Advanced books/papers on derivation of Newtonian mechanics from GR

[Juan R.]

Forgive me if I'm missing something obvious, but if you can recover NG from asymptotic flatness, can't you then also recover NG when the system of interest is sufficiently approximable by an asymptotically flat space-time?

"Asymptotic flatness" and "an asymptotically flat space-time" are the same.

 

[Hurkyl]

Yes. My point is that it appears to me that you shouldn't need to assume space-time is asympotitcally flat, just that the system of interest is sufficiently approximable by some space-time that is asymptotically flat.

 

[Juan R.]

Yes. My point is that it appears to me that you shouldn't need to assume space-time is asympotitcally flat, just that the system of interest is sufficiently approximable by some space-time that is asymptotically flat.

I will explain again

In Ehlers theory one needs fix the "gauge" via a boundary. The boundary IS and only IS

R ---> infinite Phy(x, t) = 0

This is asymptotic flatness and mean that when you look very far (R ---> infinite) in the cosmos, the density of matter may be less and less and less and less (Phy(x, t) = 0). Until that at very far distance universe may be basically a vacuum with no matter-energy. This is called an island universe model

but an island universe is

etc ____________________XXXXXXXXXXXXX_________________ etc

and our universe does not look that!

Christian is crystal clear

universe is not "an island of matter surrounded by emptiness"

Penrose is also clear

Our universe is not of island type

To assume the system of interest is sufficiently approximable by some space-time that is asymptotically flat does not work by two motives. 1) First that boundary is not a approximation, it is the needed boundary for working in NC theory. 2) Experimentally is false.

But if you take any other boundary (e.g. obtained from experimental cosmology) for example if you take any other boundary as

etc X___X__X_____X___X_____X___X_ etc

then you cannot obtain NG (exactly the 4D version). It is so simple like that!

GR derivation of NG is no rigorous

Alternative derivation via Cartan theory only work for

island universes

etc ____________________XXXXXXXXXXXXX_________________ etc

or for our universe if one introduces ad hoc equation that cannot be derived from GR


Again i ask (this is the #53 post)

any textbook or paper on GR where the Newtonian limit was rigorously derived from GR. I mean "derivation". That is, without unphysical boundaries, ad hoc equations outside from GR, and incorrect derivations like that of typical textbooks?

 

[JesseM]

False, universe is not spatially flat at large distances (i think that you are mixed by homogeneity and isotropy at large distances which are OTHERS concepts)

Huh? I thought all the astronomical data from WMAP and so forth suggested space was as close to flat as the resolution of the data allowed them to conclude. For example, this section of the WMAP homepage says:

The WMAP spacecraft can measure the basic parameters of the Big Bang theory including the geometry of the universe. If the universe were open, the brightest microwave background fluctuations (or "spots") would be about half a degree across. If the universe were flat, the spots would be about 1 degree across. While if the universe were closed, the brightest spots would be about 1.5 degrees across.

Recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know that the universe is flat with only a 2% margin of error.

Wouldn't an island of matter surrounded by emptiness imply negative curvature rather than flatness? Don't you need a certain density of matter/energy spread throughout all of space in order to keep the universe flat?

Please explain what's wrong with this argument instead of just bugging your eyes out. Like I said, I acknowledge I'm not a GR expert. The only places I've seen anyone discuss a universe empty of matter and energy is the DeSitter cosmology which is negatively curved, although it also has a nonzero cosmological constant. So what would a universe empty of matter/energy but with no cosmlogical constant look like? Perhaps it would be flat, but that doesn't necessarily demonstrate that the assumption of asymptotic flatness is equivalent to saying that the rest of the universe outside the system you're considering is empty, since flatness is also compatible with a nonzero density of matter/energy throughout space, as in the flat case of the FRW cosmological model. Again, if I'm misunderstanding something here, please explain why instead of ridiculing me.

Do you have any references where other physicists treat asymptotic flatness as equivalent to the "island universe" assumption, or is this purely your own way of thinking about it?

 

[Juan R.]

Still i would add that if we travel to any "parallel" universe where asymptotic flatness was the correct boundary. Ehlers works would continue to be wrong.

1) There are violations of causality and many mathematical and physical errors in his approach.

2) taking the limit c --> infinite, curvature of space time is zero.

g00 --> 1 and gRR --> -1, therefore, Rab = 0.

Acording to Newton theory there is gravity, According to GR cannot there exist gravity in a non-curved spacetime.

 

[Juan R.]

Please explain what's wrong with this argument instead of just bugging your eyes out. Like I said, I acknowledge I'm not a GR expert. The only places I've seen anyone discuss a universe empty of matter and energy is the DeSitter cosmology which is negatively curved, although it also has a nonzero cosmological constant. So what would a universe empty of matter/energy but with no cosmlogical constant look like? Perhaps it would be flat, but that doesn't necessarily demonstrate that the assumption of asymptotic flatness is equivalent to saying that the rest of the universe outside the system you're considering is empty, since flatness is also compatible with a nonzero density of matter/energy throughout space, as in the flat case of the FRW cosmological model. Again, if I'm misunderstanding something here, please explain why instead of ridiculing me.

I'm sorry.


____i___________________XXXXXXXaXXXXXXXXX_____________________

In (i) curvature is zero, there is no matter. In (a) curvature is non zero, there is matter. X does not mean "uniform" matter.

Again you are fixed in specific cosmological models when i and other are talking of boundaries.

boundaries boundaries boundaries boundaries boundaries boundaries boundaries boundaries boundaries boundaries boundaries boundaries

It is irrelevant what cosmological model you prefer the boundary condition that may be verified is the same and EXPERIMENTALLY is false. It is not a bout any specific cosmological model or theory is about experiment.

Do you have any references where other physicists treat asymptotic flatness as equivalent to the "island universe" assumption, or is this purely your own way of thinking about it?

I already did that and cited. It is really obvious that asymptotic flatness is equivalent to the "island universe" assumption.

