Large diffeomorphisms in general relativity

[tom.stoer]

"Large" diffeomorphisms in general relativity

We had a discussion regarding "large diffeomorphisms" in a different thread but it think we should ask this question here.

For a 2-torus there are the so-called "Dehn twists"; a Dehn twist is generated via cutting the 2-torus, rotating one of the two generated circles by some angle theta and gluing the two circles together again.

Two things are interesting:

A) using an arbitrary angle theta this is not a homeomorphism (and therefore not a diffeomorphisms either) as neigboured points are not mapped to neighboured points. Nevertheless the torus is mapped to a torus.

B) using an angle theta which is amultiple of 360° this is a diffeomorphism, but it seems that it should be called a "large" diffeomorphism as the two ccordinate systems are not transformed into each other via l"local" deformations.

Now in GR we expect everything to be invariant regarding diffeomorphisms. The Dehn twist is a rather simple example but one can easily construct similar transformations in higher dimensional spaces.

Questions:

In the case A) the twist is not a diffeomorphism, therefore we need not expect invariance; but can this case A) be generated via dynamics in GR? Or are there "diffeomorphic superselection sectors"?

In the case B) we have a diffeomorphism, but nevertheless it seems that there is a discrete structure regarding the different N*360° rotations labelling "different" (but diffeomorphic) tori. Again: are such "different" but diffeomorphic manifolds of any relevance.

General question: is there some topology of the diffeomorphism group in n dimensions which is related to these Dehn twists and other "lagre diffeomorphisms" in higher dimensions?

Thanks
Tom

 

[bcrowell]

Tom, could you provide a link to the thread where the notion of large diffeomorphisms was developed?

Suppose I take the unit square in the (x,y) plane and glue opposite edges together so that it makes a torus. Then anything of the form (x,y)->(f(x,y),y) is a diffeomorphism if f is smooth, 1-1, and onto.

It seems to me that you are arbitrarily singling out some specific y₀ and talking about the properties of f at y₀, but why is that interesting?

In other words, when you describe a diffeomorphism in terms of gluing, you're implicitly assuming that in areas away from the cut-and-glue line, we distort the torus in some smooth way -- but if I only provided you with information on the distortions (the diffeomorphism itself), I don't see how you would even know where the cut-and-glue line was.

-Ben

 

[tom.stoer]

We didn't develop that anywhere else. I am discussing this just here. Perhaps this will help: http://en.wikipedia.org/wiki/Dehn_twist

Looking at the torus in the (x,y) plane I take one y₀ where I (Dehn :-) define the cut and the twist. But the y₀ is arbitrary and after the gluing it is no longer visible. Any other y₁ would do as well, it wouldn't affect the result of the gluing, neither the torus nor the curves on the torus.

And please keep in mind that this is just one simple example in low dimensions. I think you can do something like that for arbitrary n-tori and possibly other compact manifolds.

 

[PAllen]

But at the cut and glue line, you are making points into neighbors that weren't before? I don't really know the rules of diffeomorphisms versus coordinate transforms, but is topology change really a diffeomorphism (e.g. Ben's example is a topology change)? Certainly, changing how you glue coordinate patches together produces different manifolds. Is diffeomorphism meant to include substantively different manifolds?

 

[tom.stoer]

But at the cut and glue line, you are making points into neighbors that weren't before?
...
changing how you glue coordinate patches together produces different manifolds.

It depends; as I said in case A) where I allow for an arbitrary angle you are right, but in case B) where the angle of the twist is restricted to 360°*n the transformation is topology-preserving and is definately a diffeomorphism. Le's focus on that case B) Can we say something regarding this "large" diffeomorphism?

 

[tom.stoer]

I think this may be relevant: http://en.wikipedia.org/wiki/Large_diffeomorphism

My question is, which role do such large diffeomorphisms (case B) play in GR?

 

[PAllen]

It depends; as I said in case A) where I allow for an arbitrary angle you are right, but in case B) where the angle of the twist is restricted to 360°*n the transformation is topology-preserving and is definately a diffeomorphism. Le's focus on that case B) Can we say something regarding this "large" diffeomorphism?

