Posts Tagged ‘teleparallel gravity’

Teleparallel gravity and Poincaré symmetry

19 April 2012

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 \mathbb{R}^{3,1}, 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 \mathbb{R}^4.

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 \mathbb{R}^{3,1} 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 G/H with G=\mathbb{R}^{3,1} and H = 0, 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 \mathbb{R}^{3,1} bundle and pick a section y, which specifies a reduction to the trivial subgroup. If we’ve got a connection, say A, then we can compose it with the differential dy of the section to get a map

e= A\circ dy \colon TM \to \mathbb{R}^{3,1}.

The connection A has a curvature which we will denote by T. One can then write down the teleparallel gravity action, which begins like this:

\displaystyle \int d^4x \det(e)\; T^a{}_{\mu\nu} T_a{}^{\mu\nu} + \cdots

An attractive feature of this is that it looks roughly like Yang-Mills theory with gauge group G=\mathbb{R}^{3,1}, 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 T. In Yang-Mills, there’s a background metric, which could just as well be described by a coframe field e, and the volume form corresponding to this metric looks like d^4x \det(e). But here, e 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 e 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 e can be viewed as a trivialization of TM, i.e. a vector bundle isomorphism

TM \to M \times \mathbb{R}^{3,1}

The Weitzenböck connection is just the pullback of the standard flat connection on the trivial bundle M \times \mathbb{R}^{3,1}. The reason we use T for the curvature of A 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, \det(e)
  • the metric: the pullback of the obvious metric on the trivial \mathbb{R}^{3,1} bundle
  • the torsion of the Weitzenböck connection

The first two of these things are invariant under local Lorentz group gauge transformations acting on \mathbb{R}^{3,1}. But what about the third? The torsion of the Weitzenböck connection (i.e. the curvature of the original translation group connection A) 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 \mathbb{R}^{3,1} 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 M\times \mathbb{R}^{3,1}? 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

TM \to \mathcal{T}

where \mathcal{T} is some vector bundle, which clearly must be isomorphic to TM, but not in any canonical way, and not necessarily trivial, in general. John and I like to call \mathcal{T} a “fake tangent bundle”, a name I probably picked up from him, long ago.

If \mathcal{T} is equipped with both a metric and a connection, these pull back to a metric and connection on TM. If the connection on \mathcal{T} 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.

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.