"Flat" there mean flat Robertson Walker line. Do not asymptotic flat spacetime like in SR for large distances.



The problem with flatness to R ---> infinite continues to be correct, independently of cosmological model used. Would i to say another 20 times.

i see that you do not understand , but is not my problem!

 

[JesseM]

I'm sorry.


____i___________________XXXXXXXaXXXXXXXXX_____________________

In (i) curvature is zero, there is no matter. In (a) curvature is non zero, there is matter. X does not mean "uniform" matter.

Is this a representation of the island universe? Why is curvature zero at i, then? Again, in the DeSitter model you have no matter in the entire universe yet the curvature is nonzero, while in the flat case of the FRW model you have matter throughout space yet the curvature of space (though not spacetime) is zero. I'm not saying it's wrong that an empty universe with no cosmological consant would be flat, but I just want to check and make sure that this is actually what GR predicts, since you haven't directly addressed this question yet. This section from Ned Wright's cosmology tutorial says that "For Omega less than 1, the Universe has negatively curved or hyperbolic geometry", where Omega is the ratio between the universe's matter/energy density and the critical density needed to prevent collapse--wouldn't an empty universe have Omega=0, and therefore be negatively curved rather than flat?

Again you are fixed in specific cosmological models when i and other are talking of boundaries.

But boundary conditions at infinity involve implicit assumptions about the cosmology, don't they? After all, if you just consider a local region of space a few light-years across that contains the solar system and nothing else, then the solar system is an "island universe" within this limited region, but that's not enough to tell you what boundary condition to use at infinity, is it? If space is flat at the largest scales, as the WMAP data I mentioned in my last post suggests, why doesn't that justify the assumption that space approaches flatness at infinity when making calculations involving the solar system? (Then to justify the assumption that spacetime approaches flatness you could just note that the expansion of space is pretty negligible on the scale of the solar system, so it's no wonder Newton didn't need to take it into account.) Or if it's true that a universe with no matter and no cosmological constant would be spatially flat, then does that mean that even in a universe with overall positive or negative curvature, if you have a large region empty of matter the inside of the region would be close to flat? If so, in that case perhaps you could justify the assumption of asymptotic flatness without reference to cosmology, just by considering a system in such an empty region.

It is irrelevant what cosmological model you prefer the boundary condition that may be verified is the same and EXPERIMENTALLY is false.

What experiments prove it false? If you're talking about proving the "island universe" false, aren't you referring to astronomical observations of how matter is distributed on the largest scales--ie observations about cosmology?

Do you have any references where other physicists treat asymptotic flatness as equivalent to the "island universe" assumption, or is this purely your own way of thinking about it?

I already did that and cited.

Which specific post/citation are you referring to?

"Flat" there mean flat Robertson Walker line. Do not asymptotic flat spacetime like in SR for large distances.

Again with the ridicule. By "there" do you mean the reference to WMAP I gave? I understand the distinction between flat spacetime and flat space if that's what you're talking about (I specifically used the words 'flat space' many times to avoid confusion), but do you agree that the evidence supports the idea that space is flat? As for spacetime, like I said you can just consider the limit as cosmological time approaches infinity in a spatially flat universe with no cosmological constant. In this case, I didn't think the density of matter/energy approaches zero since I know two observers in such a universe can communicate forever which I thought meant the distance between them would approach some finite value, although I may be misunderstanding something there (I suppose it might be that although the distance between any two observers is increasing without bound, the rate of expansion is shrinking fast enough so that they are never moving apart faster than light no matter how far apart they get...the amount that the distance between them increases in a given unit of time would have to be a decreasing series with no upper bound, like 1/2 + 1/3 + 1/4 + 1/5 + ...).

 

[Hurkyl]

To assume the system of interest is sufficiently approximable by some space-time that is asymptotically flat does not work by two motives. 1) First that boundary is not a approximation, it is the needed boundary for working in NC theory. 2) Experimentally is false.

(1) So what? The boundary is far, far away from the system of interest -- we don't care if the actual space-time occupied by the system of interest has a boundary that resembles that of an asymptotically flat space-time -- all we care about is that the region of the actual space-time occupied by the system of interest resembles some region of the asymptotically flat approximation.

(2) How can the existence of a mathematical approximation be experimentally false?

 

[pervect]

Forgive me if I'm missing something obvious, but if you can recover NG from asymptotic flatness, can't you then also recover NG when the system of interest is sufficiently approximable by an asymptotically flat space-time?

Probably you can, if you can untangle the limits. One is already taking a limit to define asymptotic flatness, to approach asymptotic flatness as a limit is to take the limit of a limit. It's probably not impossible, but I don't know of anyone who has done it and written it up in a paper.

 

[Juan R.]

Is this a representation of the island universe? Why is curvature zero at i, then? Again, in the DeSitter model you have no matter in the entire universe yet the curvature is nonzero, while in the flat case of the FRW model you have matter throughout space yet the curvature of space (though not spacetime) is zero. I'm not saying it's wrong that an empty universe with no cosmological consant would be flat, but I just want to check and make sure that this is actually what GR predicts, since you haven't directly addressed this question yet. This section from Ned Wright's cosmology tutorial says that "For Omega less than 1, the Universe has negatively curved or hyperbolic geometry", where Omega is the ratio between the universe's matter/energy density and the critical density needed to prevent collapse--wouldn't an empty universe have Omega=0, and therefore be negatively curved rather than flat?

I cannot study for you.

But boundary conditions at infinity involve implicit assumptions about the cosmology, don't they?

I (curiously like others: e.g. Penrose, Ehelers, Crhstians, etc.) are talking about boundaries. It involves boundaries of our universe. The specific cosmological model taked says little about that. It is a pure question of observation.

After all, if you just consider a local region of space a few light-years across that contains the solar system and nothing else, then the solar system is an "island universe" within this limited region, but that's not enough to tell you what boundary condition to use at infinity, is it?

