Lifting General Relativity to Observer Space

[marcus]

On 2 October, 5 days from now, Derek Wise will give an online international seminar talk which will be a followup to his June 2012 paper with Stefen Gielen---as I recall a lot of us found that paper very interesting and it topped the second quarter MIP poll.
So I want to review that paper, the better to understand what Derek has to say on Tuesday.

http://arxiv.org/abs/1206.0658
Linking Covariant and Canonical General Relativity via Local Observers
Steffen Gielen, Derek K. Wise
(Submitted on 4 Jun 2012)
Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.
8 pages; Essay written for the 2012 Gravity Research Foundation Awards for Essays on Gravitation

Yeah, on our 2nd quarter 2012 MIP poll eleven of us responded and 5 out of 11 voted for Derek and Stefen's paper.
This is to say that if we collectively are reasonably good judge of interesting signficant QG papers then we are probably going to hear more about "Observer Space"
and how the novel concept of a field of observers can help "bridge" between Hamiltonian approach and covariant phase space methods (maybe even get the best of both worlds.)
Lifting to a new contextual structure that may prove more universal/convenient to work in--it sounds like a move mathematics should try on physics now and then.

Derek Wise is a John Baez PhD from UC Riverside who went postdoc to UC Davis (Steve Carlip's group) and thence to Erlangen (Thomas Thiemann's group)

 

[marcus]

Here's a brief quote from the bottom of page 6 of the Gielen Wise paper:
"...observer space, the space of all possible observers, has manifest physical meaning, and simple topology: it is a
7-dimensional manifold isomorphic to the ‘unit future tangent bundle’ of spacetime, locally a product of spacetime with velocity space H³. In [10] we reformulate general relativity directly on observer space, essentially by pulling fields back along the natural projection
observerspace → spacetime.
A connection pulled back to observer space will be flat in the ‘velocity’ directions, reflecting the symmetry under changes of observer."

And the reference is to work in preparation:
[10] S. Gielen and D. K. Wise, Lifting general relativity to observer space, in preparation.

So that is precisely what the ILQGS (International LQG Seminar) talk on Tuesday 2 October is going to be about.

(I'm really pleased that Gielen Wise June paper topped our poll. Here is a link to the poll:
https://www.physicsforums.com/showthread.php?t=617250
in case anyone wants to see what all 20 papers were. There were a lot of good ones to choose from, including several I think are outstanding.)

 

[strangerep]

OK, so I read the paper and the message I got from G+W is "let's start from a velocity phase space", i.e., the usual 3+1 spacetime, together with a space of 4-velocities which, because of the usual SR normalization $v^2=c^2$ restricts the overall velocity space to a 3D hyperboloid, giving us a 7D phase space overall. G+W call this a "space of observers", iiuc.

Something puzzles me however: why is it reasonable to call such a space of observers "fundamental" when SR was implicitly invoked to restrict the velocity part of the space from 4D to 3D ?

 

[marcus]

Hi Strangerep, it's good to have some company looking at this! As you read the June 2012 G&W paper you undoubtedly noticed the frequent references to [4] which is their longer paper covering much of the same material with the same notation in greater detail. The longer paper was published May 2012 in PRD. It's helpful to refer to and has clarified things for me. If anyone else is reading here is the link:
http://arxiv.org/abs/1111.7195
Spontaneously broken Lorentz symmetry for Hamiltonian gravity

In three days Derek Wise will be online talking about this, in the international LQG seminar. If anyone plans to listen, be sure to download the slides PDF ahead of time so you can scroll through the slides along with the audio:
http://relativity.phys.lsu.edu/ilqgs/

 

[George Jones]

OK, so I read the paper and the message I got from G+W is "let's start from a velocity phase space", i.e., the usual 3+1 spacetime, together with a space of 4-velocities which, because of the usual SR normalization $v^2=c^2$ restricts the overall velocity space to a 3D hyperboloid, giving us a 7D phase space overall. G+W call this a "space of observers", iiuc.

Something puzzles me however: why is it reasonable to call such a space of observers "fundamental" when SR was implicitly invoked to restrict the velocity part of the space from 4D to 3D ?

