Einstein was wrong, and should be Cartanized! ( this is professional research)

marcus

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Einstein was wrong, and should be Cartanized! (:D this is professional research)

that's what I get from Derek's paper----classical 1915 Gen Rel is wrong because it uses a vintage 1850s manifold which has flat tangent spaces that don't roll.

It should be moved over and put on a Cartan manifold with a deSitter model geometry that can roll around on it up and down over the hills and thru the valleys.

Minkowski flat is a bad local model because it doesn't even expand so it can't adhere right to realistic spacetime---and Minkowski flat needs to be replaced by deSitter.

Going to a Cartan manifold will improve classical Gen Rel and it has some associated perks. Plus then quantizing will go better-----and the classical limit of QG will not be Gen Rel but will be a kind of corrected Cartanized Gen Rel.

Overall that is what Derek's paper suggests to me. I want to put in this thread some of the advantages that seem to accrue to cartanizing GR. If you have some different interpretations, please contribute them!

what got me excited just now was something on page 28----this is just one of many things but I'll quote it:

"...The former seems superficially rather different: a 1-form not on spacetime M, but on some principal bundle P over M, with values not in a vector bundle, but in a mere vector space g/h.

To understand the relationship between these, we first note that from the Cartan perspective, there is a natural choice of fake tangent bundle T . To be concrete, Consider the case of Cartan geometry modeled on de Sitter spacetime, so G = SO(4, 1), H = SO(3, 1). The frame bundle FM → M is a principal H bundle,..."

source:
http://arxiv.org/abs/gr-qc/0611154
MacDowell-Mansouri gravity and Cartan geometry
Derek K. Wise
34 pages, 5 figures

having a natural choice for the "internal space", or fake tangent space at each point, relieves a concern I had at the very outset learning LQG in 2003---if I understand this correctly, it offers to resolve an awkwardness in the QG formulation that has bothered me, at least, and possibly others---having to do with "internal indices" and the need for arbitrary choice of basis at every point. this looks like it might be cleaner.

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I could, of course, be wrong about this, but I tend to suspect that people are on the right track with COVARIANT LQG (people like Sergei Alexandrov, Etera Livine, Philippe Roche, Eric Buffenoir ) which uses the Lorentz group instead of the compact group SU(2) and which gets away from the Immirzi. There is also the possibility that they will not stop at the Lorentz group but will go over to the deSitter group. Could they connect then with what Derek is talking about?----Cartan gravity.
 
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marcus

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The connection and the vielbein are treated as independent quantities. Their relation is one of the equations of motion. When the coupling with spinor matter is present, this relation differs from that in Riemann geometry. Thus one obtains the Einstein-Cartan theory, in which torsion may be nonzero at points at which fermionic matter is present. Indeed, in the paper, the authors explicitly claim that they obtain the Einstein-Cartan theory in the limit in which the Immirzi parameter goes to infinity.
 

marcus

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thanks, Demy!
 

marcus

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I am very proud that I can explain something about LQG to the best LQG expert on this forum. :smile:
Demy, instead of expert, I believe I'm what's sometimes called a "science-watcher"----a kind of amateur journalist, news-gatherer, and librarian of source material. I don't feel as if I have to apologize. The observers on the side-lines play a role.

Just so we avoid any chance of confusion, sometimes real experts come here-----by that I mean people actually doing QG research. Etera Livine, Derek Wise, Garrett Lisi most recently.
Maybe you count as a QG researcher. I can't keep track of what the various people are doing. Anyway Etera, Derek and Garrett certainly count as QG researchers and experts---and there may be other recent visitors. Hope I haven't forgotten anyone who will feel disappointed.

That said, thankyou for the compliment! It feels good to be called expert, even if we both know it is only half-seriously.
 
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