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General covarianceby zonde
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#91
Oct2612, 03:50 PM

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Of course, because the field equations of NC just reproduce Newtonian gravity, there will be no time dilation. It also depends on what one calls "dynamics"; usually metric compatibility is not considered to be dynamics. http://arxiv.org/abs/1206.5176 These theories are not the usual Newtonian limits of GR, so in that sense GR (with the possible additional terms to the Einstein Hilbert action as Bien Niehoff mentions) doesn't seem to be the only BI theory. 


#92
Oct2612, 04:12 PM

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Anyway, I have the ambition to, once I thoroughly understand all this stuff, put it in some notes without all the usual explicit (often coordinatefree) mathematical mumbojumbo and vague terminology obscuring for me personally what's really going on. Somehow I still haven't found a nice and clear overview of the meaning of covariance, the meaning of and relation between active and passive coordinate transformations, etc. Even good books like Carroll couldn't really satisfy my needs. I sometimes have the feeling that a lot of physicists don't really care, and a lot of "philosophers of physics" make the discussion so obscure that it makes me wanting to run back to the "shut up and calculate" mentality :D



#93
Oct2612, 07:33 PM

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http://arxiv.org/abs/grqc/0603087 This makes a better attempt than most I've seen to try formalize what distinguishes GR from e.g. NewtonCartan (for example). Unfortunately, its overall conclusion is that the matter is not yet resolved, after all these years; that ultimately, background independence, no prior geometry, no absolute structures, etc. is not yet subject to any rigorous, problem free definition. 


#94
Oct2712, 12:16 AM

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What I'm not sure about is: does the field on flat spacetime contain cosmology? No need to include the "big bang singularity", but just the physically relevant bits that present observations constrain? 


#95
Oct2712, 03:39 AM

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One of the reasons why I got so interested in this whole notion of "background independence" was because I heard the claim from some string theory critics that "any good theory of quantum gravity should be BI" and "string theory is not BI". But the meaning of this becomes, after these discussions, a bit blurry to say the least. (The primary reason was actually that for my master thesis I had to read Wald's article on "black hole entropy is Noether charge". It then occured to me I'd never really understood this whole business.) 


#96
Oct2912, 06:06 AM

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Note the difference between the two following definitions... 1) Background dependance is tantamount to using the background field method for gravity, and ONLY for gravity (eg the metric tensor is split into a classical but arbitrary fixed background metric + a small perturbation). The approximation is valid up to some cutoff, whereupon the backreaction of the pertubation on the background can no longer be ignored. 2) Background independance is like asking whether the metric field is dynamical or not in the Lagrangian of the theory. In the sense that if you look at the variation in the action and consider (d/d&G), then you look for something that vanishes. So for instance, coupling a topological field theory to a theory with curvature invariants is clearly background independant in this definition. The terms with curvature invariants, owing to their general covariance, will integrate out any metric dependance, and terms that are topological have no metric dependancy at all. Contrast that with something like a Maxwell term, which when acted with the operator, will instantly pull out the nondynamical and absolute fixed structure. Both definitions (as well as anyone that you can think off) are not going to generalize universally, or serve as a theory 'filter'. The first problem is that the word 'background' is often generalized in the literature to mean something more than just a classical solution of Einstein's equations. Second, its a little bit unclear what physical principle you are trying to capture that is so damn important, considering that even classical GR can be written in ways that make it look background dependant. (Consider writing GR like field theorists for the first case and consider the pure connection formalism for the 2nd) 


#97
Dec1412, 09:07 AM

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I finally read the paper by Giulini, and it is nice. He defines NC gravity being not background independent because of the appearence of "absolute structures": most of the metric components of NC gravity only have 1 solution (modulo gct's), whereas all the other components are gathered into the Newton potential.
I think I found some nice insights in the paper :) It's also nice to compare the generalcovariantization of the Poisson equation (giulini does it for the Schrodinger equation, but the difference is only a time derivative) with the formulation of NewtonCartan. The latter can be seen as a much less trivial generalcovariantization of Newton. 


#98
Dec1712, 12:13 PM

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You can formulate just about any theory of physics in a generally covariant form, which really means writing it in terms of geometric objects that can be defined independently of a choice of coordinates: Scalar fields, vector fields, tensor fields. The theory is "background free" if there are no nondynamic scalar, vector or tensor fields. By "nondynamic", I mean a field that appears in the equations of physics (when written in generally covariant form) but which is not itself governed by the physics. For example, in Newtonian physics, universal time is a scalar field that is nondynamic. In Special Relativity, the metric tensor is a tensor field that is nondynamic. General Relativity has no nondynamic fields. 


#99
Dec1812, 01:52 AM

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What is true, is that up to gct's this temporal metric (with lower indices) only has one solution. That's a hint that it is nondynamical, and that the theory has been "Stückelberged". All the metric components of NCtheory turn out to be nondynamical this way, except for a combination of components which form the Newton potential. This potential does have more solutions up to gct's, and as such is the only dynamical field in the theory. In your definition I can always postulate EOM for the nondynamical field. The simplest example for this was already given; if one has a Minkowksi metric and partial derivatives in a theory, just generalcovariantize this and impose as extra EOM that the Riemann tensor for this metric vanishes. 


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