Posts Tagged ‘Klein geometry’

Explaining Cartan geometry

26 September 2011

I recently got back from a week’s visit with Julian Barbour, which we spent talking about geometric foundations of Machian physics in general, and shape dynamics in particular.

Much of what Julian and I are discussing isn’t to the level of blog-worthy material yet, but one of the key ideas is Cartan geometry, particularly in its various “conformal” incarnations. So, one thing I did during the visit was to explain a bit of Cartan geometry to him. Explaining Cartan geometry is always fun for me: it’s an incredibly beautiful subject, can be understood on an intuitive level, and yet still seems to be rather underappreciated. We thought it would be fun to record part of our discussion here for others to read.

Essentially, Cartan geometry is a way of studying geometry by rolling one object around on another: the geometry of the one serves as a “prototype” for the geometry of the other, and the nontriviality of the rolling—i.e. the failure to come back to the same configuration after rolling around a loop—measures the geometric deviation from that of the prototype. I’ll explain this in more detail in a moment.

Physically speaking, Cartan geometry is all about gauge theory of geometry. Gravity is a kind of gauge theory, but unlike the gauge theories of particle physics, it is a gauge theory that determines the geometry of the space it lives on. This is precisely what Cartan geometry is good for. I’ve explained this elsewhere, where I’ve used the idea of “rolling without slipping” to study spacetime geometry by rolling a copy of, say, de Sitter space along it.

Julian isn’t immediately interested in spacetime geometry. As I mentioned before, he’s interested in physics where “time” plays no primary role: there is only space, or more precisely, only configuration space. But, in broad terms, he is currently studying a kind of “geometric gauge theory.” Cartan geometry should still be the most natural language for it.

So, we recently spent a day struggling to form some sort of synthesis of my work and his, using Cartan geometry and rolling without slipping to understand physics in a world without time. Here is what we came up with:

Oh, OK, so we got a bit further than that…

Cartan versus Levi-Civita

We really did use the globe as a prop for discussing Cartan geometry. Unfortunately, Julian’s book is too slick and too bulky to effectively simulate rolling a plane on a sphere without slipping, but this drink coaster with cork backing worked beautifully:

Here, Julian is experimentally verifying my claims about Cartan geometry. By rolling the coaster around a cleverly chosen loop on the globe, carefully avoiding any slipping or twisting, he can get back to a configuration with the same point of tangency on the globe, but where the coaster has been both rotated and translated.

This “rolling without slipping” of the coaster along a path on the globe by is one of the simplest examples of a Cartan connection. It’s a rule for moving a homogeneous space like a plane, represented here by the coaster, around on a not necessarily homogeneous space, represented here by the globe. (While Julian’s globe looks pretty spherical at the scale of the picture, it’s actually a relief globe, which made it convenient to remember which space was supposed to be the homogeneous one.) The deviation from the homogeneous geometry is measured by the failure to come back to the same configuration after going around some loop using this rule.

Notice that this kind of “parallel transport” that is qualitatively quite distinct from that done by the more familiar Levi-Civita connection. In particular, while the Levi-Civita connection transports tangent vectors in a linear way, “rolling” the tangent plane gave us translations, which are not linear transformations.

To perform the Levi-Civita parallel transport using our drink coaster model, place one finger in the middle of the coaster, right at the point of tangency with the sphere:

Then move the coaster around with just that finger. To do this, you of course have to slide the coaster—a forbidden maneuver in the Cartan version—but you should still be careful not to twist the coaster relative to the globe.

Actually, you can think of the Levi-Civita transport via rolling without slipping, if you keep making corrections as you go. Suppose we fix the origin on our drink coaster so that we can think of it as a vector space. Then draw a vector on it. To transport the vector along a path, first break the path up into small steps. After rolling along the first bit, the coaster’s origin will no longer be at the point of tangency with the globe. This is unacceptable, since we’re supposed to be carrying our vector along by a linear transformation! So, make a correction: fortunately, there is a canonical way to slide the coaster without rotating it, maintaining the point of contact on the globe, so that the origin goes back where it should be.

Now make these little corrections after each little step along the total path. If your steps were sufficiently small, once you finally arrive at your destination, you’ll have a very good approximation to the Levi-Civita transport. In the limit of infinitely many infinitesimal steps, you get the Levi-Civita transport exactly.