In this case, it occurs to me that the 'large' aspect is an artifact. The same result could be achieved with no cut and glue. Just, a smooth, in place twist will have an identical result. Hypothesis (no proof): if we ban topology change and non-smooth transform, then every 'large' diffeormorphism (that qualifies as a diffeomorphism) can be recast as an ordinary diffeomorphism.

 

[tom.stoer]

In this case, it occurs to me that the 'large' aspect is an artifact. The same result could be achieved with no cut and glue.

If you look at a drawing of a torus and a closed curve C with winding numbers (m,n) = (1,0) according to the fundamental group Z² of the torus T², then you see that a Dehn twist with 360° changes this to (m,n') = (m,n+1) = (1,1), right?

Therefore this is not an artifact.

 

[PAllen]

If you look at a drawing of a torus and a closed curve C with winding numbers (m,n) = (1,0) according to the fundamental group Z² of the torus T², then you see that a Dehn twist with 360° changes this to (m,n') = (m,n+1) = (1,1), right?

Therefore this is not an artifact.

Ok, I see, very interesting. No matter how I define a smooth twisting without cut and re-attach, it won't change the winding number. On the other hand, a smooth point mapping function can achieve this. So the idea is to distinguish this type of diffeomorphism.

 

[tom.stoer]

So the idea is to distinguish this type of diffeomorphism.

Yes, exactly. And not only that - the idea is to understand the physical meaning :-)

Origininally Dehn twists were studied in string theory (world sheet transformations); so my idea was that if these large diffs. exist in dim=2 they may also exist in dim=n (dim=4 especially) and they may play a role for the large scale structure / topology of spacetime. The T² case was only a warm-up.

 

[bcrowell]

Thanks for the further information, Tom.

Something similar to this happens with discrete symmetries. For example, a closed FRW solution has a discrete symmetry under time-reversal. It seems to me that discrete symmetries don't integrate as cleanly into the structure of GR as they do into the structure of a subject like Newtonian mechanics or QFT, because there is not even a general way to define them. For example, a manifold describing a spacetime in GR may not even be time-orientable, so there may be no way to define a time-reversal operator.

This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. There are arguments that general covariance is trivial. There are arguments that it's not. There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.

 

[tom.stoer]

This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. ... There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.

For "small" diffeomorphisms it's rather clear (as long as you do try to quantize them :-) they are afaik local coordinate transformations. For "large" diffeomorphisms it becomes more complicated as one mixes the topology of the manifold with the topology of the diffeomorphism group. I studied similar aspects of quantum gauge theories where the topological structure of the gauge group (or bundle) plays a central role. There are rather well-known effects related to this topology (Aharonov-Bohm, instantons, ..., Gribov copies, ...)

But I haven't found similar studies (topological structure of the diffeomorphism group in n dimensions), neither in GR nor in QG. String theorists have studied the 2-dim. diff. invariance of the string world sheet, but this is very different from the target space (in addition the diff. group in 2 dim. is a very special case).

So basically it boils down to a better undersanding of the topological structure of the diffeomorphism group in n dimensions. Any further ideas?

 

[atyy]

Which metrics can one twist? Just guessing, Minkowski seems ok, but how about if the manifold is geodesically incomplete like in the Schwarzschild or FRW solutions?

 

[TrickyDicky]

This whole topic of how to physically interpret general covariance has a muddy and inconclusive history. There are arguments that general covariance is trivial. There are arguments that it's not. There are arguments that it's not the interesting notion to study, and the interesting notion is really something more like background independence.

Yes, there has been some confusion from the beguinning, starting with Einstein that mistakenly thought that General covariance was related to the General principle of relativity, but as experts in the field (like Michel Janssen or Norton,etc) have shown, the physical meaning of General covariane is actually the Equivalence principle, it is just the mathematical way to implement it. Interpreted this way is certainly not trivial.
There has been further confuson in posterior years because people with ulterior motives have tried to interpret general covariance as absolute freedom to change coordinates.

 

[TrickyDicky]

Tom, connecting with your question about large diffeomorphism's physical meaning in GR, we could say that general covariance in the last sense I mentioned allow us to make all kinds of those large diffeomorphisms, gluing, twisting and patching without any concern for the topological structure implications of those changes that are certainly topologically non-trivial.

 

[tom.stoer]

Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?

