Advanced books/papers on derivation of Newtonian mechanics from GR

[Juan R.]

During many time i have searched a complete and rigorous derivation of Newtonian limit from GR but i found none. I suspect that it does not exist!

I do not refer to that "supposed derivation" that appears in many textbooks of GR. I refer to a rigorous and unambigous derivation of Newtonian mechanics from first principles of GR.

Please do not cite Cartan-like derivation, because that one obtains there is a modified (geomtrized) version of Newtonian mechanics after using additional asumptions like the "island asumption" used by Ehlers, etc.

I refer to derive the exact Newtonian mechanics from GR alone.

Please do not cite usual textbooks. It is true that Wald manual is more rigorous that others books on the topic. Wald, for example, clearly states that Newtonian mechanics does not follow from GR in the linear regime, since one needs, in rigor, higher order terms outside of the linear regime. In the strict linear regime there is no gravity and motion of test particle is free. In the linear regime there is not Newtonian gravity even if many textbooks claim the contrary.

I said this in a reply to pmb_phy in the photon's mass thread and he/she replied "wrong". I write that because if pmb_phy or any other guy think that i say is "wrong" would read Wald p.78 about derivation of Newtonian limit first

... but, strictly speaking, we went beyond the linear approximation to show this.

before reply here.

Thanks in advance!

 

[robphy]

During many time i have searched a complete and rigorous derivation of Newtonian limit from GR but i found none. I suspect that it does not exist!

I do not refer to that "supposed derivation" that appears in many textbooks of GR. I refer to a rigorous and unambigous derivation of Newtonian mechanics from first principles of GR.

Please do not cite Cartan-like derivation, because that one obtains there is a modified (geomtrized) version of Newtonian mechanics after using additional asumptions like the "island asumption" used by Ehlers, etc.

I refer to derive the exact Newtonian mechanics from GR alone.

Please do not cite usual textbooks. It is true that Wald manual is more rigorous that others books on the topic. Wald, for example, clearly states that Newtonian mechanics does not follow from GR in the linear regime, since one needs, in rigor, higher order terms outside of the linear regime. In the strict linear regime there is no gravity and motion of test particle is free. In the linear regime there is not Newtonian gravity even if many textbooks claim the contrary.


Thanks in advance!

What would you mean by a "complete and rigorous derivation of Newtonian limit from GR"? Do you want field-equation to field-equation? Or solutions to solutions? Or both?

What starting point do you permit? For example, do I get to choose the initial spacetime manifold in GR?

By "Cartan-like", are you talking about the four-dimensional formulation using (for example) a degenerate metric? It seems to me if you don't permit a four-dimensional formulation, then it might not make a lot of sense to map a [differential-geometric] structure in GR to a corresponding one in Newtonian gravity. [I'm not sure what the "island assumption" is.]

Maybe one needs to pose the question as a specific mathematical statement to be proven or disproven (in the spirit of the big theorems in GR like the positive energy theorem or the singularity theorems).

 

[Juan R.]

What would you mean by a "complete and rigorous derivation of Newtonian limit from GR"? Do you want field-equation to field-equation? Or solutions to solutions? Or both?

All if they is possible. But derivation of "solutions" could be sufficient at first step.

What starting point do you permit? For example, do I get to choose the initial spacetime manifold in GR?

GR. You may begin from GR. If you are dicusing Solar system a good GR begin could be Schwartzilkd metric.

By "Cartan-like", are you talking about the four-dimensional formulation using (for example) a degenerate metric? It seems to me if you don't permit a four-dimensional formulation, then it might not make a lot of sense to map a [differential-geometric] structure in GR to a corresponding one in Newtonian gravity. [I'm not sure what the "island assumption" is.]

Yes, i refer to 4D geometrized version of Newtonina mechanics. The island asumption is asymptotic flatness. It is no my problem that original Newtonian mechanics was not 4D! Still standard textbooks claim for derivation of Newton 3D gravity from GR.[/QUOTE]

 

[robphy]

Yes, i refer to 4D geometrized version of Newtonina mechanics. The island asumption is asymptotic flatness.

What is your issue with this boundary condition?

 

[DrChinese]

I refer to derive the exact Newtonian mechanics from GR alone.

In GR, a clock's ticking is altered in the presence of mass (at least to some observers). Under Newtonian mechanics, there is nothing that alters the ticking of a clock. So it would not be possible to do as you are asking.

There are terms in GR which are assumed to be insignificant when moving towards a Newtonian presentation. The devil is in the details of what one agrees is acceptable to approximate. If you don't see what you are looking for in existing textbooks like MTW or whatever, you may need to derive it yourself.

The scientific community is satisfied that we live in a universe most accurately described by GR but approximated by NM in many situations. What would you hope to gain by further analysis of the correspondence of these 2 theories?

 

[Berislav]

GR. You may begin from GR. If you are dicusing Solar system a good GR begin could be Schwartzilkd metric.

You couldn't derive Newtonian physics even from the Minkowski metric. At least not directly by setting c to be infinite. Time is not a dimension in Newtonian physics, it's a parameter.

 

[Stingray]

Not this again ... There was already a thread where I spent way too much time arguing with Juan on this. It was eventually moved to theory development, and then locked.

