Can Loop Quantum Gravity Dismiss the Manifold Structure of S^4?

[kakarukeys]

It seems to me that Loop Quantum Gravity has already ruled out the possibility that the spacetime could be a manifold like S^4, by making the assumption that the tangent bundle is trivial, or M = R x Sigma. TS^4 is non-trivial implies we cannot find a basis of tetrad fields that span the module of vector fields (hairy ball theorem), hence there is no LQG theory. Any opinion?

 

[Careful]

It seems to me that Loop Quantum Gravity has already ruled out the possibility that the spacetime could be a manifold like S^4, by making the assumption that the tangent bundle is trivial, or M = R x Sigma. TS^4 is non-trivial implies we cannot find a basis of tetrad fields that span the module of vector fields (hairy ball theorem), hence there is no LQG theory. Any opinion?

Perhaps the topology of this manifold is nonsensical ?!

 

[marcus]

I'm not sure that is a flaw, kakarukeys.
One doesn't necessarily WANT a QG formulated on a spacetime that is topologically S⁴. This may be the message of C.'s post, in which case I concur.

 

[kakarukeys]

Please elucidate. Classical G.R. does not put restriction on the topological structure of spacetime and some cosmological models suggest what the global topology of universe might be. Sometimes the differential structure tells you about the topological structure, as in de Rham cohomology. Topology of spacetime may be not so nonsensical. I'm curious why LQG has to do that.

 

[marcus]

Please elucidate. Classical G.R. does not put restriction on the topological structure of spacetime and some cosmological models suggest what the global topology of universe might be. Sometimes the differential structure tells you about the topological structure, as in de Rham cohomology. Topology of spacetime may be not so nonsensical. I'm curious why LQG has to do that.

C., who replied first, has priority about answering. I will just make a few comments and then wait to see what C. says.

1. your original question was about S⁴. To me that does not make sense as a spacetime even for classical Gen Rel. Where is the big bang singularity, where is the big crunch singularity? Without even beginning to talk about LQG, and only thinking classical, I am dubious of S⁴. This is why I am not inclined to think of it as a flaw that canonical LQG does not handle that spacetime case.

2. OTOH it IS a limitation of canonical LQG that it does not handle topology change. Or that more generally its spacetime has to be of the form R x M where M is some 3-manifold.

Canonical LQG has origins in ADM and Ashtekar formulation of classical Gen Rel----these are based on a 3-manifold---same picture: R x M.
It certainly is some kind of a limitation. I don't know how serious that limitation is. Perhaps it depends on what application one has in mind, or what one wants to study.

3. non-string Quantum Gravity has now become a variety of different approaches----including others besides canonical LQG. Often people call the whole group of them "LQG" because they have no other familiar term for the whole bunch. Some of these approaches are path-integral type. I would not be able to say right off which approaches can accommodate how much difference in topology from the R x M picture.
Do you happen to know? I think not all are so restricted.

 

[kakarukeys]

Okay...

One more question: ADM's original paper does not seem to assume M is R x S. They simply pick an arbitrary coordinate system and single out the time coordinate, and define the space-like hypersurfaces to be t = const. time vector field to be \partial_t, and go on to develop the formulation. Their canonical variables are 3-metric on & extrinsic curvature of t=const.

Just when did canonical gravitists start to make that assumption?

 

[marcus]

Okay...
One more question:
Just when did canonical gravitists start to make that assumption?

I suppose you mean when did gravitists start to make that assumption explicitly.
Offhand I don't know, kakarukeys. Maybe someone else will discuss.
Have you tried looking it up in, e.g., Rovelli's section on the history of the subject?BTW I haven't looked at ADM original paper and am not sure about this, but wouldn't there be be some IMPLICIT assumptions about the topology already taken for granted in their construction?
If you have a global foliation into non-singular hypersurfaces by setting t=const and the time never loops back on itself then that looks like you are already assuming something. Like, it rules out a PANTS picture because at some t = const there is the crotch----and the hypersurface has an irregular point where it is not a manifold---where it is almost but not quite disconnected into two leg-slices. There might be a THEOREM saying that if ADM can actually perform their construction globally that spacetime is already implicitly assumed to be such-and-such. I am not answering your question---just speculating for fun.
One would have to look at ADM original. Might be tantamount to assuming M = R x S.