 

[PAllen]

Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?

I'll throw out a few thoughts, but based back on the torus example, which I can picture better.

1) Modeling the physical action of cutting, twisting, mending the torus. Here, lengths, adjacency, geodesics, and obviously winding number change. One way of describing mathematical operations would be: imagine the torus is embedded in 3-space. Before, we have a set coordinates for the torus, and also the torus is described in Euclidean 3-space coordinates. Distances, etc. computed in toroidal coordinates match those of the relevant curve embedded in the 3-space. Now we do the surgery. The embedding space remains the same, with the same metric. Pre-surgery, the torus was described by some x(a,b), y(a,b), z(a,b). Post surgery, it is defined by some x(a',b'), y(a',b),z(a',b'), such that a'(a,b), b'(a,b) causes (x,y,z) points to move as specified by the surgery . However, the way we impute a metric onto the torus remains to use the unchanged Euclidean metric applied directly to x(a',b'), y(a',b),z(a',b'). As a result, distances, geodesics, etc. all change. That is, a curve that was geodesic no longer is. In effect, the metric has not been transformed as if this was a coordinate transform.

2) As above, but starting by constructing a metric for (a,b) such as to capture the toroidal geometry , while distances also come out the same as using Euclidean metric on x(a,b),y(a,b),z(a,b). Now treat the transform as coordinate transform, mapping (a,b) to (a',b'), transforming the imputed metric by standard rules. Now, I think all intrinsic geometric calculations come out the same, despite the 'large' change. In x,y,z coordinates, you see all the changes, but using (a',b') with the properly transformed metric, you don't. Note that using Euclidean calculations on x(a',b'), etc. will now *not* agree with computations using (a',b') with transformed metric.

 

[tom.stoer]

I think 1) is irrelevant as we know that the embedding of T² in R³ does itself change the metric; T² admits a flat metric, but the embedding in R³ does not. So 1) is an artefact of the embedding.

 

[TrickyDicky]

Let's make an example. Assume R*T³ solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as T³ is flat and therefore a straight line with (0,1) should work.

No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions:
a) does this generate a new, physically different spacetime?
b) does this generate a different path of a test object on the same spacetime?
c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?

Mainstream says none of the above, I'd say a)
Normally you would have b) if you have a)

 

[tom.stoer]

I tend to agree with a) But that means that GR is NOT invariant w.r.t. all diffeomorphisms but only w.r.t. "restricted" diffeomorphisms (like small and large gauge transformations, where large gauge trf's DO generate physical effects)

 

[PAllen]

I tend to agree with a) But that means that GR is NOT invariant w.r.t. all diffeomorphisms but only w.r.t. "restricted" diffeomorphisms (like small and large gauge transformations, where large gauge trf's DO generate physical effects)

I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities. Otherwise it is just arithmetic that two curves of some length, and orthogonal to each other, and with one intersection, preserve all those feature in new coordinates with properly transformed metric.

 

[tom.stoer]

I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities.

Why? It is a diffeomorphism and does not create a singularity

 

[PAllen]

Why? It is a diffeomorphism and does not create a singularity

Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.

My understanding is that topology of a differentiable manifold is encoded in how coordinate patches overlap. So, if we don't change this (and we don't need to for the Dehn twist), and we don't change anything computable from the metric, what can change?

In my (1) and (2) I was trying to get at the idea of making the operation 'real' so it does change geometry, versus treating as a pure coordinate transform, such that the corresponding metric transform preserves all geometric facts. I've heard the terms active versus passive difffeomorphism. I don't fully understand this, but I wonder if it is relevant to this distinction.

 

[tom.stoer]

Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.

Du you agree that it changes the winding number of a closed curve?

 

[PAllen]

Du you agree that it changes the winding number of a closed curve?

Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships, then it would seem mathematically impossible. If this is an example of geometrical fact independent of the metric and patch relationships, then we would need some definition how to compute it intrinsically, and it would seem to necessitate adding some additional structure to the manifold. In this case, it may well be possible, having specifically introduced non-metrical geometric properties not preserved by coordinate transforms.

Then, the physics question becomes that conventionally formulated GR would attach no meaning to this additional structure, it would become physically meaningful only in the context of an extension to GR that gave it meaning. This is what some of the classical unified field theory approaches did.

