Advanced books/papers on derivation of Newtonian mechanics from GR

[Berislav]

I'm sorry. I didn't have time to respond to your entire post at once.

what t may i take the Newtonian clock the Einstein clock?

Proper time is Newtonian time. Time as a coordinate is non-Newtonian.

What physical mechanism explain the transition from D'alembert to Poisson equations?

I don't have that and I'm not sure how one would make such a mapping.
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 cited Dirac talking about the limit c --> infinite, and now you claim that Dirac was not talking about that

I still don't know what he is referring to.

 

[Juan R.]

OK, I admit I'm not a GR expert, but how would you describe the physical meaning of asymptotic flatness? Isn't it just saying that the further you move away from the system you're considering, the closer you get to flat minkowski spacetime? If so, it seems to me like that would be at least related to the idea that you can consider the system in isolation and don't have to worry about other distant gravitating bodies. When physicists do make the assumption of asymptotic flatness, what physical justification do they give for the assumption? You didn't provide any detailed quotes from Penrose, so I have no idea if Penrose is actually objecting to the idea that Newtonian mechanics can be derived from GR, or if he just objects to asymptotic flatness in some other unrelated context (a cosmological one, perhaps, as might be suggested by the 'island universe' comment).

Ok, you are not expert (i am not of course). i will explain why again you are confounding asymptotic flatness with clister principle and why your appeal to cosmological context is wrong.

You are working with solar systems test and you want obtain a Newtonian representation of Jupiter orbit. Since relativistic effect from Sun are insignificant you want obtain NG from GR. From usual GR you cannot do it in rigor -even if textbooks claim the contrary- then other "option" is via Cartan theory. You obtain a set of equations and for "total" -so say- compatibility with NG you need fix the "gauge". Then

i) you use new equations does not contained in GR or

ii) you fix the "gauge" via a boundary.

What boundary? Look for numerical coincidence with NG, Ehlers and other people does R --> infinite Phy = 0 because that is numerical valid for NG. But in NG "that" is the principle of decomposition of clusters, which is experimentally proven. Phy in Eherls theory is Phy(x,t) and asumption to R--> infinite is not fixing behavior of custer like in NG, Ehlers is fixing distribution of matter of the universe. Even if you are interested in solar systems tests you are doing an asumption about universe asa whole when take R--> infinite. Then you put telescope and discover that asumptions you are using is simply false.

However, physical evidence clearly suggests that we are not living in an ‘island universe’ (cf. Penrose 1996, 593-594) – i.e., universe is not ‘an island of matter surrounded by emptiness’ (Misner et al. 1973, 295).

Extracted from Christian preprint.

The island universe asumption is not valid on our universe. This is reason that limit R--> infinite is unphysical in GR but physical in NG. Are two diferent things, the physics is different.

There are more difficulties with asymptotic flatness but i believe that experimental data would be sufficient for any physicist to believe that Ehlers approach is invalid.

The thrick of GR textbooks is amazing "to use the same notation" and, therefore, ingenuous students see Wald equation and believe that it IS the Newton law when is an equation with a physical contain completely different.

 

[Juan R.]

There's also the local problem that if the sun disappeared then NG says that the effect would be felt instantly, whereas GR says that the effect would propagate at c, but it seems to me that this would agree in the limit c->infinity.

Therefore,

a = - GRAD (Phy) in NG

a = - GRAD (Phy) in GR (example equation 4.4.21. in Wald)

are two different things. In NG i would compute instanteously Phy --> Phy/2 but in GR i would work with Phy the first 8 minutes. The predicted orbits for Earth are, of course, very different.

I agree that if one take c--> infinite, one would wait instantaneous propagation in GR and both descriptions agree but.

g00 --> 1 when c --> infinite and

gRR --> 1 when c --> infinite

therefore the GR geodesic equation is like clearly stated by Wald just before section 4.4b would be equivalent to the SR metric geodesic motion

partial^a T_ab = 0 then (p78)

one predicts that test bodies are unaffected by gravity

a = 0

BUT according to NG which is valid for c --> infinite

a = - GRAD (Phy) =/= 0

Again GR is not compatible with NG.

Any textbook or paper where NG was derived -no supposedly derived- from GR?

 

[Juan R.]

