## Archive for the ‘Gravity’ Category

### Quantum Spacetime in Bad Honnef

6 March 2012

Right now, I’m staying in this mansion:

This is the Physikzentrum in Bad Honnef, Germany. It’s a great place to spend time talking about physics, with an atmosphere that tastefully blends old and new. Inside, there’s a nice modern lecture hall:

but also—and probably even more important—comfortable spaces to talk … or to sit and write after most everyone else has gone to bed, like the room I’m in now:

Anyway, the reason I’m here is a workshop called

This is the kind of quantum gravity conference I like best—one that brings people together who have different viewpoints, and different approaches to the same questions.

Today we heard talks by Jan Ambjørn, Daniel Litim, Petr Hořava, Dario Benedetti, and Gianluca Calcagni. There were some common themes running through several of these talks, especially concerning renormalization group flow and asymptotic safety, and it was nice to see different perspectives. But for me, one the most interesting things was hearing a bit about the relationship between causal dynamical triangulations (which was the subject of Ambjorn’s talk) and Hořava-Lifshitz gravity (the subject of Hořava’s talk, though he modestly didn’t call it that himself).

Both causal dynamical triangulations and Hořava-Lifshitz gravity make a sharper distinction between space and time than general relativity does. The first introduces a fixed slicing of spacetime into discrete time-steps, while the second discards Lorentz symmetry at short distance scales. Since such theories that treat space and time anisotropically seem to be popping in from various starting points (another is “shape dynamics,” which I’ve mentioned here before) it is natural to wonder about relationships between them.

And, it’s good to see that some people are doing more than just wondering:

C. Anderson, S. Carlip, J. H. Cooperman, P. Horava, R. Kommu, P. R. Zulkowski, Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations.

Some of these folks are friends of mine from when I was at UC Davis.

Anyway, today was just the first day of talks, and I’m looking forward to the rest of the conference. Right now, I should probably get some sleep so I don’t doze off during Stefano Liberati’s talk in the morning!

### 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 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… )

### 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.