 

[tom.stoer]

Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships ...

It does even in case (2)

I found an explanation on Baez "this week's finds", week 28:

http://math.ucr.edu/home/baez/week28.html

Now, some diffeomorphisms are "connected to the identity" and some aren't. We say a diffeomorphism f is connected to the identity if there is a smooth 1-parameter family of diffeomorphisms starting at f and ending at the identity diffeomorphism. In other words, a diffeomorphism is connected to the identity if you can do it "gradually" without ever having to cut the surface. To really understand this you need to know some diffeomorphisms that aren't connected to the identity. Here's how to get one: start with your surface of genus g > 0, cut apart one of the handles along a circle, give one handle a 360-degree twist, and glue the handles back together! This is called a Dehn twist.

...

In other words, given any diffeomorphism of a surface, you can get it by first doing a bunch of Dehn twists and then doing a diffeomorphism connected to the identity.

So we can now concentrate on the physical role of these "large" diffeomorphisms.

 

[PAllen]

It does even in case (2)

I found an explanation on Baez "this week's finds", week 28:

http://math.ucr.edu/home/baez/week28.html



So we can now concentrate on the physical role of these "large" diffeomorphisms.

This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?

If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant. Is it some form of coordinate dependent torsion?

 

[tom.stoer]

This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?

Yes, not a single word.

If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant.

I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.

 

[PAllen]

Yes, not a single word.


I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.

Thankyou! Very interesting. Then I spout my opinion of the physics issue (assuming something like this is what is going on). Conventional GR only gives meaning to metrical quantities, so this aspect of the diffeomorphism would have no physical significance, and anything metrically defined would be preserved. And I come back to the idea that this sort of thing provides an opportunity to extend conventional GR- without changing any of its predictions, you can add new content.

 

[PAllen]

I found two possibly relevant papers, both focusing on 2+1 dimensions:

http://relativity.livingreviews.org/Articles/lrr-2005-1/

http://matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3928.pdf

If I am reading section 2.6 of the Carlip paper (first above) correctly, it suggests that GR is invariant under large diffeomorphisms, as I guessed above.

 

[tom.stoer]

Thanks a lot for checking that and providing the two links. Looks very interesting.

 

[Haelfix]

This material has always confused me, and its hard to find good references. Over the years I've asked a few specialists but it hasn't helped me much.

In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime.

Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations (incidentally, to avoid confusion, the notion of a large gauge transformation is afaik typically done where you fix a spacetime point, fix a vielbein and treat the diffeomorphism group acting on these elements in an analogous way to intuition from gauge theory)

 

[tom.stoer]

This material has always confused me, ...

In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime.

Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations

I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).

 

[TrickyDicky]

I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).

It seems clear to me that large difeomorphisms in the GR context are not well understood, and yet all the mainstream experts have decided that spacetimes are invariant under these large diffeomorphisms without offering any real reason.
And yet I'd say this is a vital point, the theoretical base of many yet unobserved physics, such as that of black holes(see the Carlip cite in section 2.6) depends on whether spacetimes are considered as invariant or not for these large diffeomorphisms.
Of course some people that are not very fond of thinking for themselves would rather just obey the conventional opinion on this and let it be like that, so your question Tom, touches a very sensitive spot.

 

[tom.stoer]

This is similar to "QCD is invariant w.r.t. SU(3) gauge transformations". As long as one studies infinitesimal ones everything is fine, but as soon as you try to study large gauge transformations on different topologies it becomes interesting (in the axial gauge "A³(x)=0" you cannot gauge away a dynamical zero mode a³=const., Gribov ambiguities, winding numbers, instantons and merons, center symmetry, ...).

There is an important question: gauge transformations arise due to unphysical degrees of freedom (in contradistinction to other global symmetries like flavor) which have to be gauge-fixed (e.g. via Dirac's constraint quantization in the canonical formulation). But it seems that in gauge theories unphysical local and physical global aspects are entangled).