Yes the solar system is an "island universe" for finite radius, but if you continue to see beyond you again find matter. Therefore that is not Ehlers boundary.

If space is flat at the largest scales, as the WMAP data I mentioned in my last post suggests, why doesn't that justify the assumption that space approaches flatness at infinity when making calculations involving the solar system?

No, space is not flat at "large distances" (you say is wrong) and that data says nothing about boundaries. You do not understand diference between a boundary, the Ehlers boundary (which is, -would i say again?-, experimentally false: read Penrose, read Christyan read, read) and the fact that average density of matter is close to zero in a homogeneous isotropic cosmological model of universe.

Christian is crystal clear

universe is not "an island of matter surrounded by emptiness"

Penrose is also clear

Our universe is not of island type

I also am being clear. Do you not know that is a boundary, what is a cosmological model, what is the RW line element...

What experiments prove it false?

Direct observation.

Which specific post/citation are you referring to?

It is obvious. No?

Again with the ridicule. By "there" do you mean the reference to WMAP I gave? I understand the distinction between flat spacetime and flat space if that's what you're talking about (I specifically used the words 'flat space' many times to avoid confusion), but do you agree that the evidence supports the idea that space is flat? As for spacetime, like I said you can just consider the limit as cosmological time approaches infinity in a spatially flat universe with no cosmological constant. In this case, I didn't think the density of matter/energy approaches zero since I know two observers in such a universe can communicate forever which I thought meant the distance between them would approach some finite value, although I may be misunderstanding something there (I suppose it might be that although the distance between any two observers is increasing without bound, the rate of expansion is shrinking fast enough so that they are never moving apart faster than light no matter how far apart they get...the amount that the distance between them increases in a given unit of time would have to be a decreasing series with no upper bound, like 1/2 + 1/3 + 1/4 + 1/5 + ...).

Sorry, is not ridicule, simple you are very amazing.

"I understand the distinction between flat spacetime and flat space if that's what you're talking about"

No, you do not understand, it is clear that i was talking. My phrase was precise and unambigous "flat Robertson Walker line"

The evidence supports idea that an average metric of the whole universe (i.e a homogeneous isotropic cosmology) is close to that of a flat universe, but say nothing about boundaries.

 

[Juan R.]

(1) So what? The boundary is far, far away from the system of interest -- we don't care if the actual space-time occupied by the system of interest has a boundary that resembles that of an asymptotically flat space-time -- all we care about is that the region of the actual space-time occupied by the system of interest resembles some region of the asymptotically flat approximation.

(2) How can the existence of a mathematical approximation be experimentally false?

(1) There is no experimental evidence. In fact, the experimental evidence indicates no sign of a island type model of universe, that is, with matter vanishing more and more. The rest of your claim is wrong.

(2) Is not a mathematical approximation, it is the boundary in NC gravity for it works. Experimentally is false, since our universe is not of island type.

 

[JesseM]

Juan R. refuses to answer my question...is anyone else willing to field this one? I just want to know what GR predicts about the curvature of a universe wholly empty of matter and energy, since it would probably predict something similar about an "island universe" which was wholly empty except for one clump of matter.

I'm not saying it's wrong that an empty universe with no cosmological consant would be flat, but I just want to check and make sure that this is actually what GR predicts, since you haven't directly addressed this question yet. This section from Ned Wright's cosmology tutorial says that "For Omega less than 1, the Universe has negatively curved or hyperbolic geometry", where Omega is the ratio between the universe's matter/energy density and the critical density needed to prevent collapse--wouldn't an empty universe have Omega=0, and therefore be negatively curved rather than flat?

 

[Berislav]

Of course, but Wald does not take the limit c --> infinite, because then the metric g00 = 1 gRR = 1. That is FLAT and cannot explain gravitation.

Please check the metric again. You will see that that is not what happens to it. And I think that Wald didn't take that limit because he dealt with SR and Newtonian approximations in one of the previous chapters, so he assumed that the reader understands that that is how one reduces to Newtonian physics from relativity.

Perhaps i explained bad. Precisely in NG one would do the instantaneous change Phy --> Phy/2. Whereas in GR only after of 8 minutes one would change the potential, of course in GR the change (after of the 8 minutes) is not Phy --> Phy/2 it would be more gradual. Any case both description are different and this is reason that Wald equation is not Newtonian equation.

Yes, of course, that will happen in pure GR and in reality, but if you take c--> infinity, then it will propagate instantly.

I can accept the gauge of Phy in GR but Phy in NG is rather physical at least if one take the integration constant equal to zero which is always done. I believe that "Unphysical" is not the correct expression, because in NG the potential is Energy by unit of mass of test body and that is physical, of course i know that one could redefine energy using a new zero for the scale, but one definition would not be more physical that other and one take the integration constant zero by commodity.

The gradient of the potential is what is physical, not the potential, and hence you can add any constant to the potential; and that's what any leftover constant after you take a limit is.

Dirac is cristal clear. There exit two inconsistent theories: one for non relativistic phenomena, other for certain relativistic phenomena.

The basics of QED, which Dirac mentions, are derived from Maxwell's laws and quantum physics (and a second quantization). You don't even have to mention SR, per se, as Maxwell's equations are relativistically covariant. Furthermore, the Klein-Gordon equation, for instance, is just a relativistic version of the Schrödinger equation, and a reduction from the former to the latter is simple. So, I don't really think that Dirac was talking about what you point out as a problem. So, please, if you could quote Dirac on what the exact problem is, that would be great.

Probably you can, if you can untangle the limits. One is already taking a limit to define asymptotic flatness, to approach asymptotic flatness as a limit is to take the limit of a limit. It's probably not impossible, but I don't know of anyone who has done it and written it up in a paper.

I don't understand. Could you please elaborate for me?

I just want to know what GR predicts about the curvature of a universe wholly empty of matter and energy

If it has a non-zero cosmological constant then this is a de Sitter universe, otherwise it is a Minkowski spacetime, meaning, among other things, that it is static.