So, the Cartan connection knows about the Levi-Civita connection, but it also knows more: in fact, the additional information we’ve suppressed in forcing the origin to remain in contact with the globe is enough to reconstruct the metric on the sphere, up to a constant global scale, or, if you prefer, up to a global choice of unit of length. For details, see Proposition 3.2 in Sharpe’s book on Cartan geometry for details. The equivalence of a Riemannian metric (up to global scale) with this type of Cartan geometry is one of the most basic applications of Cartan’s method of equivalence.

In any case, rolling a plane around on a sphere is just one kind of Cartan geometry—there’s really a different flavor of Cartan geometry for each kind of Klein geometry. For example, we could also talk about spherical Cartan geometry by rolling a ball on Julian’s globe:

Even though the globe and the ball are both spheres here, the “rolling distribution” is nontrivial (and would be even if the globe were perfectly spherical), because they are spheres of different diameter. Rolling around a loop on the globe, we can get any transformation of the ball we wish.

If, on the other hand, the globe were a perfect sphere and we had a ball that was an exact mirror image of it, parallel transport by rolling would would be completely trivial: starting out in a configuration where, say, Nairobi, Kenya touches its mirror image, and going around any loop from Nairobi back to itself, no matter how convoluted, the two Nairobis always come back in contact in the end.

The easy intuitive proof of this fact is to imagine rolling the globe on an actual mirror.

Hamster geometry

By now you may be wondering if I’ll get beyond the picture of rolling homogeneous spaces and tell you more precisely what a Cartan geometry is. I will—at least up to a few details that you can look up.

I’ll assume you already know some Klein geometry, or that you at least have vague impressions of it and can fake the rest. Briefly, a (smooth) Klein geometry is a manifold Y equipped with a Lie group G of symmetries acting transitively: there is at least one g\in G taking me from any point in Y to any other point. Picking any point y \in Y, we can identify Y with the coset space G/H. So, abusing terminology a bit, we often refer to “A Klein geometry G/H,” forgetting the name of the original homogeneous space Y.

Now, if M is a manifold of the same dimension as the Klein geometry G/H, then a Cartan geometry on M, “modeled on G/H,” has two basic ingredients:

  1. a principal H bundle P \to M,
  2. a \mathfrak{g}-valued 1-form on P (the Cartan connection)

satisfying some properties that I won’t bother writing down here. Instead, I just want to describe the geometric meaning of these ingredients. I’ve explained this in my papers using what I call “hamster geometry,” and it is perhaps worth reiterating that explanation here.

In the example of a ball rolling on a surface, the ball has symmetry group G= \mathrm{SO}(3) and point stabilizer H= \mathrm{SO}(2). So, Cartan geometry for this model involves an \mathfrak{so}(3)-valued 1-form on a principal {\rm SO}(2) bundle over a 2d manifold, namely a surface. To understand the geometric meaning of these things, think of the ball as being controlled by a hamster inside of it. Here is a hamster in a hamster ball on a clearly non-homogeneous torus:

Forgetting about the ball itself for the moment, a hamster can be placed at any point on the surface, facing in any of an \mathrm{SO}(2)‘s worth of directions. So, the configuration space of a hamster on a surface is a principal \mathrm{SO}(2) bundle over the surface. That is the geometric meaning of the bundle.

Now what about the “Cartan connection”? In this case, it should be an \mathfrak{so}(3)-valued 1-form on our hamster configuration space.

The key to understanding this one form is to realize that, so long as there is no slipping of the ball on the surface, the motion of the ball is completely determined by the motion of the hamster. The \mathfrak{so}(3)-valued 1-form just describes the rotation of the ball as the hamster moves: it takes tangent vectors to hamster configuration space—”infinitesimal changes” in hamster configuration—and gives elements of \mathfrak{so}(3)—”infinitesimal rotations” of the model sphere.

Most importantly, you can “integrate” these infinitesimals (using the path-ordered exponential) to get actual rotations of the sphere from actual paths through hamster configuration space. It’s all just a precise setup for describing how the hamster drives the ball around.

For more general Cartan geometries, I often imagine a “generalized hamster” running around on my base manifold, pushing a copy of G/H around as he goes. You may have a hard time visualizing a generalized hamster, but with a little practice, you can do it! I’ll explain how in an upcoming post.

