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Different Kinds of Metric

by waterfall
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tom.stoer
#19
Mar4-12, 03:31 AM
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P: 5,364
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.
waterfall
#20
Mar4-12, 04:01 AM
P: 381
Quote Quote by tom.stoer View Post
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.
So the solution to this is that Strings don't need to first be Ricci-flat and add spin-2 field. The solution is to directly solve for the FRW from first principle without going to this Ricci-flatness?

The FRW thing also proves that the plain particle physics approach of using flat spacetime plus spin-2 fields = curved spacetime wont work. Therefore at least String theory and LQG need to be the minimum approach of quantum gravity?
tom.stoer
#21
Mar4-12, 04:11 AM
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P: 5,364
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).
waterfall
#22
Mar4-12, 04:38 AM
P: 381
Quote Quote by tom.stoer View Post
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).
Are you saying that at this point in time in our String Theory. We still don't know how to even derive at the FRW metric using Perturbative String Theory and it can only be solved by going into the ongoing non-perturbative AsD/CFT path?
tom.stoer
#23
Mar4-12, 08:42 AM
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P: 5,364
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.
waterfall
#24
Mar4-12, 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 + spin-2 field. Exact pages please. Thanks.
atyy
#25
Mar4-12, 06:00 PM
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P: 8,332
Quote Quote by waterfall View Post
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 + spin-2 field. Exact pages please. Thanks.
The reference I gave was Weinberg, not MTW. But in fact, I misread it, so I don't know if the FRW solution can be expressed in harmonic coordinates. Also, it seems that in 1963, Feynman was unsure if gravity as a field on flat spacetime could handle cosmology http://books.google.com/books?id=tnp...gbs_navlinks_s (p187).
atyy
#26
Mar4-12, 07:40 PM
Sci Advisor
P: 8,332
Quote Quote by tom.stoer View Post
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.
So the FRW for k=0 can be described as large perturbations from flat spacetime (at least classically)? And the other two recovered as large perturbations from de Sitter and Anti de Sitter spacetimes?
tom.stoer
#27
Mar5-12, 12:55 AM
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P: 5,364
Quote Quote by atyy View Post
So the FRW for k=0 can be described as large perturbations from flat spacetime (at least classically)? And the other two recovered as large perturbations from de Sitter and Anti de Sitter spacetimes?
I am not sure if you understand correctly.

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
Mentz114
#28
Mar5-12, 03:16 AM
PF Gold
P: 4,087
Quote Quote by atyy View Post
The reference I gave was Weinberg, not MTW. But in fact, I misread it, so I don't know if the FRW solution can be expressed in harmonic coordinates. Also, it seems that in 1963, Feynman was unsure if gravity as a field on flat spacetime could handle cosmology http://books.google.com/books?id=tnp...gbs_navlinks_s (p187).
In this paper arXiv:gr-qc 0705.0080 Michael Ibison finds a transformation to harmonic coords for the FLRW k=1, k=-1. I can follow the paper but I found that the k=-1 case in harmonic coords admits a vacuum solution with expansion ( like the Milne chart). The corresponding standard k=-1 FLRW does not have a vacuum solution. So it looks as if the change of coords has changed the spacetime. Is this possible ? This is the metric equation (73) in Ibison's paper.


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