P.S.
This 'island universe' assumption sounds to me like just a logical assertation that nothing outside our cosmological horizon can affect us.

 

[pervect]

Juan R. refuses to answer my question...is anyone else willing to field this one? I just want to know what GR predicts about the curvature of a universe wholly empty of matter and energy, since it would probably predict something similar about an "island universe" which was wholly empty except for one clump of matter.

Interesting question. I think the answer is that you can view a universe that's totally empty of matter either as a non-expanding Minkowsky universe, or an expanding Milne universe.

Obviously the solution must be homogeneous and isotropic, because a vacuum is homogeneous and isotropic. So you should get some sort of FRW cosmology.

A static Minkowski space-time will satisfy this, and so will an expanding Milne universe, which has a spatial curvature k=-1 and a uniform scale factor a(t) = t. IIRC the two are equivalent, they represent a different coordinate system for the same space-time.

I haven't worked this out as carefully as I might, the answer is a bit "off the cuff".

 

[JesseM]

If it has a non-zero cosmological constant then this is a de Sitter universe, otherwise it is a Minkowski spacetime, meaning, among other things, that it is static.

But isn't it true that in the FRW cosmological model, a universe with zero cosmological constant will be negatively curved if Omega is less than 1? The diagram at the top of this page from Ned Wright's cosmology tutorial shows a universe with Omega<1 having negative curvature, and in the paragraph below he says "These a(t) curves assume that the cosmological constant is zero". What am I misunderstanding here?

 

[Berislav]

or an expanding Milne universe.

I didn't mention that spacetime because I never heard about it.
I apologize.

 

[pervect]

I don't understand. Could you please elaborate for me?

The notion of how to make this statement mathematically rigorous bothers me a bit.

The limit of f(x) as x-> a is well definied. But how do you take the limit of all possible maniolds as they "approach flatness"? I suppose we can do this if we have a distance measure between manifolds. How do we construct this distance measure?

Perhaps part of the answer is that we are assuming we have a map from from the manifolds to a single scalar number, which is a "measure" of the "flatness" of the manifold. Can we really rigorously construct this measure? How do we go about it, exactly? I.e. I give you a manifold, and you take out your measuring instrument and you say "The flatness of that manifold is 22" - how do we accomplish this?

 

[JesseM]

Interesting question. I think the answer is that you can view a universe that's totally empty of matter either as a non-expanding Minkowsky universe, or an expanding Milne universe.

Obviously the solution must be homogeneous and isotropic, because a vacuum is homogeneous and isotropic. So you should get some sort of FRW cosmology.

A static Minkowski space-time will satisfy this, and so will an expanding Milne universe, which has a spatial curvature k=-1 and a uniform scale factor a(t) = t. IIRC the two are equivalent, they represent a different coordinate system for the same space-time.

I haven't worked this out as carefully as I might, the answer is a bit "off the cuff".

Interesting, so there are two separate solutions to this problem (the diagram on Ned Wright's page which I mentioned above seems to show the expanding universe with a(t)=t), but they can be made equivalent by a coordinate transformation? In the coordinate system that treats this as an expanding Milne universe, is space indeed negatively curved rather than flat?

Also, does the expanding Milne universe have an initial singularity, and if so, does the fact that it can be transformed into a minkowski spacetime mean this is just a coordinate singularity rather than a "real" singularity?

 

[pervect]

Interesting, so there are two separate solutions to this problem (the diagram on Ned Wright's page which I mentioned above seems to show the expanding universe with a(t)=t), but they can be made equivalent by a coordinate transformation? In the coordinate system that treats this as an expanding Milne universe, is space indeed negatively curved rather than flat?

That's what I read on the internet http://web.mit.edu/8.286/www/quiz00/e6qs3-1.pdf

but I haven't double checked this or thought about it much yet. Since this is a student-written "quiz response" it's worth double checking it, though it seems right on this point. I'm not sure how long it will be up, hopefully for a few days at least, these sort of things tend to disappear without notice.

The quiz response above also made some interesting statements about Birkhoff's theorem and how it applies to Juan's dilema, but I'm not sure I believe them yet as an accurate statement of the theorem.

Hopefully I'll post more later, after I've had coffee, breakfast, and bashed a few metrics through GRTensor.

 

[Berislav]

I'll attempt to fill in for pervect in the mean time.

The Robertson-Walker metric is derived by assuming homogenity and isotropy (nothing else about the content of the universe):

ds^2=-dt^2+R^2(t) (\frac{dr^2}{1-kr^2}+r^2d\Omega^2),

where k can be anything, but we can redefine it as being either -1,0, or 1.

Now it follows from the Einstein field equations that:

(R&#039;/R)^2=-k/R^2+\frac{8 \pi \rho}{3},
where R' denotes the derivative with respect to time and \rho the density of the perfect fluid in the universe. Now if k=0 and there is nothing in the universe it easily follows that R'=0 and the universe is static and flat. If k is something else it will be positively curved and shrinking, or negatively curved and expanding.

I should have added that de Sitter universe assumes a positive cosmological constant.

Oops. Notation mistake.

 

[George Jones]

Milne's universe is just an interesting coordinate system on a proper subset of Minkowski spacetime that"splits" the subset into time and space. In this splitting, the spatial curvature for space at any "instant" of time is negative, but spacetime curvature is zero, as it must be. It is impossible to transform zero spacetime curvature into non-zero spacetime curvature by a change of coordinates.

Regards,
George

 

[Hurkyl]

The limit of f(x) as x-> a is well definied. But how do you take the limit of all possible maniolds as they "approach flatness"? I suppose we can do this if we have a distance measure between manifolds. How do we construct this distance measure?

Well, the question of interest is the predictions of the two space-times on the system of interest -- the appropriate distance would be the deviation of the predictions... maybe the maximum of the differences of the appropriate derivatives over some compact subset as defined by your favorite coordinate charts.

