Large diffeomorphisms in general relativity

[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?

 

[crackjack]

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

If you are looking for topological indices that characterize gravitational instantons ( ~ Asymptotically Locally Euclidean solutions), the following might help:
http://empg.maths.ed.ac.uk/Activities/GT/EGH.pdf

 

[atyy]

Carlip comments explicitly on this in http://arxiv.org/abs/gr-qc/0409039 , section 2.6.

Also interesting are his comments in http://arxiv.org/abs/gr-qc/0501033 about gauge/real symmetries depending on boundary.

 

[atyy]

More interesting comments on diffeomorphisms when a boundary is present: "But the presence of a boundary alters the gauge invariance of general relativity: the infinitesimal transformations must now be restricted to those generated by vector fields with no component normal to the boundary, that is, true diffeomorphisms that preserve the boundary of M. As a consequence, some degrees of freedom that would naively be viewed as "pure gauge'' become dynamical, introducing new degrees of freedom associated with the boundary. http://math.ucr.edu/home/baez/week41.html"

A similar thing happens with other sorts of gauge structures: "The failure of gauge invariance under large gauge tranformations is also reflected in the properties of Chern-Simons theory on a surface with boundary, where the Chern-Simons action is gauge invariant only up to a surface term. http://arxiv.org/abs/0707.1889"