When I do that, I also want to discuss some more particular examples: mainly various versions of conformal Cartan geometry, which I’ve been thinking about more lately, partly because of the discussions with Julian. In fact, this post was supposed to be about conformal Cartan geometry until I got carried away. It will have to wait for another time.

The 2-Erlangen Program and Mach’s Principle

3 September 2011

This summer I’ve had two visitors: John Baez and Julian Barbour.

My discussions with John and Julian were different, but there were also some eerie parallels, even beyond them having the same initials. With each of them, I spent time discussing math and physics in various cafés in downtown Erlangen. My conversations with each of them, appropriately for the location, centered on aspects of Felix’s Klein’s Erlangen Program. And with each of them, I discussed some alternative theory of gravity whose conceptual foundations differ from those of Einstein’s general relativity.

Julian Barbour: Shape dynamics, Machianism and Kleinianism

I had never met Julian Barbour until he arrived in Erlangen, but we had some great discussions during his brief visit. Here’s a picture of us talking at a café near the Schlossgarten:

Julian is interested in quantum gravity, but his approach involves serious rethinking of classical gravity. This is an attitude I can really relate to. I’m not necessarily convinced by any of the best developed approaches to quantum gravity, though I like aspects of several approaches. My feeling is that progress in quantum gravity may ultimately require some conceptual revision of classical gravity, or quantum field theory, or both.

So, it’s great talking to Julian Barbour. He has a deep understanding of general relativity and its historical and philosophical roots. But his research on foundational issues calls into question some ideas modern relativists take for granted.

Julian’s current research is on what he calls “shape dynamics”. To see what that’s about, I recommend first reading this nice short article:

Julian Barbour and Niall Ó Murchadha, “Conformal Superspace: the configuration space of general relativity.” arXiv:1009.3559.

Then, if this whets your appetite, Julian has a new expository introduction:

Julian Barbour, “Shape Dynamics. An Introduction.” arXiv:1105.0183.

This paper also lists all of the technical papers where you can go for more details.

But briefly, what is this theory is about? First of all, you can think of shape dynmaics as one way to realize Wheeler’s idea of geometrodynamics: the description of gravity as “evolving spatial geometry”, rather than as “spacetime geometry”. Here’s how Wheeler himself described this idea:

Give the fields that generate mass-energy, and their time-rates of change, and give the 3-geometry of space and its time-rate of change, all at one time, and solve for the 4-geometry of spacetime at that one time … And only then let one’s equations for geometrodynamics and field dynamics go on to predict for all time … both the spacetime geometry and the flow of mass-energy throughout this spacetime. (Misner, Thorne, and Wheeler, Gravitation (p. 484))

The now standard realization of geometrodynamics is the ADM formulation of general relativity, in which spacetime is equipped with spacelike foliation, and the Einstein equations split accordingly into a part that describes the geometry of space and a part that describes the time evolution of that geometry. Shape dynamics is related to ADM, but there are some key differences.

The first way that shape dynamics differs from the ADM picture, and indeed from the geometrodynamics picture as stated in Wheeler’s quote above, is that in shape dynamics there is no time. Or at least, time is not put in from the outset, but is rather a derived concept. The elimination of time is part of Barbour’s pursuit of a completely Machian theory.

Why is that? You might think of “Mach’s principle” as the assumption that a lone body in an otherwise empty universe cannot experience acceleration or angular momentum; that these concepts make sense only in relation to other material bodies. And indeed, this is something like what Einstein had in mind when he coined the term “Mach’s principle” for one of his guiding ideas in developing general relativity.

But in Barbour’s view, there are really two “Mach’s principles”: one for space and one for time. He quotes Ernst Mach himself on the temporal version:

It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive by means of the changes of things.

In shape dynamics, one initially uses an arbitarily parameterized path in configuration space. Time then becomes a particular choice of parameter that is selected by the theory itself. In Barbour’s words, “The universe is its own clock!

The other main way that shape dynamics differs significantly from the ADM formulation of general relativity is in what is meant by “spatial geometry”. Again, we can refer to Mach’s principle—this time in its more familiar “spatial” incarnation. I’ll pretend the following is a direct quote of Mach; it isn’t, but it matches his ideas, and it is the logical parallel of his quote above, obtained simply by trading spatial for temporal notions:

It is utterly beyond our power to measure the positions of things in space. Quite the contrary, space is an abstraction at which we arrive by means of the relative positions of things.

