Posts Tagged ‘general relativity’

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 geometric role of symmetry breaking in gravity

30 January 2012

I suffered a blogging derailment at the end of 2011. Now I’m eager to get back on track, and I’ve got a bunch of stuff I want to write about.

Right at the end of the year, I wrote a short conference proceedings article:

The geometric role of symmetry breaking in gravity

The point of the paper is that the mathematics physicists are most familiar with because of "spontaneous symmetry breaking" plays a somewhat different role in gravitational physics, as a key ingredient of Cartan geometry.

Anyway, this paper is at a level of detail that could work just as well as a blog article, and it fits with what I’ve been talking about on this blog, so I’m putting a version here.


The success of spontaneous symmetry breaking in condensed matter and particle physics is famous. It explains second order phase transitions, superconductivity, the origin of mass via the Higgs mechanism, why there must be at least three generations of quarks, and so on. These applications are by now standard material for modern textbooks.

Much less famous is this: broken symmetry links the geometry of gauge fields to the geometry of spacetime. This, in my view, is the main role of symmetry breaking in gravity.

An early clue came in 1977, when MacDowell and Mansouri wrote down an action for general relativity using a connection for the (anti-) de Sitter group, but invariant only under the Lorentz group. Though their work was surely inspired by spontaneous symmetry breaking, it was Stelle and West who first made their action fully gauge invariant, breaking the symmetry dynamically using a field y locally valued in (anti-) de Sitter space.

Whether one breaks the symmetry dynamically or `by hand,’ the broken symmetry of the MacDowell–Mansouri connection plays the geometric role of relating spacetime geometry to the geometry of de Sitter space. This is best understood using Cartan geometry, a generalization of Riemannian geometry originating in the work of Élie Cartan, in which the geometry of tangent spaces is generalized—in this case, they become copies of de Sitter space. But to explain how this works, and how symmetry breaking is involved, it helps to back up further.

In geometry, inklings of spontaneous symmetry breaking date from at least 1872, in the work of Felix Klein. Ironically, to study a homogeneous space Y, with symmetry group G, one first breaks its perfect symmetry, artificially giving special significance to some point y\in Y. This gives an isomorphism Y \cong G/G_y as G-spaces, where G_y is the stabilizer of y, allowing algebraic study of the geometry. While Y itself has G symmetry, this description of it is only invariant under the subgroup G_y. Different algebraic descriptions of Y are however related in a G-equivariant way, since G_{gy} = gG_y g^{-1}.

This is strikingly similar to spontaneous symmetry breaking in physics. There, one really has a family of minimum-energy states, related in a G-equivariant way under the original gauge group G. Singling out any particular state |0\rangle as `the’ vacuum breaks symmetry to G_{|0\rangle}.

Cartan took Klein’s ideas a dramatic step further, getting an algebraic description of the geometry of a nonhomogeneous manifold M, by relating it `infinitesimally’ to one of Klein’s geometries Y. Just as Klein geometry uses broken symmetry to get an isomorphism Y\to G/G_y, in Cartan geometry, the broken symmetry in a G connection induces an isomorphism e\colon T_xM \to \mathfrak{g}/\mathfrak{g}_y for each tangent space. This is just the coframe field, also called the soldering form since identifying T_xM with \mathfrak{g}/\mathfrak{g}_y\cong T_yY effectively solders a copy of Y to M, at each point x. These copies of Y are then related via holonomy of the Cartan connection, which can be viewed as describing `rolling Y along M without slipping’ (see my paper on MM gravity, and also Appendix B of Sharpe’s book).

Physics history unfortunately skips over Cartan geometry. The Levi-Civita connection is adequate for the standard metric formulation of general relativity, and more general kinds of connections played no vital role in physics until some time later. When these eventually were introduced in Yang–Mill theory, they served a purpose far removed from spacetime geometry. Yang–Mills gauge fields are really just the principal connections of Ehresmann, who, building on Cartan’s ideas, liberated connections from their bondage to classical geometry. Ehresmann’s definition, which lacks the crucial `broken symmetry’ in Cartan’s original version, has just the flexibility needed for gauge fields in particle physics, which are concerned only with the geometry of an abstract `internal space’—a bundle over spacetime, rather than spacetime itself. On the other hand, Cartan’s original version is better when it comes to studying gravity.