 

[George Jones]

But isn't it true that in the FRW cosmological model, a universe with zero cosmological constant will be negatively curved if Omega is less than 1? The diagram at the top of this page from Ned Wright's cosmology tutorial shows a universe with Omega<1 having negative curvature, and in the paragraph below he says "These a(t) curves assume that the cosmological constant is zero". What am I misunderstanding here?

Bersislav was talking about empty universes while standard FRW universes are not empty.

Also, I am not sure what you mean by curvature of the universe. In FRW models, k = -1, 0, 1 refers to curvature of spatial sections, not to curvature of spacetime.

Regards,
George

 

[JesseM]

Milne's universe is just an interesting coordinate system on a proper subset of Minkowski spacetime that"splits" the subset into time and space. In this splitting, the spatial curvature for space at any "instant" of time is negative, but spacetime curvature is zero, as it must be. It is impossible to transform zero spacetime curvature into non-zero spacetime curvature by a change of coordinates.

But what you and pervect's reference are saying is that spatial curvature, as opposed to spacetime curvature, can be transformed by a change of coordinates, correct? When I talked about an empty universe being negatively curved I was just talking about the spatial curvature. Anyway, thanks for the answers guys, it helped clear up my confusion on this.

 

[George Jones]

But what you and pervect's reference are saying is that spatial curvature, as opposed to spacetime curvature, can be transformed by a change of coordinates, correct?

Yes, at least in some cases. Spatial curvature depends on the choice that specifies the "nows".

When I talked about an empty universe being negatively curved I was just talking about the spatial curvature.

Sorry - I wasn't sure.

Regards,
George

 

[pervect]

With no cosmological constant, the only homgeneous and isotropic vacuum solutions are indeed the Milne solution and the Minkowski solution.

As Berislav points out, when one allows a cosmological constant, there are other solutions such as the DeSitter solution.

The metric for the most general homogeneous and isotropic soultion is

variables = [t,\chi,\theta,\phi]

<br /> \left[ \begin {array}{cccc} <br /> -1&amp;0&amp;0&amp;0\\<br /> 0 &amp; a^2 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; a^2 \Sigma^2 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; a^2 \Sigma^2 sin(\theta)^2 \\<br /> <br /> \end {array} \right] <br />

where a = a(t) is the expansion factor and \Sigma = \Sigma(\chi) is a different function depending on the spatial curvature

k=1, \Sigma = sin(\chi)

k=0, \Sigma = \chi

k=-1, \Sigma = sinh(\chi)

[/tex]

Solving for an all-zero Riemann or Einstein tensor (a vacuum solution with no cosmological constant), real solutions only exist for k=0 and k=-1.

Basically we have (da/dt)^2 + k = 0

So when k=-1, (da/dt) = +1 or -1, and we have the Milne solution for an expanding universe, or a "big crunch" time-reversed Milne universe.

When k=0, (da/dt)=0, so a(t) is constant, and we have the familiar Minkowski metric.

When k=1 there is no solution (as I mentioned previously).

While the Minkowski metric and the Milne metric appear different on the surface, either one can be transformed into the other by a change of variables, so they are not really "different" solutions.

[add]
Ways to see the equivalence between the Mline metric and the Minkowski metric

1) The Riemann of the Milne metric is zero
2) Substitute t&#039; = -t*cosh(\chi), \chi&#039;= t* sinh(\chi) into the metric -dt'^2 + dx'^2 (use the chain rule).

 

[Berislav]

So when k=-1, (da/dt) = +1 or -1, and we have the Milne solution for an expanding universe, or a "big crunch" time-reversed Milne universe.

Are you sure that the Riemann tensor of this metric is zero?

In my previous post I used:
ds^2=-dt^2+R^2(t) (\frac{dr^2}{1-kr^2}+r^2d\Omega^2)
There still is a coefficient multiplying dr^2.

When k=1 there is no solution (as I mentioned previously).

I think one should say that it isn't physical as R (or a, in your notation) will become negative because the integration constant must be finite, rather than there's no solution.

 

[pervect]

The Milne metric is, if you look at the first post and make k=-1 so that \Sigma(\chi) = sinh(\chi) and at a(t)=t

<br /> ds^2 = -dt^2 + t^2 d \chi^2 + t^2 sinh(\chi)^2 d \theta^2 + t^2 sinh(\chi)^2 sin(\theta)^2 d \phi^2<br />

the Riemann of the above metric is identically zero, and the variable substitution below, equivalent to the one I mentioned earlier (but with time running forwards!)

t1 = t*cosh(\chi), r1 = -t*sinh(\chi)

will convert the standard Minkowski metric below

-dt1^2 + dr1^2 + r1^2(d \theta^2 + sin(\theta)^2 d \phi^2)

into the Milne metric (first line of the post).

 

[Berislav]

Yes, my mistake. The appearance of the metric tricked me. I should have checked the curvature 2-form before I said anything.

 

[pervect]

Yes, my mistake. The appearance of the metric tricked me. I should have checked the curvature 2-form before I said anything.

It's a funny looking beast, alright.

 

[Juan R.]

Please check the metric again. You will see that that is not what happens to it.

Yes i forgot a minus sign. In solar systems test the limit c --> infinite gives

g00 = 1 and gRR = - 1

passing to cartesian coordinates again the metric is (1, -1, -1, -1) I am using trace -2.

And I think that Wald didn't take that limit because he dealt with SR and Newtonian approximations in one of the previous chapters, so he assumed that the reader understands that that is how one reduces to Newtonian physics from relativity.

If he take the limit curvature is zero and geodesic equation read a = 0 whereas Newtonian gravity says a =/= 0. Then Wald is forced to use an inconsistent hibrid.

Yes, of course, that will happen in pure GR and in reality, but if you take c--> infinity, then it will propagate instantly.

But, then, metric is FLAT and according to GR there is not gravitation which contradict Newtonian gravity.