If there were perfect symmetry between the temporal and spatial aspects, I would now say “in shape dynamics, there is no space.” But this is stretching the truth. What I can say is that the “space” of shape dynamics starts out as something much less rigid than its ADM analog. While the ADM formulation is about evolving Riemannian geometry of space, in shape dynamics, space starts out with only a conformal geometry. The basic objects in shape dynamics are “shapes of the universe,” by which one technically means diffeomorphism classes of conformal structures on a fixed manifold representing space.

I’d like to say more about shape dynamics, and specifically about how both “time” and a more rigid “space” emerge from an initially timeless setup with spatial conformal symmetry. But for now, I just want to say just a bit about what Julian and I are actually discussing. In particular, I so far haven’t said anything about what our conversations in Erlangen had to do with Klein’s Erlangen program, as I mentioned at the beginning of this article.

In fact, “Machianism” in physics and “Kleinianism” in mathematics have a lot in common. Kleinian geometry is all about the duality between structure and symmetry. One studies the structure of a homogeneous space by studying the symmetries that preserve that structure, and conversely. Machian physics, on the other hand, especially in the extreme taken by Barbour, is all about tension between things and relationships between things. We arrive at abstract concepts like space and time as a way of describing the relationships between things. But these abstractions, to whatever extent they “really” exist, are viewed as floppy, bendable, stretchable entities—indeed, highly symmetric entities—that only acquire some kind of rigidity because of the “physical things” that inhabit them and give them structure.

So, part of the discussion Julian and I are now in the midst of concerns the question “what is structure?“—from both mathematical and physical standpoints. I’ve discussed the mathematical side of this question repeatedly with Jim Dolan. Julian, on the other hand, has thought about this question from a more physical perspective than Jim and I. So, it was interesting to discover that we have similar ideas about the answer. I’m visiting Julian next week, in part to continue talking about this.

That’s all I’ll say at least until after my trip to see Julian, since we’ve really just gotten started on some topics that we want to dig deeper into.

But hold on a moment! Did I just say Mach’s principle was about “things” …

… and “relationships” …

between things? Hmm. That starts to look a lot like category theory, no?

Indeed, my discussions with Julian are leading us into parts of category theory as well, which, after all, can be viewed as some kind of generalization of Klein geometry. Just to mention one way that category theory shows up in our discussion, there are very precise categorical ways of describing “stuff, structure, and properties“, thanks to an idea of Jim Dolan.

In fact, after talking to Julian for a while, I could tell that he already loves category theory … he just doesn’t know it yet! Should I try to prove to him that he likes it, or just use it as my secret weapon in our discussions?

John Baez: Klein 2-geometry and teleparallelism

Just a couple of weeks before Julian’s visit, John Baez was here. This was great: I hadn’t seen John in some time, until the conference in Zürich the week before, and now we had a week to sit around in cafés in Erlangen and talk about math and physics.

As usual, we talked about lots of different stuff. But one of our discussions really took off, and we’re now writing a couple of papers. A draft of the first one is available here:

John Baez and Derek Wise, Teleparallel gravity as a higher gauge theory I (Draft version)

This project is turning out to be a lot of fun, partly because several ideas that John and I have discussed over the years all seem to converge here.

For one thing, John and I have both been interested for some time in categorifying Klein’s Erlangen Program. So, John’s visit to Erlangen seemed like the perfect occasion to finally pursue the “2-Erlangen Program” in earnest … at least for a few days.

The idea here is that, just as spaces (sets with certain structure) have groups of symmetries, “2-spaces” (categories with certain structure) have “2-groups” of symmetries and “2-symmetries”, or symmetries of symmetries. And just as the Erlangen Program is about describing homogeneous spaces as quotients of groups, there should be some “2-Erlangen Program” in which “homogeneous 2-spaces” are described as “quotients” of 2-groups.

In fact, we’ve long been interested in carrying this idea further to categorify Cartan geometry. Cartan generalized Klein’s Erlangen Program in a powerful way to consider spaces that only look “infinitesimally” like homogeneous spaces. Lots of spaces don’t have much global symmetry, but can still be treated as “infinitesimally modeled on” homogeneous spaces, using Cartan geometry. Similarly, lots of 2-spaces don’t have much global 2-symmetry, but might still be viewed as “infinitesimally” like homogeneous 2-spaces. So there should be not only “Klein 2-geometry”, but also “Cartan 2-geometry.”