I guess that soemthing similar will happen in GR as well. Many aspects in gauge theory become visible during quantization. So as long as we do not fully understand QG, some aspects may be hidden or irrelevant; example: what is the physical meaning of Kruskal coordinates? we don't care classically - but we would have to as soon as during BH evaporation the whole Kruskal spacetime has to be taken into account in a PI or whatever. If spacetime will be replaced by some discrete structure many problems may vanish, but if spacetime as a smooth manifold will survive quantization than these issues become pressing (diffeomorphisms in 4-dim. are rather complicated - see Donaldson's results etc. )

 

[TrickyDicky]

what is the physical meaning of Kruskal coordinates? we don't care classically - but we would have to as soon as during BH evaporation the whole Kruskal spacetime has to be taken into account in a PI or whatever. If spacetime will be replaced by some discrete structure many problems may vanish, but if spacetime as a smooth manifold will survive quantization than these issues become pressing (diffeomorphisms in 4-dim. are rather complicated - see Donaldson's results etc. )

Here is a slide presentation that treats the Kruskal coordinates in terms of gauge and local versus global gauge freedom that can be related to the small vs. large diffeomorphisms, see for instance slides 27-28.
http://www.sam-jordan.ch/download/physics/ba_presentation.pdf

 

[PAllen]

This material has always confused me, and its hard to find good references. Over the years I've asked a few specialists but it hasn't helped me much.

In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime.

I think this is the key point to understand. The Dehn twist example does not. As you noted, there seems to be little or no useful material about the 4-d case (I looked hard in tracking down the two links for the 3-d case).

Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations (incidentally, to avoid confusion, the notion of a large gauge transformation is afaik typically done where you fix a spacetime point, fix a vielbein and treat the diffeomorphism group acting on these elements in an analogous way to intuition from gauge theory)

 

[PAllen]

It seems clear to me that large difeomorphisms in the GR context are not well understood, and yet all the mainstream experts have decided that spacetimes are invariant under these large diffeomorphisms without offering any real reason.
And yet I'd say this is a vital point, the theoretical base of many yet unobserved physics, such as that of black holes(see the Carlip cite in section 2.6) depends on whether spacetimes are considered as invariant or not for these large diffeomorphisms.
Of course some people that are not very fond of thinking for themselves would rather just obey the conventional opinion on this and let it be like that, so your question Tom, touches a very sensitive spot.

I think the sensitive point referenced is whether there is a possible problem of this type with the transform from Schwarzschild to Kruskal–Szekeres (TrickyDicky thinks there may be). (Arriving at, e.g., region I+II of the KS geometry, before maximally extending to add III+IV).

It seems clear to me that this transform is smoothly reachable from the identity transform, so it does not qualify under the definition used by referenced sources in this thread.

However, a very key point noted in intro to section 2.6 of the Carlip paper, and consistent with all other references introduced in this thread, is that large diffeomorphisms are possible whenever you have topologically non-trivial manifolds. It seems clear to me that the maximally extended KS geometry is topologically non-trivial so such diffeomorphisms should exist. Further, I wonder about even just I+II? Just the existence of the black hole throat (and the need to remove the singular point from the manifold) seems like it could allow a 4-d analog of the Dehn twist. The critical aspects of such a diffeomorphsim would seem to be limited to interior region of the black hole, which, of course, is the more physically problematic region anyway. It really would be interesting if someone found and analyzed en example of such diffeomorphism in 4-d.

 

[atyy]

Possibly relevant

http://www.math.uchicago.edu/~mduchin/teich.html

Fischer, A, and Moncrief, V (1997), Hamiltonian reduction of Einstein's equations of general relativity, in Nuclear Physics B (Proceedings Supplement) 57, in Proceedings of the Second meeting on Constrained Dynamics and Quatum Gravity QG96, Santa Margherita Ligure, Italy, 17 Spetember 1996, edited by J Nelson, NorthHolland, Elsevier Science B.V., The Netherlands, pp. 142-161, abstract at http://www.elsevier.nl/ge-jng/29/35/28/36/9/26/abstrac.html , full article at http://www.elsevier.nl/gejng/29/35/28/36/show/Products/NPE/toc.htt#iss:1-3 , MR99a:83005. Available at http://www.math.ucsc.edu/faculty/fischer.html

 

[atyy]

I found two possibly relevant papers, both focusing on 2+1 dimensions:

http://relativity.livingreviews.org/Articles/lrr-2005-1/

http://matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3928.pdf

If I am reading section 2.6 of the Carlip paper (first above) correctly, it suggests that GR is invariant under large diffeomorphisms, as I guessed above.