Here are a few more references that I've googled.
They do address some issues with more care than [can be included] in standard textbooks.

http://edoc.mpg.de/60619 (Bernard F. Schutz, "The Newtonian Limit")
http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104270381 (Alan D. Rendall, "The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system")
http://arxiv.org/abs/gr-qc/9506077 (Simonetta Frittelli and Oscar Reula, "On the Newtonian Limit of General Relativity") [I see now that this is the CMP 166, 221-235 (1994) reference.]


Good luck.

Thanks by your link but the rigor is small. none of those papers is deriving the Newtonian limit. It is really interesting says Schuzt document.

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

It is really interesting like the author is supporting a point i said was incorrect many time ago and some PF people said "you wrong".

In fact, a week ago or so, in the photon mass' thread, a guy -with no idea of nothing- said "you wrong" regarding this matter.

Yes, Schuzt is more rigorous, but again i see no rigor in his work. I begin to think that nobody has derived, in rigor, the NG from GR still and all is a kind of myth.

Of course the arrogant claim in Baez page that Newtonian limit is derived in any textbook on GR is just another of examples of how arrogant is many relativistic people. Unfortunately almost all that is said in Baez page (cited above in PF in the Wrong claims thread) is, at the best, non rigorous.

I would recommend to the PF staff the elimination of the "Wrong claim" thread, or, at least, to add comment saying that several things stated in Baez page are simply wrong.

About the last link, Yes is the same CMP paper i have the final journal article. Well i do not check if there is any diference with the preprint.

 

[Juan R.]

Yes. Because the Dirac bracket disappers and canonical pairs commute. The wave equation becomes infinite. Hence, it's classical.

Then you newer has obtained the h--> 0 limit of Klein/Gordon equation. It does not offer the correct classical limit, what is well known, due to the Zitterbewegung problem.

Potential is not physical, it's gauge.

Again my advice you would read with care my posts before reply.

limit, limit, limit, limit, limit...

limit R --> infinite of Phy(x, t) is unphysical

limit R --> infinite of Phy(R(t)) is physical

The first limit is asimptotic flatness which is experimentally false. The second limit is principle of decomposition of cluster which is well tested.

If you want a GR calculation you will have to construct a non-static spacetime metric. If we know the nature of explosion we could find a new vacuum solution to the Einstein field equations. Using Newtonian physics would be simpler.

Bla, bla, bla, bla. Chronon offered correct reply.

I really don't understand.

Either you use unphysical boundaries (Ehlers approach) or you need add ad hoc equations to GR, for example vanishing of dreivative of Newtonian connection, which does not follow from the GR field equations and is used ad hoc.

 

[Juan R.]

But Newton's theory has never been used in these contexts, so violating it doesn't matter (if you disagree, give an example). The limit of GR with asymptotic flatness does work. With the proper identifications, all of the equations in the restricted NC theory are the same as the standard Newtonian ones. This is in some sense a formality. It does, however, show that the practical implementations of both theories are identical.

Do not matter the context. If you are working in solar system test, you need fix the gauge In NC theory and either you use ad hoc equations or you use boundary condition.

About asymptotic flatness i (and others) already said

for lovers of experimental verification alone i can say that the “island universe” assumption, Misner, Thorme, and Wheeler (1973, p.295), is not physical because cosmologists claim that all the matter in the universe is not concentrated in a finite region of space, therein the name "island asumption". I think that Joy Christian (arXiv:gr-qc/9810078 v3) is clear

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

Also Penrose has claimed that our universe is not of island type.

In fact, last observations claim a dodecaedrical structure for cosmos, therefore, asymptotic flatness is experimentally wrong.

Moreover, even if our universe was of island type, You could do not take that limit because is unphysical. I remember that already explained to you why but i think that you do not understand still and then ignore it.

 

[JesseM]

Ok, you are not expert (i am not of course). i will explain why again you are confounding asymptotic flatness with clister principle and why your appeal to cosmological context is wrong.

You are working with solar systems test and you want obtain a Newtonian representation of Jupiter orbit. Since relativistic effect from Sun are insignificant you want obtain NG from GR. From usual GR you cannot do it in rigor -even if textbooks claim the contrary- then other "option" is via Cartan theory. You obtain a set of equations and for "total" -so say- compatibility with NG you need fix the "gauge". Then

i) you use new equations does not contained in GR or

ii) you fix the "gauge" via a boundary.

