people call the "Principle of Equivalence" or "minimal coupling".]
-Carroll: "Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard."

Thank you. You have pretty much convinced me that general covariance is indeed a red herring.

And thank you for the link to the Rovelli paper you provided. In that paper Rovelli is much clearer than in his book! He makes very explicit his definitions (and does not talk about general covariance. He may have received comments and questions after the publication of his book and rephrased his presentation to make it more transparent).

So let me set aside the discussion in his book and use the paper instead. There he says explicitly that all theories can be made invariant under passive diffeomorphisms (i.e. arbitrary changes of coordinates) but only GR (and not QED, QCD, etc) is invariant under active diffeomorphisms. And he defines clearly active and passive diffeomorphisms. In a passive diffeomorphism, any given point in the manifold is assigned to the same value of the field. In an active diffeomorphism, each point in the manifold is assigned to a *different* value of the field, so the mapping T:M to R (let's consider a scalar field) is changed. He says that GR is special in that it is invariant under those changes, and this is what implies that we cannot assign any significance to the points in the manifold.

I have to say that his presentation in that paper makes a whole lot of sense to me (much more than in his book, although I am sure he meant the same thing).



Now, Carroll says that active diffeo are completely equivalent to coordinates changes.
But in that section he talks only about diffeo of the manifold, there is no field. I would love to know what he does with fields under active diffeomorphisms. If he uses the same definition as Rovelli, then one cannot say anymore that active diffeo are equivalent to coordinates transfos. Maybe he uses a different definition. But then I am curious about how he would call what Rovelli defines as active diffeo.

That the metric is a dynamical field is the definition of no prior geometry.

Yes. I think I used "no prior geometry" incorrectly. I am probably confused about the meaning of the term. It seems to me that there are two issues at hand (maybe they are facets of the same thing and I don't realize it)

a) The metric is dynamical so it does not make sense to talk about spacetime intervals between points on the manifold as being defined prior to solving the dynamics.

b) But it seems that Rovelli (and the hole argument) seems to be implying more than that.
That one cannot assign a physical meaning to the points in the manifold. Isn't it what the hole argument is about?

In other words, that a) implies that we cannot talk about the distance between two points in a manifold until we solve the dynamics. But b) states that we should ot even talk about points in the manifold has having a physical reality in the first place. Am I wrong?

The most common definition of "active diffeomorphism" is Carroll's and Wald, which is just a "diffeomorphism" and has nothing to do with "no prior geometry". Rovelli has used a differrent definition of "active diffeomorphism" by which he means "no prior geometry" (eg. last paragraph of section 4.1 of http://arxiv.org/abs/gr-qc/9910079). Anyone is allowed to make up their own terminology, no matter how confusing. In his 2003 book, he claims to be using the Carroll and Wald definition of "active diffeomorphism", but the passage makes no sense to me unless he is not using it. I personally think it best to avoid Rovelli's definition.

Thank you again for this useful reference. As I said earlier in my post, I can't really compare Carroll's definition of active diffeomorphism with Rovelli's because Carroll did not define how fields (or tensors) transform under his active diffeomorphism. As long as we just look at how the manifold changes (and the coordinates, which are defined through a pullback), then it seems to me that they are saying the same thing. What is needed, however, is to compare how they define the transformation of fields.


Thank you again for all your feedback. It is very much appreciated.

 

[nrqed]

Ok the following is a personal view.


I don't think that the *best* way to think about gravity is that it is the curvature of space-time. Although this interpretation is the most straight forward mathematically i think it some what misses the real meaning of general relativity. Instead if you think back to Einstein's original motivation, before he realized that Riemannian geometry was the language was elegantly expressed in, he was trying to generalise the idea that all motion is relative. So that all that is physical is the interaction between matter and forces. In this way gravity is just a force interacting with other forces and matter fields. What is special about gravity is that it actually "gauges away" space-time; it removes space-time as being physical in any sense. There is no space-time. I find it much more intuitive to think of gravitational interactions this way; thinking of the force of gravity acting directly on matter. What is physical then is just the relation of the matter to the gravitational force both of which are dynamical and not to any idea of space-time.


Viewing it this way then explains why the idea of space-time being emergent seems to be imbedded, already, in classical general relativity e.g. its relation to thermodynamics. The idea that there is no space-time is already there in the original motivations of Einstein it was just lost in the language of Riemannian geometry.

Very interesting. But if there is no spacetime, how do gravitational waves propagate?
How is the information about the presence of the Sun over there conveyed to Saturn, say?
I read the hole argument as saying that we cannot identify spacetime points, i.e. we cannot think of a given spacetime point as having an independent existence. However, I think that the actual manifold exists. Is that your interpretation too?