 

[kakarukeys]

Hmm, will check the book.
ADM did not perform their construction (foliation of spacetime) globally but on a local chart that's what I'm thinking. And since the constraints, equations derived have the same form if we change to other local charts, the theory is 4-covariant, although they singled out the time direction.

 

[kakarukeys]

Hmm, will check the book.
ADM did not perform their construction (foliation of spacetime) globally but on a local chart that's what I'm thinking. And since the constraints, equations derived have the same form if we change to other local charts, the theory is 4-covariant, although they singled out the time direction.

 

[selfAdjoint]

Hmm, will check the book.
ADM did not perform their construction (foliation of spacetime) globally but on a local chart that's what I'm thinking. And since the constraints, equations derived have the same form if we change to other local charts, the theory is 4-covariant, although they singled out the time direction.

I've been wanting to post on this, but I see you have reached the same (and I believe the right) conclusion I had. That the foliation is only done locally and has no implications for the global topology of the spacetime manifold. This has always been my sense of the papers I have looked at in the canonical tradition of LQG, for example Thiemann's. Never never do they consider anything global, and even their work on black hole thermodynamics rests on "typical" vertexes piercing the horizon locally.

Now LQC is a different matter, but it is manifestly a simplified, enhanced symmetry model derived from LQG, not intended as a full-dress global theory.

 

[kakarukeys]

1. your original question was about S⁴. To me that does not make sense as a spacetime even for classical Gen Rel. Where is the big bang singularity, where is the big crunch singularity? Without even beginning to talk about LQG, and only thinking classical, I am dubious of S⁴. This is why I am not inclined to think of it as a flaw that canonical LQG does not handle that spacetime case.

Could you give me some references about this? I have been looking for this in GR books but can't find any discussion or proof that there exists a closed time-like curve in compact space-time and there is no big bang singularity in S^4.

 

[ccdantas]

Canonical LQG has origins in ADM and Ashtekar formulation of classical Gen Rel----these are based on a 3-manifold---same picture: R x M.
It certainly is some kind of a limitation. I don't know how serious that limitation is. Perhaps it depends on what application one has in mind, or what one wants to study.

Hello all,

I do not know if this helps. According to:

@Article{smo04,
author = "Smolin, L.",
title = "An invitation to Loop Quantum Gravity",
year = "2004",
eprint = "hep-th/0408048"
}

Pages 7-8:

"To define the kinematics of a loop quantum gravity theory, pick one from each of the following

- A topological manifold, Sigma, say S^3. (...)

There are a number of variants, depending on the specification of Sigma. We may specify that Sigma is a differential manifold, in which case the classes {Gamma} are equivalent up to diffeomorphisms (or, in some formulations, piecewise diffeomorphisms) of Sigma. If Sigma has a boundary, that can indicate an asymptotic region, or a horizon of a black hole. Or one can drop Sigma and build the theory just from combinatorial graphs."

Best wishes
Christine

 

[marcus]

thanks Christine! Nice to hear from you.

selfAdjoint has suggested one way of replying to kakarukeys concern: not to worry about the restriction to M = R x S because it is only LOCAL in the first place.

... That the foliation is only done locally and has no implications for the global topology of the spacetime manifold. ...

Now LQC is a different matter,...

So not a flaw in the sense of necessarily restricting the global topology.

OTOH you quote Smolin that there are several different developments of LQG including where there is no manifold S----no M = R x S---but instead the theory is built on combinatorial graphs. This also suggests no necessary restriction on global topology.

=================

my opinion is slightly different----I speculate that LQG is a a step in the right direction and something to learn from but has limitations to be overcome and one of these limitations IS IN FACT THE RESTRICTED TOPOLOGY
and I personally suspect that time is emergent as a largescale phenom and that gravitational collapse can fork time and so in effect GRAVITATIONAL COLLAPSE CAN CHANGE TOPOLOGY. but this is not known. To me it seems like a big unknown.
And so I conjecture that an especially good kind of LQG research to do is to model black hole gravitational collapse in LQG and I think this will eventually force the researchers to consider other models besides M = R x S
and the researcher will have to INVENT a non-cylinder model that looks more like "Y x S"
where time forks and one leg goes down the hole and re-expands to make a new section of the universe
where "Y" is a graphic symbol like the fork of a tree, and where
"YxS" is NOT ACTUALLY A CARTESIAN PRODUCT BUT SOMETHING ELSE

So I am not making statements here---these are just my speculations, if one can be permitted to speculate sometimes.

