Lately John Baez and I have been thinking a bit about teleparallel gravity, from a somewhat esoteric point of view based on 2-groups. We’re just about to finish up a paper on that.
Right now, though, I just have a few thoughts about one of the more usual ways of thinking about teleparallel gravity.
If you asked me what what teleparallel gravity is about, the first thing I’d tell you is that it is a rewriting of general relativity so that torsion takes the lead role, rather than curvature. But, not everyone motivates it in that way. One often hears, in particular, this statement:
Teleparallel gravity is a gauge theory for the translation group.
What does this mean? The isometries of Minkowski spacetime form the Poincaré group, and the “translation group” means the subgroup consisting of just translations by vectors. Let’s call this group , just to emphasize that we’re working in three “space” dimensions and one “time” dimension, but it’s really just the abelian group underlying the vector space .
From a certain point of view, it’s understandably tempting to try describing gravity as a gauge theory for the group of translations of Minkowski spacetime. After all, the tangent bundle is the bundle with the lead role in general relativity, but a principal bundle on (3+1)-dimensional spacetime can start to look a lot like the tangent bundle, at least once you pick a section, so that all of those affine Minkowski fibers become vector spaces.
If you believe that Cartan geometry underlies any “geometric gauge theory” of gravity, as I do, then this suggests you are modeling gravity using the homogeneous space with and , the trivial subgroup. This works OK, but it’s a bit strange geometrically: by ignoring the Lorentz transformations we’re treating Minkowski spacetime as being completely anisotropic. Reducing the symmetry from the Poincaré group to just the translation group is like adding some sort of structure that lets us distinguish absolute directions in space.
But Minkowski space itself doesn’t have preferred directions. The key property of Minkowski space that we want to mimic is its “distant parallelism”—the ability to compare vectors at distant points and decide whether they are parallel—which is something that’s preserved not only under translations but also under Lorentz transformations. So, it seems weird to throw out the Lorentz symmetry from the outset! What’s going on here, geometrically?
What I want to discuss now is this: Even though you can start off thinking of teleparallel gravity as a gauge theory for the translation group, if we think about the geometry a bit, and listen to the lessons of gauge theory, Lorentz symmetry is easily restored.
I guess now I should just come out and say what people actually do to think of teleparallel gravity as a gauge theory for the translation group. It’s pretty clever.
Say we start with a principal bundle and pick a section , which specifies a reduction to the trivial subgroup. If we’ve got a connection, say , then we can compose it with the differential of the section to get a map
The connection has a curvature which we will denote by . One can then write down the teleparallel gravity action, which begins like this:
An attractive feature of this is that it looks roughly like Yang-Mills theory with gauge group , at least if you squint until those T’s start to look like F’s. I’ll say why we used “T” in a minute.
Of course, it’s not really Yang-Mills theory, and not just because the field strength is called . In Yang-Mills, there’s a background metric, which could just as well be described by a coframe field , and the volume form corresponding to this metric looks like . But here, isn’t a background field, but a dynamical field—it is equal to the “connection” in the alleged Yang-Mills theory! Plus, there are more terms in the action, which I haven’t written, that can’t be written down in an ordinary Yang-Mills theory. These terms can only be written because of the peculiar double role of the connection as a coframe field. So, the resemblance to Yang-Mills is actually somewhat superficial. But, it’s still cute.
Anyway, on with the story.
While is really just the translation group connection, written in a particular gauge, it’s related to a certain connection on the tangent bundle called the “Weitzenböck connection”. For this, we note that can be viewed as a trivialization of , i.e. a vector bundle isomorphism
The Weitzenböck connection is just the pullback of the standard flat connection on the trivial bundle . The reason we use for the curvature of is it is naturally identified with the torsion of the Weitzenböck connection.
The action for teleparallel gravity can then be written using just the following ingredients:
- the determinant of the coframe,
- the metric: the pullback of the obvious metric on the trivial bundle
- the torsion of the Weitzenböck connection
The first two of these things are invariant under local Lorentz group gauge transformations acting on . But what about the third? The torsion of the Weitzenböck connection (i.e. the curvature of the original translation group connection ) is invariant not under arbitrary Lorentz gauge transformations, but only covariantly constant gauge transformations.
In other words, as we’ve described it so far, teleparallel gravity has a “global Lorentz symmetry” that is not a “gauge symmetry”.
The lesson of gauge theory, though, is that we should generalize any global symmetry we find to a local gauge symmetry that can vary from point to point. How do we do this?
The trick is fairly obvious from my description of the coframe field as a vector bundle isomorphism. The reason the Weitzenböck torsion isn’t obviously invariant under Lorentz gauge transformations is that the connection is the pullback of a fixed connection on the trivial bundle. Of course, saying it this way makes it sound a bit silly: if we’re transforming everything else by a gauge transformation, why are we not also transforming this connection on ? Once we do that, everything behaves much better under Lorentz gauge transformations.
In fact, there’s really no a priori reason to think of the coframe as setting up a trivialization. It’s more natural to think of a coframe as a vector bundle isomorphism
where is some vector bundle, which clearly must be isomorphic to , but not in any canonical way, and not necessarily trivial, in general. John and I like to call a “fake tangent bundle”, a name I probably picked up from him, long ago.
If is equipped with both a metric and a connection, these pull back to a metric and connection on . If the connection on is flat, then so is its pullback, and this pullback is every bit as good for teleparallel gravity as the Weitzenböck connection, so we might as well call it the Weitzenböck connection—this is what we do in that paper we’re finishing up.
But, this version of the Weitzenböck connection is invariant under local Lorentz gauge transformations, since such gauge transformations act on both the coframe and the connection on the fake tangent bundle.
Lorentz gauge symmetry in teleparallel gravity is restored.
In fact, we then get teleparallel gravity, not as a gauge theory for the translation group, but rather as a gauge theory for a Cartan connection modeled on Minkowski space. That is, Cartan geometry based on the Poincaré group with the Lorentz group as stabilizer subgroup. Some of this is implicit in the new paper with John Baez (update: that paper is now done), though there the emphasis is rather on Cartan 2-geometry. I should perhaps write up the 1-geometry version more explicitly elsewhere.