The Giulini reference is about 3+1D. In the discussion around Eq 1.10, he says:

-The diffeomorphism constraint generates the identity component of asymptotically trivial diffeomorphisms
-The Hamiltonian formalism requires to regard two points in the same orbit of the group generated by the diffeomorphism constraint as physically identical.
-Strictly speaking it is not required by the Hamiltonian formalism to also identify [large diffeomorphism - my interpretation - please read the text and tell me if this is wrong] equivalent points. This is completely analogous to the situation in Yang-Mills theories, where ‘large’ gauge transformations are not generated by the Gauss constraint.
-In the classical theory it would be appropriate to quotient by the [large diffeomorphism] action if we agreed that the functions on phase space which we want to use as observables cannot separate any two points in one orbit of [large diffeomorphism]. This is in fact usually assumed ...

 

[tom.stoer]

sounds familiar from gauge theories and the LQG approach

 

[atyy]

Why does it seem different from the 4D and 3+1D views?

From the 4D viewpoint, it is the 4D spacetime geometry that's physical. So given a manifold of fixed topology and a diffeomorphism, I can generate a physically equivalent metric by using the pullback corresponding to that diffeomorphism. I think this is what PAllen has repeatedly argued.

 

[tom.stoer]

I could only think about one major difference, namely if one restricts the theory four-manifold to globally R*M³.

 

[atyy]

I could only think about one major difference, namely if one restricts the theory four-manifold to globally R*M³.

What's that difference?

 

[PAllen]

Earlier in this post I asked if anyone knew how to compute winding number (or, more generally, any measure related to the number of 'large' moves in a diffeomorphism) from within the differentiable manifold. None of the references up to then discussed this, and I haven't seen it in the newer ones either (though I have only skimmed the Giulini paper, as much of it is beyond my current background). Tom Stoer proposed that maybe you can't compute this within the manifold, you can only describe the large moves made by the diffeomorphism as part of the definition of the diffeomorphsim.

With a bit of hand waving, I propose you can construct an internal measure of something like winding number in a torus, but that such an internally measured quantity does not change after the diffeomorphism even if it performs twists (! if I'm right). More precisely, I don't propose a way to measure winding number of an arbitrary curve, but instead propose to construct a curve that we may agree has a given winding number, and then argue that this internally specified winding number does not change even if Dehn twists are included in a diffeomorphism. This does not dispute what Tom proposed, but argues the position that for classical GR, large vs small diffeomorphism is physically irrelevant.

The first step is to specify the geometry of the torus (saying torus only specifies topology). I propose a geometry where there is a one parameter family of shortest, equal length closed geodesics, and that they don't intersect each other (call these w geodesics). Among geodesics orthogonal to any of these, there is a unique shortest closed geodesic (call this the u geodesic). The u geodesic intersects all of the w geodesics. This is obviously just one possible geometry, and definitely not the simplest, but it is plausible and matches the conventional image of a donut. Construct a curve based on the u geodesic plus, say, 3 w geodesics, with smoothings at the intersections, to make a single smooth closed curve. We say this has a winding number of 3.

We now perform a diffeomorphism with a Dehn twist, transforming the metric with it, according to the standard rules. These metric transform rules guarantee that lengths, angles, and geodesics are preserved. Thus our winding number 3 curve still appears to be a winding number 3 curve, constructed the same way.

[EDIT: I should add, that if, in the alternative, we cannot adjust the metric to preserve these quantities, then we have a contradiction: a diffeomorphism for which we can't provide diffeomorphism invariance!]

 

[tom.stoer]

What's that difference?

R*M³ means a direct product. This is frequently used in the canonical approach were a global time coordinate is required. It excludes from the very beginning a spacetime like a 4-sphere or a Goedel universe with closed timelike curves. Even if it seems physically reasonable it has been questioned quite frequently if such a global foliation shall be imposed by hand or if it introduces a kind of "background independence". In the canconical framework one can derive an expression that guarantess invariance under 3-diffeomorphisms; the 4th coordinate behaves differently and instead of an additonal diffeomorphsims one gets something like reparametrization invariance (known from thr relativistic particle) which is expressed as Hamiltonian constraint H~0

I do not say that it's physically wrong, but it's certainly not the most general mathematical setup.