 

[marcus]

I also opened a topic about this a while ago:

https://www.physicsforums.com/showthread.php?t=280232

Also, the paper of Norton is mentioned; he wrote a lot about the interpretation of passive and active coordinate transfo's and the history of it.

I still don't understand it completely, and I don't think I've ever met someone who does.

That's good, haushofer. Your thread in the GR forum is a good place to go into technical detail about GR.

This thread has focused on a few pages of Rovelli's QG book, and involved different people interpreting and understanding it different ways. Probably at times misinterpreting or simply not understanding. But the thread is too bulky to scrutinize systematically, so I will try to make a clean start with a fresh summary.

Rovelli's conceptual framework here is pretty standard. A less condensed treatment is in his 1999 paper with Marcus Gaul. "LQG and the meaning of diffeomorphism invariance." Really good clear introductory paper. The book gives a compressed version which is more difficult to follow.

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∏∑

 

[marcus]

Very interesting. But if there is no spacetime, how do gravitational waves propagate?
How is the information about the presence of the Sun over there conveyed to Saturn, say?
I read the hole argument as saying that we cannot identify spacetime points, i.e. we cannot think of a given spacetime point as having an independent existence. However, I think that the actual manifold exists. Is that your interpretation too?

Nrqed you are getting to the heart of it!

In GR there is no manifold, no physical objective existing continuum. There is geometry. Waves propagate in the geometry.

People get the idea from popular/simplified accounts of GR that there is a fixed spacetime manifold and the geometry is a metric g_mu_vu on that manifold.

That is not true. In GR the gravitatonal field is an equivalence class of metrics under diffeomorphism.

This effectively denies the manifold (and spacetime points) any objective physical existence.
Einstein remarked on this very explicitly in a couple of 1915 quotes.

The gravitational field is not some particular metric, but rather the whole class of metrics which you can get by mooshing and mapping the manifold around with diffeomorphisms.
And of course the matter gets mooshed around when you do that, as well.

For an online source of the 1915 Einstein quotes, see page 43 of this pdf at a University of Minnesota website
www.tc.umn.edu/~janss011/pdf%20files/Besso-memo.pdf[/URL]

==quote from the source material==
...In the introduction of the paper on the perihelion motion presented on 18 November 1915, Einstein wrote about the assumption of general covariance “by which time and space are robbed of the last trace of objective reality” (“durch welche Zeit und Raum der letzten Spur objektiver Realität beraubt werden,” Einstein 1915b, 831). In a letter to Schlick, he again wrote about general covariance that
“thereby time and space lose the last vestige of physical reality” (“Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität.” Einstein to Moritz Schlick, 14 December 1915 [CPAE 8, Doc. 165]).
==endquote==

In fact it was the GR invariance under diffeomorphisms that led to those radical conclusions. You could moosh and morph a solution and it would still be a solution.
Coord change is different. The manifold and its shape stay the same and you just used different coords.

With coord change you change coordinates but you still have the same rubber sheet :biggrin: but [B]with GR there is no rubber sheet[/B].

 

[atyy]

Thank you. You have pretty much convinced me that general covariance is indeed a red herring.

And thank you for the link to the Rovelli paper you provided. In that paper Rovelli is much clearer than in his book! He makes very explicit his definitions (and does not talk about general covariance. He may have received comments and questions after the publication of his book and rephrased his presentation to make it more transparent).

So let me set aside the discussion in his book and use the paper instead. There he says explicitly that all theories can be made invariant under passive diffeomorphisms (i.e. arbitrary changes of coordinates) but only GR (and not QED, QCD, etc) is invariant under active diffeomorphisms. And he defines clearly active and passive diffeomorphisms. In a passive diffeomorphism, any given point in the manifold is assigned to the same value of the field. In an active diffeomorphism, each point in the manifold is assigned to a *different* value of the field, so the mapping T:M to R (let's consider a scalar field) is changed. He says that GR is special in that it is invariant under those changes, and this is what implies that we cannot assign any significance to the points in the manifold.

I have to say that his presentation in that paper makes a whole lot of sense to me (much more than in his book, although I am sure he meant the same thing).



Now, Carroll says that active diffeo are completely equivalent to coordinates changes.
But in that section he talks only about diffeo of the manifold, there is no field. I would love to know what he does with fields under active diffeomorphisms. If he uses the same definition as Rovelli, then one cannot say anymore that active diffeo are equivalent to coordinates transfos. Maybe he uses a different definition. But then I am curious about how he would call what Rovelli defines as active diffeo.

