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Different Kinds of Metric 
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#19
Mar412, 03:31 AM

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There are three classes of FRW metrics, namely k=±1,0; only k=0 corresponds to a flat spacetime. You cannot describe a globally (topologically) different spacetime e.g. the closed k=+1 type by considering a flat k=0 type plus small oscillations.
Think about a function (for k=0) f(t) = ωt and try to find a small fluctuation that deformes it to (for k=+1) f(t) = sin(ωt) This is what you need to deform the k=0 to the k=+1 type. Of course this is impossible with a small fluctuation. 


#20
Mar412, 04:01 AM

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The FRW thing also proves that the plain particle physics approach of using flat spacetime plus spin2 fields = curved spacetime wont work. Therefore at least String theory and LQG need to be the minimum approach of quantum gravity? 


#21
Mar412, 04:11 AM

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When consideruing classical FRW I was talking about classical GR; it shows that even classical "flat spacetime + small oscillations" does not work for all classes of solutions of GR. It's the same with strings: they find a huge number of classes of "classical solutions" and consider small oscillations for each class. The big difference is that for different "classical solutions" one has to consider different oscillations, i.e. the particle content is different.
It's hard to compare ST and LQG here. ST always uses these "classical solutions" and constructs a quantum theory on top of them. So for each class you get a quantum theory, but in a sense you don't have the big picture, i.e. no unique quantum theory (string theory) describing all these classes at once. In LQG it's exactly the other way round. They claim to have a unique quantum theory of gravity (w/o particle physics) w/o the need to introduce these classes. But at the samepoint this is the weak point of LQG b/c they are not able to derive the classical solutions, i.e. they don't have a way to extract the classical physics with different spacetimes we know from GR (there are only very restricted models or methods from which they get rather specific spacetimes). 


#22
Mar412, 04:38 AM

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#23
Mar412, 08:42 AM

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The FRW metric can definitly not be derived using flat space + perturbative strings; I don't know which realistic vacuum solutions have been constructed so far, but AdS is not the right way to go b/c AdS is not realistic; with a positive cosmological constant one needs dS instead, for FRW w/o cc again something different.



#24
Mar412, 05:28 PM

P: 381

atyy, I can't send private message to you because you somehow disable the feature. I just wanted to ask you what exact pages in the MTW book Gravitation one can find the statements that the FRW Universe satisfy harmonic coordinates and that all spacetime that is covered by harmonic coordinates can be decomposed into flat spacetime + spin2 field. Exact pages please. Thanks.



#25
Mar412, 06:00 PM

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#26
Mar412, 07:40 PM

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#27
Mar512, 12:55 AM

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If you want to study flat spacetime + small fluctuations you can start with FRW (k=0) + small gravitational waves; you will be able to explore the k=0 sector. If you want to study a universe with big crunch you have to study the k=1 sector, again with some fluctuations. You will never be able to study a big crunch by starting in the k=0 sector and introduce small fluctuations. k=±1 have nothing to do with dS or AdS; FRW (k=±1) are two solutions w/o cosmological constant, i.enot (A)dS; in order to study (A)dS you have to introduce a cosmological constant but this is not what I would call FRW. So you get new topologically different spacetimes 


#28
Mar512, 03:16 AM

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