Concretely, a Cartan geometry may be thought of as a connection on a principal bundle (with Ehresmann’s now standard definition) together with a section y of the associated Y bundle. As an example, let us write a version of the MacDowell–Mansouri action, using de Sitter space Y\cong G/H = \mathrm{SO}(4,1)/\mathrm{SO}(3,1) as the corresponding Klein geometry:

I[A,y]  = \int \mathrm{tr}(F_y \wedge {\star}_y   F_y)

The Cartan connection (A,y) consists of an \mathrm{SO}(4,1) connection A and a locally de Sitter-valued field y. F is the curvature of A, calculated by the usual formula, and F_y is its \mathfrak{g}_y-valued part, where \mathfrak{g}_y \cong \mathfrak{so}(3,1) \cong \Lambda^2\mathbb{R}^{3,1} has Hodge star operator \star_y.

I have described additional examples of Cartan-geometric formulations of various gravity theories elsewhere, and there are many more. But besides the diversity of specific examples, there are deep reasons that gravity, or any related "gauge theory of geometry," should be framed in the language of Cartan geometry. This is the subject of geometric "equivalence theorems."

In fact, if one believes semi-Riemannian metrics are fundamental in classical gravity, one is forced to accept Cartan connections as equally fundamental. The reason for this is Cartan’s method of equivalence, a process for proving that specified kinds of "raw geometric data" are equivalent to corresponding types of Cartan geometry. In the case of Riemannian geometry, solving the "equivalence problem" leads to the following theorem:

Theorem: A Riemannian metric determines a unique torsion-free Cartan geometry modeled on Euclidean space; conversely, a torsion-free Cartan geometry modeled on Euclidean space determines a Riemannian metric up to overall scale (on each connected component).

Proof: See Sharpe’s book.

Physically, the "overall scale" in the converse just represents a choice of length unit. One can also show that deformed versions (or "mutations") of Euclidean geometry, namely hyperbolic and spherical geometry, lead to Cartan geometries that carry the same information. The Lorentzian analogs of these results are the real reason de Sitter and anti de Sitter geometries work in MacDowell–Mansouri gravity.

Riemannian geometry is but one application of the equivalence method. There are analogous theorems, for example, in conformal geometry or Weyl geometry, relating various types of conformal structures to Cartan geometries that take the model Y to be an appropriate kind of homogeneous conformal model. Sharpe’s book contains some such theorems, and some significant work has been done on applications of conformal Cartan geometry—which often goes by the name "tractor calculus"—in physics. (See, e.g. this paper and references therein.)

For now, I just want to describe one more application of Cartan geometric thinking in gravitational theory. Besides spacetime geometry, one can also use Cartan’s ideas to describe the geometry of space.

Wheeler’s term "geometrodynamics" originally referred to the of evolution of spatial geometries in the metric sense. This has sometimes been contrasted with "connection dynamics" (see e.g. here or here). In light of the above equivalence theorem, however, there seems little point in establishing any technical distinction between geometrodynamics and connection dynamics, at least if we mean connections in the Cartan-geometric sense. The metric and connection pictures have their own advantages, but the equivalence theorem suggests we should be able to translate exactly between the two.

In recent work with Steffen Gielen, we take an explicitly Cartan-geometric approach to evolving spatial geometries. In this case, the symmetry breaking field y lives in 3d hyperbolic space \mathrm{SO}(3,1)/\mathrm{SO}(3), and can be interpretated as a field of observers, since the spacetime coframe field converts it into a unit timelike vector field. This can be dualized via the metric to a unit covector field, which we might call a field of co-observers. Just as observers determine a local time direction, co-observers determine local space directions, by taking their kernel. Our strategy in the Hamiltonian formulation is to fix a field of co-observers—the infinitesimal analog of picking a spacetime folitation—but let the field of observers be determined dynamically, as part of determining the metric.

The result is a model in which the observer field plays a two part symmetry breaking role: first splitting spacetime fields into spatial and temporal parts, but then also acting as the symmetry breaking field in Cartan geometry of space. This gives a Cartan-geometric Hamiltonian framework in which the spatial fields fit neatly and transparently into their spacetime counterparts and transform in an equivariant way under local Lorentz symmetry.

Thanks to the equivalence theorem, this may be viewed as a concrete link between connection dynamics and geometrodynamics in the original sense.

It is conceivable that gravity descends from a more fundamental theory with larger gauge group, and so fits into the tradition of symmetry breaking in gauge theories. Such ideas are clearly worth pursuing (see, e.g. papers by Percacci or Randono). At the same time, we should not ignore the lesson of Cartan geometry: broken symmetry is the means to establishing exact correspondence between geometric structures living on tangent spaces on one hand and connections on the other.


Thanks to John Baez, Julian Barbour, James Dolan, Andy Randono and Steffen Gielen for helpful discussions.

(By the way, someone might still remember that in the last post on this blog, I promised to explain some things. I still plan to do that later… )