Quick check on terminology:

1) Which paper of Rovelli's do you mean for definitions http://arxiv.org/abs/gr-qc/9910079 or http://relativity.livingreviews.org/Articles/lrr-2008-5/ ?

2) In special relativity, eg. Maxwell's equations on flat Minkowski spactime, do you consider the metric a field or not?

 

[nrqed]

That's good, haushofer. Your thread in the GR forum is a good place to go into technical detail about GR.

This thread has focused on a few pages of Rovelli's QG book, and involved different people interpreting and understanding it different ways. Probably at times misinterpreting or simply not understanding. But the thread is too bulky to scrutinize systematically, so I will try to make a clean start with a fresh summary.

Rovelli's conceptual framework here is pretty standard. A less condensed treatment is in his 1999 paper with Marcus Gaul. "LQG and the meaning of diffeomorphism invariance." Really good clear introductory paper. The book gives a compressed version which is more difficult to follow.

I agree completely. Now I use only the paper to understand Rovelli's arguments.


But even then, things are not as clear cut. Carroll says

Let's put some of these ideas into the context of general relativity. You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold M with metric g_{\mu \nu} and matter fields \psi, and \phi : M \rightarrow Mis a diffeomorphism, then the sets (M, g_{\mu \nu},\psi)and (M, \phi_{*}g_{\mu \nu}, \phi_{*} \psi)represent the same physical situation. Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard. When people say that GR is diffeomorphism invariant, more likely than not they have one of two (closely related) concepts in mind: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime.

So Carroll does not use the same definition for active diffeomorphisms as Rovelli does!
You see how hard it is for a neophyte like me not to get confused!


I *think* that the difference between the two views is in the definition of how the fields transform. Carroll transform them by pulling them back using the mapping. I think that Rovelli does not pull them back.

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.

All this is (finally!) clear to me now. Nice summary.

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.

I understand the idea and it makes sense to me.
In order for me to *really* understand, though, I would like to see the precise (i.e. mathematical) transformation of the metric under such a diffeomorphism.

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.

Yes, it is finally sinking in now!

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?

Thanks!

 

[nrqed]

Nrqed you are getting to the heart of it!

In GR there is no manifold, no physical objective existing continuum. There is geometry. Waves propagate in the geometry.

People get the idea from popular/simplified accounts of GR that there is a fixed spacetime manifold and the geometry is a metric g_mu_vu on that manifold.

That is not true. In GR the gravitatonal field is an equivalence class of metrics under diffeomorphism.

This effectively denies the manifold (and spacetime points) any objective physical existence.
Einstein remarked on this very explicitly in a couple of 1915 quotes.

The gravitational field is not some particular metric, but rather the whole class of metrics which you can get by mooshing and mapping the manifold around with diffeomorphisms.

...

Interesting stuff deleted

...

In fact it was the GR invariance under diffeomorphisms that led to those radical conclusions. You could moosh and morph a solution and it would still be a solution.
Coord change is different. The manifold and its shape stay the same and you just used different coords.

With coord change you change coordinates but you still have the same rubber sheet but with GR there is no rubber sheet.

That's extremely interesting. But it seems to me that this may be overstating the case.
I mean, if there is no manifold, then we cannot even define a diffeomorphism to start with!
I thought that the conclusion was that we cannot think of points in the manifold as having any physical meaning. In other words, we could not imagine making dots on the manifold and thinking of these dots as representing points in spacetime. Because diffeomorphism can move them around. But the manifold itself still exits, doesn't it? Maybe we are saying the same thing but using a different language. If there is no manifold, I don't see how to define a diffeomorphism to start with.

 

[nrqed]

Quick check on terminology:

1) Which paper of Rovelli's do you mean for definitions http://arxiv.org/abs/gr-qc/9910079 or http://relativity.livingreviews.org/Articles/lrr-2008-5/ ?

I apologize. I meant the paper on the archives. His section 4.1 seems to give the clearest exposition of his argument that I have seen.

2) In special relativity, eg. Maxwell's equations on flat Minkowski spactime, do you consider the metric a field or not?

I don't consider it a field, no.

Let me make my position more clear.
Let's say that we see spacetime as a rubber sheet. I identify points in spacetime by marking a few dots with a pen. Now we do GR. GR says that the metric is dynamical, its value depends on the energy/mass/momentum distribution. Ok, so I picture this by deformations of the rubber sheet. If a mass is present, the rubber sheet is deformed and the distance between the marked points changes. So the metric is dynamical here, but it still does not seem to imply that we must deny reality to the points in the manifold (the points marked with the pen). At least it does not seem to imply that to me. This is why I would say that having a dynamical metric does not imply that we must deny the physical reality of the points in spacetime.