The gradient of the potential is what is physical, not the potential, and hence you can add any constant to the potential; and that's what any leftover constant after you take a limit is.

Yes i agree that only gradients or diferences of potential are significative. But taking an origin for the potential the potential itself become physical.

I was expresing is that limit R --> infinite in NG is physical, called principle of decomposition of clusters and experimentally verified. Whereas limit R --> infinite on GR or Cartan theory is unphysical becasue is not experimentally verified.

The basics of QED, which Dirac mentions, are derived from Maxwell's laws and quantum physics (and a second quantization). You don't even have to mention SR, per se, as Maxwell's equations are relativistically covariant. Furthermore, the Klein-Gordon equation, for instance, is just a relativistic version of the Schrödinger equation, and a reduction from the former to the latter is simple. So, I don't really think that Dirac was talking about what you point out as a problem. So, please, if you could quote Dirac on what the exact problem is, that would be great.

Dirac explicitly attack renormalization procedure there, but does not states in its quote what is the problem with QFT and NRQM. But it is unnecesary since it is well-known that QFT is NOT QM. Therefore, in practice, like Dirac clearly states, one works with two theories, one for nonrelativistic phenomena and other for certain question of relativistic phenomena.

If you study QFT you can see differences between QFT and QM. For example there is a very basic difference regarding positions in both approaches doing both incompatible like Dirac clearly notes, x is a operator in QM but is a parameter in QFT. Also QFT is only defined for single particles and bound states are, rigorously, undefined in QFT.

It is not true that the Klein-Gordon equation was just a relativistic version of the Schrödinger equation. In fact in RQM, the KG is unphysical, and cannot be thought like a relativistic version of Schrödinguer equation, and in QFT "it" is not a wave equation and the link with NRQM is broken (like Dirac clearly states). The same about Dirac equation

In QFT (ipartial + m) Phy = 0 for electrons is not the original Dirac equation even if look like. In fact, this is like difference between Wald equation and Newtonian equation, both looks equal but are not equal.

That above QFT equation of above is NOT Dirac equation is also noted by Weinberg in his manual vol 1 on QFT. In fact, Weinberg beggins his manual critizing Dirac RQM theory.

 

[Juan R.]

The quiz response above also made some interesting statements about Birkhoff's theorem and how it applies to Juan's dilema, but I'm not sure I believe them yet as an accurate statement of the theorem.

1º problem) Eherl boundary is not supported by observation. I already cited several guys including Penrose. The fact that that boundary is unphysical has forced to people to work with other approach which is add a new equation to original GR equations, the equation with the vanising of derivation for the defined Newtonian connection. But then one is not deriving NG from GR, since that equation is added ad hoc to the field equations of GR.

2º problem) Even if boundary was supported by observation that violates causality, since is asumming that an event sited infinitely in the past, i.e. infinite time before big bang is acting on current event.

3º problem) Even ignoring all above, there is a lack of continuity by the use of Ehlers function Phy(x, t) in the transition from steady states to non ones. For mathematical details and physical insight on EM similar problem see PRE 1996, 53(5), 5373 since i would not explain better.

4º problem) One is "forced" to work with Cartan-Ehlers models since derivation of NG from GR is impossible and the textbook derivation is incorrect. Therefore from standard GR, the textbooks derivation, is clearly incorrect. I already cited on this also.

5º problem) Even ignoring that, the final result is the "geometrized version" of NG (which is not exactly original NG) which is worked in the limit c --> infinite.

In the specialized literature (Stingray may know this better) people really work with the so-called causality constant which is defined like k = (1/c), but some authors take k = (1/c^2).

therefore above limit read k --> 0. But in the limit k --> 0 even if it was well defined, one know by standard GR that metric used in solar systems test (which I'm sorry to say this JesseM is not flat and is not a Robertson Walker one which is falt in an average sense for the whole universe) that the metric is

(1, -1, -1, -1)

anyone can check this from a GR textbook since the functions

A = (1 - 2phy/c^2) and B = - (1/A) enter on the metric and

(1 - 2phy/c^2) --> 1

and for a flat metric there is no gravitation in GR.

Any textbook, online course, Arxiv, preprint, or paper where NG was rigorously derived from GR? I am thinking that derivation is a myth.

 

[pervect]

Here's the section that caught my eye:

(a) Birkhoff's theorem
Birkhoff's Teorem states that the gravitational effect of a uniform medium
external to a spherical cavity is zero." This is a theorem from general relativity,
and necessary to know in order to extrapolate our Newtonian cosmology results to
the whole universe: it might have been the case that the global curvature of space
would have interfered with our Newtonian results. The other choices in the question
were generally true statements from other areas of cosmology.

I'm still not sure if the above statement is correct or not, unfortunately - it doesn't resemble any traditional statement of Birkhoff's theorem that I recall seeing.

 

[Berislav]

Yes i forgot a minus sign. In solar systems test the limit c --> infinite gives

g00 = 1 and gRR = - 1

passing to cartesian coordinates again the metric is (1, -1, -1, -1) I am using trace -2.

g_{00}=c^2 \left( 1-\frac{2GM}{c^2 r} \right)


As you can see the metric blows up in that limit. This is because time has no geometric structure, or meaning as such, in Newtonian physics.

But, then, metric is FLAT and according to GR there is not gravitation which contradict Newtonian gravity.

No, it blows up.

But taking an origin for the potential the potential itself become physical.

No, it doesn't. That's called gauge fixing.

Dirac explicitly attack renormalization procedure there,

Which doesn't have anything to do with the limit h--->0.

But it is unnecesary since it is well-known that QFT is NOT QM.

No one said that it was. It was derived from QM as a generalization of it.

x is a operator in QM but is a parameter in QFT.

We have a lot of freedom in chosing our parameters in quantum physics. x is chosen as another parameter to put in on equal footing with time and because we're no longer dealing with a single particle but a field in space. It is possible to define both as operators, for instance, but that wouldn't change the underlaying principles of the theory.