What boundary? Look for numerical coincidence with NG, Ehlers and other people does R --> infinite Phy = 0 because that is numerical valid for NG. But in NG "that" is the principle of decomposition of clusters, which is experimentally proven. Phy in Eherls theory is Phy(x,t) and asumption to R--> infinite is not fixing behavior of custer like in NG, Ehlers is fixing distribution of matter of the universe. Even if you are interested in solar systems tests you are doing an asumption about universe asa whole when take R--> infinite. Then you put telescope and discover that asumptions you are using is simply false.

How is it false? The universe is indeed pretty close to spatially flat on large scales. Of course it's also expanding, but I think it's reasonable that a derivation of Newtonian physics from GR should be able to ignore the expansion of the space, and as I suggested to pervect you could just consider the limit as the expansion rate of a spatially flat universe approaches zero.

The island universe asumption is not valid on our universe.

But how is the term "island universe" used by physicists? Does it refer to any use of asymptotic flatness, even just as an approximation, or does it refer to a specific model of cosmology? Can you provide some quotes or online papers that use this term so I can see the context? It's also technically unphysical to assume the distribution of matter and energy is perfectly uniform as in the FRW models of cosmology, but everyone understands that this is just meant to be an approximation for a universe that is close to uniform but not perfectly so. I would imagine that asymptotic flatness is also just meant as a sort of simplification rather than an actual assumption about cosmology, a way of looking at a particular system in isolation and not worrying too much about the details of the surrounding universe besides the idea that it's close to spatially flat on large scales, and that we can ignore the expansion of space when considering small bound systems over relatively short timescales. Do you agree that the universe is close to spatially flat on large scales and that it's reasonable to ignore the expansion of space when analyzing small-scale problems like the orbits of planets?

You never addressed my question about the Penrose quote, by the way. Was he objecting to any use of asymptotic flatness regardless of the context, or was he just objecting to a specific cosmological model?

However, physical evidence clearly suggests that we are not living in an ‘island universe’ (cf. Penrose 1996, 593-594) – i.e., universe is not ‘an island of matter surrounded by emptiness’ (Misner et al. 1973, 295).

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?

 

[Juan R.]

Proper time is Newtonian time. Time as a coordinate is non-Newtonian.

Therefore in

a = - GRAD (Phy) [GR]

a is d²x/dt_E²

and in

a = - GRAD (Phy) [NG]

a is d²x/dt_N²

but dt_N =/= dt_E

because for static case

dt_N = dt_E SQR(1 + 2 Phy/c²)

I don't have that and I'm not sure how one would make such a mapping.
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.

to the first part. Imposible from GR.

To the second part i have not the book here now. I will obtain again the next Wednesday (the travel is 1 hour in Bus from here ) and i will can see that exact equation is and will reply to you. I already know that Wald will be wrong , but i cannot say now the list of errors of (6.3.15)

In Schulz pdf introduced above it is clearly stated that textbooks derivations of NG are wrong since there is no rigor. That is asuming that one may prove, textbooks follow mathmeatical steps for obtaining the result that one, know a priori, but does not prove if really the derivation is correct or only a "myth".

Textbooks derivation look like

if 2 > 4

then 2 + 10 > 4 + 10

therefore 12 < 14

nonsense!

Again i would ask, any textbook or paper where [NG] was derived from [GR]?

is there a logical connection

GR ------------> NG ?

or, in Dirac terms, (see my adaptation of Dirac thoughts to gravitation in the last part of #25)

GR -----/-------> NG

and one uses two inconsistent theories, GR for some relativistic problems and NG for nonrelativistic problems

I still don't know what he is referring to.

Dirac is saying that in the limit c--> infinite QFT does not reduce to NRQM and therefore one may use two different incompatible theories. NRQM for nonrelativistic problems and QFT for some relativistic problems. I think that his words are "cristal clear". Is correct this expresion?

 

[JesseM]

Yes, Schuzt is more rigorous, but again i see no rigor in his work. I begin to think that nobody has derived, in rigor, the NG from GR still and all is a kind of myth.