My first thought:

In relativity (special and general), an observer is a future-directed timelike curve parametrized by proper time, i.e., an observer is a mapping

$$\gamma : I \rightarrow M$$

such that $g \left( \dot{\gamma} , \dot{\gamma} \right) = c^2$ (proper time condition), where $I$ is an interval of reals, and $M$ is a Lorentzian manifold. Each $\dot{\gamma}$ lives in a tangent space that is isomorphic to Minkowski space.

In terms of coordinates, each observer has a 4-velocity that satisfies $g_{\mu \nu} u^\mu u^\nu =c^2$, even in GR.

This thought came to mind before I looked at the paper. After briefly scanning the paper, I am not sure that this thought is relevant, and I am afraid that I have misinterpreted your question.

 

[strangerep]

[...] The longer paper was published May 2012 in PRD. It's helpful to refer to and has clarified things for me. If anyone else is reading here is the link:
http://arxiv.org/abs/1111.7195
Spontaneously broken Lorentz symmetry for Hamiltonian gravity

Thanks. I'll need a while to digest that paper, but even just the introductory sections clarified a few things. In particular, the first paragraph:

Lorentz symmetry is a slippery topic in Hamiltonian formulations of general relativity and quantum gravity, for a simple geometric reason. The standard first step in Hamiltonian gravity is to pick a spacelike foliation, in order to define time evolution. Such a foliation gives a hyperplane distribution in the tangent bundle of spacetime, specifying the ‘purely spatial’ directions at each point. However, if we then perform a Lorentz gauge transformation, the spatial hyperplanes rotate in such a way that the resulting distribution is in general nonintegrable -- it need not be the tangent distribution of any foliation. Since the property of being a spacelike foliation is preserved only under very carefully chosen local Lorentz transformations, it is little wonder that introducing a foliation tends to obscure the behavior of a theory under local Lorentz symmetry.

So G+W advocate starting from a nonholonomic (a.k.a. nonintegrable) frame field (repere mobile) instead of a coordinatized manifold, iiuc. I tried to work with nonholonomic frames in a different context 30 years ago, but got nowhere. I sure hope G+W can do better.

It's certainly interesting: a nonintegrable field of frames -- interpreted as observers -- is physically more intuitive than a coordinatized manifold constructed by the usual methods like rigid rods, radar method, and similar fictions. In this sense, I now see that calling it "more fundamental" is indeed reasonable.

It will also be interesting to see to what extent such avoidance of the problems caused by global foliation must be paid for by the inconvenience of nonholonomicity.

Now to (try and) digest 1111.7195 ...

 

[strangerep]

[...]
In terms of coordinates, each observer has a 4-velocity that satisfies $g_{\mu \nu} u^\mu u^\nu =c^2$, even in GR.

Yes -- that's why I was (previously) having trouble seeing what's significantly new in the G+W approach.

This thought came to mind before I looked at the paper. After briefly scanning the paper, I am not sure that this thought is relevant, and I am afraid that I have misinterpreted your question.

Perhaps a bit, though it may be useful to have your post in this thread, in case anyone wants to go further and discuss the use of nonholonomic frame fields that seem to be novel in G+W's approach.

 

[MTd2]

Marcus,

I might be wrong though, but these theories are very similar, in principle. So, What this paper wis probably, but I am almost sure, a simplified version of Noldus' theory:

http://vixra.org/abs/1106.0029

The most recent update is 10 days old. It seems Noldus went way further, even in the 1st version.

Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.)

 

[marcus]

My big question about the Gielen Wise work:
As classical geometry I think it's great: adroit, creative, beautiful, seems to open up avenues for development.
I'm thinking mainly of the article published earlier this year in Physical Review D, namely
"Spontaneously broken..." 1111.7195

But how does it lead to a quantum theory of matter interacting with geometry?

Maybe I'm missing something. Please, if anyone has a glimmering of how, or even just a firm belief that it can be done, let us know!
This Gielen Wise thing is basically a bid to REPLACE THE FOLIATION-based canonical approach with something more general which specializes back down to the conventional Hamiltonian treatment in the case where the observer-field corresponds to a foliation. And this more general "neo-canonical" treatment of GR is manifestly Lorentz. It does not have spatial slices. It does not need reality conditions or second-class constraints.