It is not true that the Klein-Gordon equation was just a relativistic version of the Schrödinger equation.

See, for instance, Griffiths' Introduction to Elementary Particles p.213-215.

 

[JesseM]

therefore above limit read k --> 0. But in the limit k --> 0 even if it was well defined, one know by standard GR that metric used in solar systems test (which I'm sorry to say this JesseM is not flat and is not a Robertson Walker one which is falt in an average sense for the whole universe)

Wait, are you saying the metric used in the GR analysis of the solar system does not approach flatness as the distance from the solar system approaches infinity? That's all that I ever said, I thought that was part of the meaning of "asymptotic flatness". I did not claim that this was a Robertson-Walker metric, although I did guess that a spatially flat Robertson-Walker universe would approach minkowski spacetime in the limit as cosmological time approached infinity and the expansion rate approached zero, and pervect seemed to say this might be correct.

 

[Juan R.]

g_{00}=c^2 \left( 1-\frac{2GM}{c^2 r} \right)

As you can see the metric blows up in that limit. This is because time has no geometric structure, or meaning as such, in Newtonian physics.

I used (ct, x). You are introduced c into the metric which is not standard, but in any case when c become more and more large your g00 defines a flat spacetime. Take c = 10^50 and after 10^500. Eact time spacetime is more flat.

Using your metric you obtain g00 = infinite and gRR = -1

No, it blows up.

Exactly does not blows, simply one need more care on work with that divergence. This is reason that research in the limit c --> infinite is done via NC theory and not from standard presentation of GR. But NC is obtained from a 're-geometrization' of GR.

No, it doesn't. That's called gauge fixing.

I was talking of Newtonian potential, which is physical once you fix the origin of the potential. This is reason that Newtonian potential is defined like

Phy = -GM/R

instead of Phy = -GM/R + cte or similar.

Of course, in field theory there is gauge theory, but taking a gauge the potential used has full physical sense (related to the choosed gauge).

No one said that it was. It was derived from QM as a generalization of it.

Hum! Weinberg is ambiguous here.

We have a lot of freedom in chosing our parameters in quantum physics. x is chosen as another parameter to put in on equal footing with time and because we're no longer dealing with a single particle but a field in space. It is possible to define both as operators, for instance, but that wouldn't change the underlaying principles of the theory.

Hum not true. x is an observable in NRQM, but only a parameter in RQFT. If you elevate time to range of observable then the structure of RQFT is very different and you are working with a different theory.

After of saying that both NRQM and QM are incompatible. Dirac asked that we need a new relativistic formulation, far from RQFT.

See, for instance, Griffiths' Introduction to Elementary Particles p.213-215.

The KG equation is not a consistent relativistic generalization of Schrödinger equation. This is the reason that was abandoned in RQFT where the evolution equation is a Schrödinguer like equation. Also the Dirac equation was abandoned in RQFT. Weinberg manual is cristal clear.

After both KG and Dirac lagrangians define fields of bosons and fermions. BUT are equations for fields, are not the original equations for wavefunctions.

For example

(i partial + m) Phy = 0

in QFT is NOT the Dirac wave equation even if looks 'close'. It is an identity for the fermion electronic field Phy(x,t). See the Weinberg volume 1.

 

[Juan R.]

Wait, are you saying the metric used in the GR analysis of the solar system does not approach flatness as the distance from the solar system approaches infinity? That's all that I ever said, I thought that was part of the meaning of "asymptotic flatness". I did not claim that this was a Robertson-Walker metric, although I did guess that a spatially flat Robertson-Walker universe would approach minkowski spacetime in the limit as cosmological time approached infinity and the expansion rate approached zero, and pervect seemed to say this might be correct.

No i am not saying that.

If you take R --> infinite the metric is (1, -1, -1, -1) and that is asymptotically flat. But that metric is defined for the Solar system and if you take R more large than solar system, then the metric is not that, because you may introduce the curvature of other sources of matter. You can understand this easily.

Take the direction on alfa Centaur from the Sun. At large distance of the Sun, but in the Solar system, the metric is valid. Now take R = distance to alpha centaur. There according to you initial metric curvature of spacetime would be zero or close to zero (because Solar system metric was derived asuming un Universe formed only by the Sun) but close to alpha centaur the real (observed) metric is very different from flat one.

Similar questions appliy to Ehler boundary. He assumes that when R is more and more great, the quantity of matter in the universe is more and more insignificant until beyonf certain limit there only vaccuum. This is the reason that is called the 'island asumption' and there is no evidence that was correct. In fact the distribution of matter is not more and more small for large distances

Ehlers universe look like

______________X_X_XXX_X_XXXXXXX_X_X_XXX_X_X_XXX_________


Our universe does not look like 'an island of matter surrounded by emptiness'. This is the reason that Ehlers work is not completely accepted. It looks 'like'

X_XXX_X_XXXXXXX_X_X_X_X_XXX_X_XXXXXXX_X__X_XXX_X_X_XXX

The Shwartzild metric asumes that our universe is

_____________________________S_____________________________

with S the Sun. Which obiously is not correct at galactic scales. But is a very good approximation inside the Solar system.

 

[Juan R.]

But what about Wald p. 138, 139? The effective potential equation (6.3.15) is a good example of how GR and Newtonian physics differ by a factor. You will notice that it doesn't contain coordinate time and hence can be reduced directly to Newtonian gravity.

i already read the Wald again. The equation (6.3.15) does not contain Newton potential, contains the retarded field that follows from GR. There is significant difference on funtional forms. For Newton Phy = Phy(R(t)). For GR Phy = Phy(x, t). From the GR functional dependence one cannot explain all phenomena (i cited on this but in similar problem on EM). Moreover, astronomers uses Newtonian potential newer GR retarded field for the computation of orbits, due to experimental absence of gravitational aberration and other issues (like stability of orbits in numerical computations).