However, the hole argument and the active diffeomorphism invariance (in the language of Rovelli) seem to imply something stronger: that we cannot even think of marking points in the rubber sheet and assigning them an objective reality (although I do not understand thsi deeply since I haven't found an explicit and clear demonstration of this). To me, this is more than just saying that the rubber sheet may be deformed (i.e. that the metric is dynamical)

I don't know what the statement "no prior geometry" means. I thought it meant simply that the spacetime interval between spacetime points depends on energy/mass, i.e. that the rubber sheet is not stiff (as in other theories) but may be deformed. However, the hole argument and the business about diffeomorphisms seem to say more: that we cannot mark points in the rubber sheet (in other words, that there is no objective meaning to a given point in the manifold).


I hope this makes sense.

 

[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?

 

[nrqed]

Now, Carroll says that active diffeo are completely equivalent to coordinates changes.
But in that section he talks only about diffeo of the manifold, there is no field. I would love to know what he does with fields under active diffeomorphisms. If he uses the same definition as Rovelli, then one cannot say anymore that active diffeo are equivalent to coordinates transfos. Maybe he uses a different definition. But then I am curious about how he would call what Rovelli defines as active diffeo.

Let me take this back. I had not realized that Carroll does define how his fields transform under active diffeomorphisms (see my post number 42 where I quote the relevant passage).
He uses the pullback of the transformation to define the fields in the original manifold. In that case I can see how such an active diffeomorphism has no more content than a simpel change of coordinate system. However, Rovelli uses a different definition, so this is one question that is answered. One down, 20 to go!

 

[atyy]

I don't consider it a field, no.

OK, so in SR the metric is not a field - but, in GR presumably it is a field?

 

[nrqed]

OK, so in SR the metric is not a field - but, in GR presumably it is a field?

Yes. And it is a dynamical field, I agree with that. However, I don't quite see how the "metric as a dynamical field" issue is related to the hole argument. To me they seem like separate issues. See my post number 44 where I try to clarify my question as much as possible.

Thanks!

NOTE ADDED: I think it would help me understand greatly if I could see a clear definition of the hole argument (not just stated n words but with clear definitions of the transformations of all the quantities involved). Any good reference anyone might suggest?

 

[atyy]

Yes. And it is a dynamical field, I agree with that. However, I don't quite see how the "metric as a dynamical field" issue is related to the hole argument. To me they seem like separate issues. See my post number 44 where I try to clarify my question as much as possible.

OK, that's all that I understand then. The difference between SR and GR is that the metric is a dynamical field which is all that I mean when I say that GR is distinguished by "no prior geometry."

I think the argument about points in the manifold having no physical existence is baseless - quite simply, we need the manifold to define GR. 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 are - but that's common sense since to start we postulated that a metric is physically important - and this is a property of both SR and GR (ie. I have never understood why need to dig ourselves into the hole problem then dig ourselves out).

With respect to the larger issue of background independence, let me note that in GR topology and signature are fixed backgrounds. And in Group Field Theory, which is a current approach to trying to define dynamics for LQG, the theory is defined on a fixed metric of the group manifold ( http://arxiv.org/abs/gr-qc/0607032 ). So I find Rovelli's motivation for LQG poorly conceived, and the criticism that string theory is not background independent also baseless (unless one is also willing to criticize group field theory in the same way).

 

[atyy]

NOTE ADDED: I think it would help me understand greatly if I could see a clear definition of the hole argument (not just stated n words but with clear definitions of the transformations of all the quantities involved). Any good reference anyone might suggest?

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

In the later chapters he does say "The fact that GTR has Diff(M) as gauge symmetry means, due to the hole argument as discussed in Chapter 4, that the view which takes the spacetime manifold to be a substance has come under pressure."

However, what he means by this has nothing to do with "points of the manifold having no physical reality". He makes clear what he means before that "Thus, although there still exists a spacetime, it is no longer absolute, but it is demoted to being just another physical field.", which is completely in line with all standard textbooks saying that the distinguishing feature of GR is that the metric is a dynamical field, which is the definition of "no prior geometry".

To make clear that there is nothing mysterious aobut being "demoted" to a field, we could just as easily have said "promoted" to a field. eg. this paper talks about the Barbero–Immirzi parameter being "promoted" to a field http://arxiv.org/abs/0902.2764. 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.

 

[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.