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.
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 equipped with a Lie group of symmetries acting transitively: there is at least one taking me from any point in to any other point. Picking any point , we can identify with the coset space . So, abusing terminology a bit, we often refer to “A Klein geometry ,” forgetting the name of the original homogeneous space .
Now, if is a manifold of the same dimension as the Klein geometry , then a Cartan geometry on , “modeled on ,” has two basic ingredients:
- a principal bundle ,
- a -valued 1-form on (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 and point stabilizer . So, Cartan geometry for this model involves an -valued 1-form on a principal 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 ‘s worth of directions. So, the configuration space of a hamster on a surface is a principal 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 -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 -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 —”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 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.