Moreover equation (6.3.15) is derived from (6.3.10) which is parametrized for kappa 1 and 0. Taking c infinite, you cannot maintain kappa timelike.

 

[Juan R.]

I posted some data on sci.physics.relativity about this.

Surprinsingly, i received two types of replies:

1) You are wrong because the limit c --> infinite cannot be taken in GR. Tom Roberts said "... and you are being too naive. c->infinity removes gravitation from GR"

2) You are wrong because the limit c --> infinite has already obtained in NC formulation. This was the point of renowed specialist S. Carlip. However, Carlip cited references (except one) that i had already studied and cited here. For example, the paper Commun. Math. Phys. 166, 221-235 (1994)

It is really interesting the confusion in this topic, one says that the limit does not exist "therefore JR wrong", other claims that limit exists, "therefore JR wrong" again. Obviously i cannot be wrong in both cases at the same time

Unfortunately there is many 'noise' in sci.physics.relativity i read many times 'Crazy moron' and similar. I launched a post in moderated sci.physics.research.

I got reply by Igor Khavkine today, unfortunately reply is wrong and even trivial. for example on

"The theorem stating that gtr does indeed go over to Newtonian gravitostatics in the very weak field, very slow motion limit is proven in detail in almost every gtr textbook."

Igor states that

That is indeed true.

simply compare with claim from an specialist in the topic (Bernard F. Schutz, "The Newtonian Limit") reference introduced in PF by robphy (Thanks!) in #30.

there are at least two reasons why the simple textbook extractions of the Newtonian limit are not rigorous.

 

[Juan R.]

Some interesting discussion on the topic began with several relativists including renowned Steve Carlip. However, in my personal opinion -please do not me atack because i am thinking this now-, Carlip is wrong in several crucial details doing his attempt to prove that Newtonian gravity is derived from General relativity wrong.

So far like i can see Carlip has not proved that curvature interpretation follows in the Newtonian limit; has not proved how spacetime quantitites transform into Newtonian potentials; has not proven that one obtain full Newtonian gravity, etc.

For example, in my prescription x^0 = ct, one obtains full physical sense for flat (Newtonian) derivatives. Carlip, by chossing x^0 = t, obtains that physical derivative is covariant one in the Newtonian regime due that 00-connection is not zero in his approach!

There exist more difficulties. More data of interest and references on sci.physics.research

http://www.lns.cornell.edu/spr/2005-10/msg0071918.html

P.S: For moderators. I do not find the direct link to sci.physics.research here in https://www.physicsforums.com/forumdisplay.php?f=123.

 

[Juan R.]

Some interesting discussion on the topic began with several relativists including renowned Steve Carlip. However, in my personal opinion -please do not me atack because i am thinking this now-, Carlip is wrong in several crucial details doing his attempt to prove that Newtonian gravity is derived from General relativity wrong.

Last news about this topic.

Some time ago i said that the curvature interpretation of general relativity is not valid. I based my claim in that when one takes the non-relativistic limit, one obtain a flat spacetime and, however, one does not obtain a zero gravity.

If curvature IS the cause of gravity and you are eliminating gravity then gravity would vanish and however it does not! This clearly indicates that curvature is not the cause of gravity. Remember, basic epistemological principle: if A is the cause of B elimination of A eliminate B.

Of course in textbooks proof, spacetime is not flat, but textbooks does not take the correct relativistic limit and final equation is NOT Newtonian equation. That is the reason that advanced research literature does NOT follow textbooks wrong derivation.

Some 'specialists' as Steve Carlip were rather hard in their replies. In his last reply, the specialist Carlip have expressed his doubts about that in the non-relativistic limit one can obtain a flat spacetime.

[quote = Carlip]
He also thinks that the Minkowski metric should apply even to Newtonian gravity (!).

I proved this time ago. Carlip simply ignores my proof. One would remember that Carlip is NOT a specialist on Newtonian limit theory and, in fact, has published nothing in this hot topic.

Now i find a recent paper claiming the same. The paper has been published in leader journal on gravity.

On (Class. Quantum Grav. 2004 21 3251-3286) the author claims the substitution (1/c) --> (epsilon/c) in GR equations, and states that epsilon = 1 is Einstein GR and epsilon = 0 is Newton theory.

I find curious as that author (working the Newtonian limit with detail) writes

The fiber epsilon = 0 is Minkowski space with a (non-degenerated) Newtonian limit.

That is, the limit epsilon = 0 of GR is Newtonian gravity and in that limit spacetime is Minkoskian, which is flat. My initial prescription that in the non-relativistic limit one obtain GRAVITY with a FLAT spacetime is correct. Therefore, that i said in page 17

of

www.canonicalscience.com/stringcriticism.pdf[/URL]

in April was mainly correct. That April comment contains some imprecision (i am thinking in rewriting again with last advances in the research), but basically it was correct regarding the geometric prescription of GR.

One may reinterpret the basic of general relativity.

I find really interesting this!

 

[jtbell]

Since Dr. Carlip does is not a member of this forum, as far as I know (at least I don't remember seeing him post here), interested readers might want to watch the thread in sci.physics.relativity where Juan also references the paper that he references here. Perhaps Dr. Carlip will respond to him there.

 

[Juan R.]

Since Dr. Carlip does is not a member of this forum, as far as I know (at least I don't remember seeing him post here), interested readers might want to watch the thread in sci.physics.relativity where Juan also references the paper that he references here. Perhaps Dr. Carlip will respond to him there.

Above link is not about scientific discusion with Carlip

Carlip (incorrect, in my opinion) post is here

http://groups.google.com/group/sci....cbd?scoring=d&&scoring=d#doc_22bf366b013f1d39

and my formal reply is here

http://groups.google.com/group/sci....cbd?scoring=d&&scoring=d#doc_ca7b1885fe389649

I am anxiously waiting his reply.

P.S: Any comment on Eric error on Minkowski metric? I have detected that is working in NASA. Perhaps he was one of those participating in those famous mission that had the problem with units