However the analysis involves a provisional (local) splitting that is constantly being used--to what looks like good effect--and which takes the place of the conventional fixed spatial slicing. And what has me wondering is that this splitting depends on the COFRAME which is the very thing you want to solve for.

If I am not mistaken what Derek Wise calls the coframe e(x) is exactly what Rovelli and others call the TETRAD. This is morally kind of like "the square root of the metric g(x)" and it is basically the geometry itself. It represents the geometry of the manifold.
So it is the main thing that appears in the Lagrangian--e, that is. The other terms are derived from the coframe, or tetrad, e. In "first-order" GR this is what takes the place of the metric.

In a quantum theory one would presumably not know the coframe precisely, because that is the very geometry that one is quantizing. So is their heavy use of the coframe (in virtuoso-style differential geometry) fully justified? How will this play out when the coframe itself becomes a blur, either a pure blur or a mixed blur?
I would like to think it is fully justified, because I find the work admirable.

Maybe at the seminar Ashtekar, or someone else, will ask Derek about this as in "Well this is all very fine but how does this relate to Quantum Gravity?"

 

[marcus]

Finally the PAPER is out that tomorrow's seminar talk is going to be about.
This paper is beautiful IMHO. The writing is clear. It seems more widely understandable than previous Gielen Wise papers. I think what the paper delivers conceptually could turn out to be important.
http://arxiv.org/abs/1210.0019
Lifting General Relativity to Observer Space
Steffen Gielen, Derek K. Wise
(Submitted on 28 Sep 2012)
The 'observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.
34 pages

To provide some context, I will quote an earlier post in this thread:

On 2 October, 5 days from now, Derek Wise will give an online international seminar talk which will be a followup to his June 2012 paper with Stefen Gielen---as I recall a lot of us found that paper very interesting and it topped the second quarter MIP poll.
So I want to review that paper, the better to understand what Derek has to say on Tuesday.

http://arxiv.org/abs/1206.0658
Linking Covariant and Canonical General Relativity via Local Observers
Steffen Gielen, Derek K. Wise
(Submitted on 4 Jun 2012)
Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.
8 pages; Essay written for the 2012 Gravity Research Foundation Awards for Essays on Gravitation
...
Derek Wise is a John Baez PhD from UC Riverside who went postdoc to UC Davis (Steve Carlip's group) and thence to Erlangen (Thomas Thiemann's group)

To ILQGS just google those letters. Links to the slides PDF and the audio file should appear tomorrow or (if there is a technical delay) in a day or two.

 

[marcus]

The slides for Derek's talk are posted online:
http://relativity.phys.lsu.edu/ilqgs/wise100212.pdf

Hard to overstate how natural the passage to 7D observer space seems. In the past the basic geometry variable has been the TETRAD: a 4d frame describing the essentials of the geometry, like a metric does but more useful than the metric (which you can anyway get easily from the tetrad).

In Greek the root word for 7 is HEPT (as in heptagon) so we just shift over to using the HEPTAD instead.

The slides point out that the idea of observer is LOGICALLY PRIOR to ideas of space and time geometry. You get to this viewpoint by a logical process of generalization, which is clearly diagrammed. The whole thing has a kind of naturalness and transparency that I find convincing. See what you think.

When the audio is posted online, the link for it will probably be this:
http://relativity.phys.lsu.edu/ilqgs/wise100212.wav
EDIT: Audio was posted around 10:30 AM Pacific, soon after the slides.

I just listened to the talk. Between minute 44 and minute 50 the telephone connection with Derek (in Germany) went down and Steffen Gielen (at Perimeter) did some filling in. This was around slide 13 thru 19. There were questions primarily from Ashtekar and from Lee Smolin, but also I think from Simone Speziale (at Marseille?) and possibly Rovelli. I couldn't always tell who was asking the questions.

Here's the main link in case you want to check out some of the other talks:
http://relativity.phys.lsu.edu/ilqgs/

 

[MTd2]

It should be interesting to read from pages 111-126, axioms I to III. Pages 129- and on, AXIOM VII.

http://vixra.org/abs/1106.0029