But actually, even though we had the idea of 2-geometry in the backs of our minds, that’s not how we got into this project. Rather, we were thinking about one particular 2-group—the Poincaré 2-group.

John discovered the Poincaré 2-group long ago, when he first got interested in the idea of higher gauge theory. One place to read about this 2-group is in the introdution to our book on 2-group representation theory. But, despite having worked together on this 2-group, its representations and potential physical applications, we still didn’t know of any convincing way it shows up in higher gauge theory. And, despite the Poincaré 2-group being related to the ordinary Poincaré group, we still didn’t know of any convincing way it shows up in established physics.

So, during John’s visit, we were happy to realize that “2-connections” for the Poincaré 2-group actually have a very nice geometric interpretation in terms of flat connections with torsion. This got us thinking about one place where a flat connection with torsion—and hence a Poincaré 2-connection—plays a key role: teleparallel gravity.

I’ve been intrigued by teleparallel gravity ever since the discussions on the newsgroup sci.physics.research that led up to what John wrote about it in TWF 176. John and I have talked about teleparallism off and on since then, partly because there are some nice things to say about teleparallel gravity and Cartan geometry which I should write up some day. (This is perhaps no surprise, given that Cartan himself invented most of the mathematics needed for teleparallel gravity.)

We never expected teleparallel gravity to involve Cartan 2-geometry. But, that is the conclusion we seem to have reached after a couple of months of working on this stuff.

So what is teleparallel gravity? On first sight, it looks radically different from general relativity. For example:

  • In general relativity, there is no canonical way to compare vectors at different points; in teleparallelism there is—this is the origin of the term teleparallelism, or “distant parallelism”.
  • In general relativity, Einstein elegantly replaced the Newtonian concept of “gravitational force” with a geometric notion: spacetime curvature; In teleparallel gravity, we “flatten” spacetime back out, and bring the gravitational force back from the grave!

But the most shocking thing about teleparallel gravity is that, as reactionary as it might sound, it is locally isomorphic to general relativity!

So how did we get from teleparallel gravity to Cartan 2-geometry? That’s part of the second paper we’re writing. There are some hints in the cliffhanger ending of the current draft of our first paper. You can also get some hints by reading the nice blog article John already wrote about our work:

John Baez, Klein 2-Geometry XII

and by reading the comments. Some people are starting to guess parts of the answer in the comments section of John’s blog post. But, anyway, I won’t say more right now, since you can already read the draft of the first paper, and right now I’d rather spend time actually finishing the second paper than telling you about what’s going to be in it!

General relativity, teleparallelism, shapes …

Sometimes alternatives to general relativity can really challenge deeply ingrained conceptions about how gravity is to be understood.

Shape dynamics arose out of Julian Barbour’s long quest for a completely Machian, or completely “relational” theory, and out of his conclusion that time does not exist. This theory makes both “duration” and “size” relative concepts, and so is more “generally relative” than general relativity. Teleparallel gravity on the other hand, first explored long ago by Einstein, Cartan and others, seems like a total Machian heresy! It reintroduces in particular the “absolute” concept of distant parallelism.

Yet both teleparallel gravity and shape dynamics are equivalent, at least under certain certain conditions, to general relativity. You can read about the equivalence of general relativity and shape dynamics in these papers by Barbour’s recent collaborators:

Henrique Gomes, Sean Gryb, and Tim Koslowski, Einstein gravity as a 3D conformally invariant theory

Henrique Gomes and Tim Koslowski, The link between general relativity and shape dynamics

The equivalence of general relativity with teleparallel gravity is a bit more direct. But, it is interesting to think about how the rest of physics fits into the teleparallel philosophy. For this, I suggest:

H.I. Arcos and J. G. Pereira Torsion gravity: a reappraisal

So far, to me, both of these theories fall under the heading “fun things to think about.” I find fascinating the wide variety of conceptual stories one can tell and end up with theories that would be hard to tell apart experimentally. Personally, teleparallel gravity seems less likely to me to lead us to any fundamental truths than some of Barbour’s ideas. But I don’t necessarily “believe in” either teleparallel gravity or shape dynamics. So far.

Maybe Julian will persuade me further when I visit.