In my personal speculative opinion, which I will not try to justify, I think kakarukeys example of classical spacetime S⁴ is an uninteresting spacetime to think about. Personally I think it is bad---I would not bother to study it. Essentially because it is compact and so it is talking about a universe with finite lifetime and if you REMOVE TWO POINTS (which are the bang and crunch singularities) YOU JUST TOPOLOGICALLY GET A CYLINDER back again!

Remember that the singularities are not part of the spacetime where the metric is defined. So if you try to make a spacetime that looks like S⁴. and then you locate the singularities, then it is PUNCTURED and you don't have S⁴ any more------you just have R x S³----the same old CYLINDER.

So in my personal opinion S⁴ is an uninteresting spacetime to explore also in the LQG context.

But I would urge a young researcher to attack the general problem of extending LQG beyond the cylinder topology so that it can accommodate a possible FORKING of time caused by gravitational collapse. I would say first read what Bojowald et al and Ashtekar et al have to say about Black Hole QG-------Renate Loll also has something but I don't know if it is going anywhere. And maybe there is a subfield developing which is "BHQG" ----black hole qg.

notice that Bojowald and Ashtekar rarely mention the possibility of a forking scenario. the tendency is to try as hard as possible to accommodate BHQG WITHOUT forking. It is a risky research venture to consider this because it does not fit the normal cylinder formalism. and possibly for other reasons. But if one wants to TAKE risks, this is one way to go "out on a limb"

 

[ccdantas]

thanks Christine! Nice to hear from you.

Hello Marcus,

I very much enjoy this forum, I lurk here everyday
(luckily, this site has not been blocked yet from the server
of the institute where I work). Reading the discussions
here is a good way to start the day. Congratulations
for those involved in keeping this forum.

Best wishes
Christine

 

[Careful]

I'm not sure that is a flaw, kakarukeys.
One doesn't necessarily WANT a QG formulated on a spacetime that is topologically S⁴. This may be the message of C.'s post, in which case I concur.

Obviously I meant this, I think I made it very clear so far how I think about the role of topology change in quantum gravity.

C.

 

[George Jones]

Just when did canonical gravitists start to make that assumption?

I don't have my books with me (just moved - they're still in boxes), but didn't Geroch prove that any globally hyperbolic spactime M, i.e., a spcetime that has a Cauchy surface, can be written is the form M = R x S.

 

[ccdantas]

Hi,

Some references are:

Geroch, R., “Domain of dependence”, J. Math. Phys., 11, 417-449, (1970).

Antonio N. Bernal, Miguel Sánchez, "Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes" [http://arxiv.org/abs/gr-qc/0401112]

Bernal, A.N., and Sánchez, M., “On smooth Cauchy hypersurfaces and Geroch’s splitting theorem”, Commun. Math. Phys., 243, 461-470, (2003).
[http://arXiv.org/abs/gr-qc/0306108]

Christine

 

[ccdantas]

This is also a very nice reference:

http://arxiv.org/abs/gr-qc/0404084

Abstract:

Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the ideas of the solution.

 

[George Jones]

Geroch, R., “Domain of dependence”, J. Math. Phys., 11, 417-449, (1970)

I had forgotten about this classic paper. I have read it, and I may even have a copy of it in one of my boxes.

Thanks for the links to other papers; they look interesting.

 

[Haelfix]

Be sure there are topological restraints in QG on the ultimate type of manifold we live in. For instance, the existence of a spin structure must be satisfied by any theory claiming relevance to the real world (this is the vanishing of the 2nd Stiefel Whitney class constraint)

Now, S^4 could have been the topology of spacetime, but it seems like observationally that is no longer the case, so for the time being that is ok.

 

[kakarukeys]

globally hyperbolic spacetime implies some interesting and convenient features... including existence of Cauchy surface, splitting into R\times\Sigma.

Is this the reason why any Hamiltonian formulation on spacetime must contain the assumption that spacetime is globally hyperbolic? Or is there any physical motivation?

 

[Haelfix]

More or less yes.

Its not limited to hyperbolic spacetimes though, but you do need a good choice of foliation that doesn't break various energy conditions.