 

[tom.stoer]

In gauge theories (e.g. SU(2)) this winding number is well-known. The Chern-Simons term "measures" the winding number W[g] of a gauge transformation (which is called large if W[g] is != 0) as it is only invariant modulo a discrete number ~n under large gauge transformations. This is also related to instantons (tunneling between different n-sectors) and theta-vacua.

The explicit expression for the winding number for an SU(2) matrix g is

W[g] = \frac{1}{24\pi^2}\int d^3x\,\epsilon_\{\mu\nu\lambda}\,\text{tr}\,(g^{-1}\partial_\mu g\;g^{-1}\partial_\nu g\;g^{-1}\partial_\lambda g\;)

which explicitly shows that the winding number is a property of the transformation g and not of the gauge field itself.

I do not know if there is an explicit formula for a "winding number" of general diffeomorphism.

 

[PAllen]

I decided to work out more concretely the simple torus example I posted previously (#45).

So, we have a torus described by x between 0 and 10, y between 0 and 5, x and x+10 being the same, y and y+5 being the same. Strictly (for manifold definition) we would need to break this up into a few overlapping open patches, but we ignore this. Now we could use the identity matrix as the metric (Euclidean), but that wouldn't give me my u geodesic. So we could use ds^2 = (1+.1y) dx^2 + dy ^2.

Now, a diffeomorphism with a double dehn twist is simply given by:

x' = x
y' = y + x

under the assumption that y' + 5k is the same as y'. We have to make some assumption like this or a closed curve of constant y ceases to be closed after the diffeomorphism. Further, we see that we couldn't make this work for e.g.

y'=y+.9x

(keeping curves connected would conflict with ensuring that y=0, y=5 etc. map to the same point).

Further, it is easy to see that, on a coordinate basis, a curve of y=constant winds around twice in the primed coordinates. That is, the twist is very clear in coordinate values. However, after transforming the metric using tensor rules, I indeed find all distances, angles, and geodesics the same. Further, my metrically defined winding number 3 curve continues to satisfy the same definition.

Thus, despite the coordinate twist, no aspect of geometry defined by the metric changes. Thus, for classical GR, it seems the only relevance of large diffeomorphismhs is the realization that they exist, and result in the fact that you can't connect all diffeomorphisms by a series of 'very small' diffemorphisms.

 

[Haelfix]

It seems clear to me that large difeomorphisms in the GR context are not well understood, and yet all the mainstream experts have decided that spacetimes are invariant under these large diffeomorphisms without offering any real reason.

I'm sure the experts know exactly what they're doing, however I suspect its kept out of most textbook treatments simply b/c it is a little bit subtle.

For instance, the Dehn twist material I believe is pretty well understood mathematically. See here: http://www.math.uchicago.edu/~margalit/mcg/mcgv50.pdf

and as you can see, the algebraic geometry is pretty thick for mere mortals!

Anyway, I am less confused about that and more confused about how far the gauge theory analogy really works in this case. Keep in mind, to make the analogy work in the first place requires a great deal of structure to begin with. Namely the existence of a suitable Velbein, a canonical foliation and I thought some sort of fixed Killing symmetries on the boundary (eg we want most of our fields to die off at infinity in analogy with gauge theory).

Diffeomorphisms that change this structure puzzles me, b/c naively I would have thought that physical observables, (say the ADM energy) would fail to be invariant under such transformations?

 

[tom.stoer]

Anyway, I am less confused about that and more confused about how far the gauge theory analogy really works in this case.

Don't get me wrong; I only wanted to point out that in gauge theory two topologically relevant objekts exit:
a) the value of the Chern-Simons functional which is a property of the gauge field configuration
b) the winding number of the large gauge transformation
(both values can be calculated via well-known formulas)

The question is how this maps to the case of diffeomorphisms
a') the winding numbers (m,n) of a curve around the torus
b') the "twist-number" of the Deh-twist

Questions:
- is there more than an analogy between these two cases?
- is there a formula to calculate a') and to check how it changes under diffeomorphisms?