Would you say that if we are allowed to assume asymptotic flatness, the derivation of Newtonian mechanics from GR can then by made mathematically rigorous, putting aside the question of whether you think asymptotic flatness is a physically justifiable assumption to make?

In fact, last observations claim a dodecaedrical structure for cosmos, therefore, asymptotic flatness is experimentally wrong.

The dodecehedral universe model just makes a new assumption about the topology of space, it doesn't contradict the idea that the curvature of space is close to flat on large scales, so I don't see why this would lead to any new problems with assuming asymptotic flatness (you can treat a topologically compact universe as an infinite universe where regions of space repeat themselves over and over in a regular pattern). In any case, the evidence for the dodecahedral universe was very tentative.

 

[Berislav]

are two different things. In NG i would compute instanteously Phy --> Phy/2 but in GR i would work with Phy the first 8 minutes. The predicted orbits for Earth are, of course, very different.

No. That's not how one would do it. The spacetime wouldn't static anymore. One would for one have to use this metric (provided by pervect and robphy):
https://www.physicsforums.com/showthread.php?t=88883
because instantaneous propagation of matter of the source of gravity would lead to a singularity.

a = - GRAD (Phy) [GR]

a is d2x/dtE2

and in

a = - GRAD (Phy) [NG]

a is d2x/dtN2

but dtN =/= dtE

because for static case

dtN = dtE SQR(1 + 2 Phy/c2)

And as c goes to infinity they become the same. See Wald's explanation of how GR and NG conceptualy differ in that section.

limit, limit, limit, limit, limit...

limit R --> infinite of Phy(x, t) is unphysical

limit R --> infinite of Phy(R(t)) is physical

The first limit is asimptotic flatness which is experimentally false. The second limit is principle of decomposition of cluster which is well tested.

I was under the impression that Phy was Phi ([itex]\phi[/tex]), the gravitational potential, which is unphysical (as it should be) in both theories. Take a physical quantity in both theories and then take that limit, see if they differ.

Dirac is saying that in the limit c--> infinite QFT does not reduce to NRQM and therefore one may use two different incompatible theories. NRQM for nonrelativistic problems and QFT for some relativistic problems. I think that his words are "cristal clear". Is correct this expresion?

I meant that I don't know what problem, exactly, is he refereing to. Where is the problem in QFT when c---> infinity.

therefore the GR geodesic equation is like clearly stated by Wald just before section 4.4b would be equivalent to the SR metric geodesic motion

No, because in that limit the two metrics aren't equivalent.

 

[Juan R.]

How is it false? The universe is indeed pretty close to spatially flat on large scales.

I can repeat and even can use a bigger font but i cannot write more clear.

for lovers of experimental verification alone i can say that the “island universe” assumption, Misner, Thorme, and Wheeler (1973, p.295), is not physical because cosmologists claim that all the matter in the universe is not concentrated in a finite region of space, therein the name "island asumption". I think that Joy Christian (arXiv:gr-qc/9810078 v3) is clear

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

Also Penrose has claimed that our universe is not of island type.

I think that you are confounding "asymptotic flatness" or "island universe" with the asumption of "homogeneus isotropic universe" used in cosmological models.

But how is the term "island universe" used by physicists? Does it refer to any use of asymptotic flatness, even just as an approximation, or does it refer to a specific model of cosmology?

This is unambigous. Island universe means

"an island of matter surrounded by emptiness"

it is equivalent to asymptotic flatness, i already explained why!

It is not an approximation, it is NOT related to a specific cosmological model (your mind may be blocked here), it is just the boundary condition used by Ehlers for "deriving" NG from GR.

Are you studied field theory guy? Do you know that a boundary of a field is? Are you studied Newtonian mechanics also? When i take the limit R --> infinite on Newton potential i am not doing allusion to a "cosmological model"...

Can you provide some quotes or online papers that use this term so I can see the context?

I already did.

It's also technically unphysical to assume the distribution of matter and energy is perfectly uniform as in the FRW models of cosmology, but everyone understands that this is just meant to be an approximation for a universe that is close to uniform but not perfectly so.

Irrelevant, isotropic models is very good, and even if "locally" universe is not homogeneous, one is globally working with the average density of matter which if is homogeneous. Still if you substitute the homogeneous density by real density you are improving the model newer doing poor.

I would imagine that asymptotic flatness is also just meant as a sort of simplification rather than an actual assumption about cosmology, a way of looking at a particular system in isolation and not worrying too much about the details of the surrounding universe besides the idea that it's close to spatially flat on large scales, and that we can ignore the expansion of space when considering small bound systems over relatively short timescales.

False, asymptotic flatness IS the boundary needed for describing NG from GR via Cartan theory even if you are working with solar system tests. Precisely is the only boundary possible for numerical compatibility with NG

Again, i remark that you are confounding asymptotic flatness with principle of cluster. Asymptotic flatness is not about "particular system in isolation"

I'm sorry to say this but i have a very distorted understanding of physics. Penrose and other no have your problem, and this is the reason that "asymptotic flatness" or also called "the island asumption" is unphysical -as Penrose and others claim- but decomposition of clusters of NG is perfectly valid and, until now, always experimentally verified.

Do you agree that the universe is close to spatially flat on large scales and that it's reasonable to ignore the expansion of space when analyzing small-scale problems like the orbits of planets?

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), in the study of orbit of planets asymptotic flatness is newer used in NG, only GR (in Cartan form) needs of it because does not work correctly.

You never addressed my question about the Penrose quote, by the way. Was he objecting to any use of asymptotic flatness regardless of the context, or was he just objecting to a specific cosmological model?

I did. He was talking about any unphysical boundary condition. You continue emphaiszing the word cosmology when it is unnecesary. In fact, i am focusing of the aplication of GR inside the solar system. What now?

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?

 

[Juan R.]

Would you say that if we are allowed to assume asymptotic flatness, the derivation of Newtonian mechanics from GR can then by made mathematically rigorous, putting aside the question of whether you think asymptotic flatness is a physically justifiable assumption to make? The dodecehedral universe model just makes a new assumption about the topology of space, it doesn't contradict the idea that the curvature of space is close to flat on large scales, so I don't see why this would lead to any new problems with assuming asymptotic flatness (you can treat a topologically compact universe as an infinite universe where regions of space repeat themselves over and over in a regular pattern). In any case, the evidence for the dodecahedral universe was very tentative.

you guy are not understanding!

It is not only a question of mathematical rigor. In fact, the criticism to Ehlers boundary condition is that is unphysical, even if Ehlers math had some minimum level of rigor.

If you want ignore physical experimentally accesible data and assume that asymptotic flatness is valid in our universe (which is experimentally false), still the derivation is both physically and mathematically incorrect. For example, there is violation of causality and standard Big Bang model, etc.

The dodecehedral universe is not an island surrounded by emptiness, precisely is a dodecehedral not asymptotically flatness!

Pictorically the observed distribution of matter look like

X___X__X_____X___X_____X___X_ etc

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"

 

[Hurkyl]

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?

 

[Berislav]

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?

I am not sure, but I think that if spacetime is approximately flat then gravity is negligable. Also, there is a problem with the fact that Newton's physics has no underlaying geometric structure (i.e, it is not a flat spacetime).
If you mean if the spacetime is asymptotically flat then most spacetimes are - either flat or asymptotically de Sitter.

 

[Juan R.]

No. That's not how one would do it. The spacetime wouldn't static anymore. One would for one have to use this metric (provided by pervect and robphy):
https://www.physicsforums.com/showthread.php?t=88883
because instantaneous propagation of matter of the source of gravity would lead to a singularity.

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.

And as c goes to infinity they become the same. See Wald's explanation of how GR and NG conceptualy differ in that section.

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.

I was under the impression that Phy was Phi ([itex]\phi[/tex]), the gravitational potential, which is unphysical (as it should be) in both theories. Take a physical quantity in both theories and then take that limit, see if they differ.

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.

That is unphysical, that is experimentally false is the limit in GR but is experimentally correct in NG.

I meant that I don't know what problem, exactly, is he refereing to. Where is the problem in QFT when c---> infinity.

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

No, because in that limit the two metrics aren't equivalent.

In the limit c ---> infinite

g00 = 1 and gRR = -1

curvature R_ab = 0

Therefore, according to geometrical interpretation of gravity in GR, cannot exist gravitational "force" a = 0

From NG, however, a =/= 0

 

[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:

$$(R'/R)^2=-k/R^2+\frac{8 \pi \rho}{3}$$,
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.