To reiterate without going into detail again, there's no good reason to have a problem with the (specialized) Newton-Cartan theory. It provides the same predictions as Newton's original theory. That's all you can possibly ask for.

Now some philosophers like to say that no two theories using different languages can ever be equivalent because a perfect translation is impossible. In my opinion, this is a ridiculously pedantic and useless point of view that would (apparently) invalidate most of science. If this is the point you want to argue, however, be clear about it.

Your issues with asymptotic flatness are also unfounded. It is clear that all predictions of Newton's theory which have ever been tested are derived assuming that there is no significant amount of matter "at infinity." It follows that Newton's theory only needs to be recovered in the asymptotically flat case, and that's exactly what is done.

 

[pervect]

Not this again ... There was already a thread where I spent way too much time arguing with Juan on this. It was eventually moved to theory development, and then locked.

To reiterate without going into detail again, there's no good reason to have a problem with the (specialized) Newton-Cartan theory. It provides the same predictions as Newton's original theory. That's all you can possibly ask for.

Now some philosophers like to say that no two theories using different languages can ever be equivalent because a perfect translation is impossible. In my opinion, this is a ridiculously pedantic and useless point of view that would (apparently) invalidate most of science. If this is the point you want to argue, however, be clear about it.

Your issues with asymptotic flatness are also unfounded. It is clear that all predictions of Newton's theory which have ever been tested are derived assuming that there is no significant amount of matter "at infinity." It follows that Newton's theory only needs to be recovered in the asymptotically flat case, and that's exactly what is done.

Perhaps I should let this thread die a natural death, but I think it's worth noting that Newton-Cartan theory gives a different prediction for gravitational lensing of the sun than General Relataivity gives (the apparent deflection of light from a distant star).

Newton-Cartan theory and it's generalization to include first-order relativistic effects (PPN theory) will only give approximately correct answers to actual experiment when the required conditions are met. These requirements include low velocities, low pressures, and weak fields. The deflection of light fails the "low velocity" condition, though it's worth noting that the error is "only" 2:1 even at lightspeed.

Early measurements of the bending of light were imprecise, but the experimental techniques have been refined, and the answers we get nowadays agree with General Relativity, not PPN or Newton-Cartan theory.

It appears that by demanding that GR reduce to Newton Carton theory under all condtions, that the OP is demanding that GR give results that are contradiction to experiment. (At least that's what I gather, the post was not terribly clear). GR is refusing to cooperate with this demand, instead giving answers that match experiment.

 

[Stingray]

pervect, I think you might have misunderstood me. I was not saying the (full) Newton-Cartan theory is equivalent to GR, and I'm pretty sure the original poster wasn't either. The argument that I had with him was over the reduction of GR to a special case of the Newton-Cartan theory in a particular limit (basically 1/c \rightarrow 0). It is known that this special case is basically Newton's original theory (as any such limit should be), although Juan disagrees because it uses a notation that looks very different from Newton's.

 

[Juan R.]

What is your issue with this boundary condition?

Several!

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.

 

[Juan R.]

In GR, a clock's ticking is altered in the presence of mass (at least to some observers). Under Newtonian mechanics, there is nothing that alters the ticking of a clock. So it would not be possible to do as you are asking.

There are terms in GR which are assumed to be insignificant when moving towards a Newtonian presentation. The devil is in the details of what one agrees is acceptable to approximate. If you don't see what you are looking for in existing textbooks like MTW or whatever, you may need to derive it yourself.

The scientific community is satisfied that we live in a universe most accurately described by GR but approximated by NM in many situations. What would you hope to gain by further analysis of the correspondence of these 2 theories?

Thanks! I replied to you

1) "So it would not be possible to do as you are asking." Then we cannot derive it!

2) "There are terms in GR which are assumed to be insignificant when moving towards a Newtonian presentation." Then we can derive it!

It would be great a link to any site, book, paper where the derivation was done with rigor. Then i could verify if the derivation is real or only ficticious. There exist examples of ficticious derivations on textbooks. The most clear is the asumption that h -> 0 quantum mechanics reduces to classical mechanics that any textbook on QM states. Which is, strictly speaking false, and this is the reason that still people is working in HOW obtain classical mechanics from quantum one.

For example, i already said that most of textbooks claim that Newtonian gravity is obtained in the linear regime (Baez page that is cited above in PF "wrong claims" thread claims that theorem of derivation of the Newtonian limit "is proven in detail in almost gr textbook"), but when one works the details of the "teorem" of those textbooks, one discovers that in the linear regime a=0 (this is the reason that Wald argues that one may go outside of the linear regime). My question is what one work ALL the details seriously? Can one really derive Newtonian gravity from GR?

3) Yes, i agree that NM is not suficient but if NM cannot be derived from GR, then one is working with a collection of theories. NM for some things, GR for others, etc. My claim is not trivial. Also Relativistic QFT is not strictly reduced to non relativistic quantum mechanics and thus Dirac was hungry that one need two theories one for studying nonrelativistic phenomena and other for studying certain relativistic phenomena, then Dirac asked what is the real relativistic generalization of quantum mechanics?

If GR cannot be reduced to NM, one is using different theories for different problems (this point has been also recently maintained, in a more general framework, by Michele Vallisneri, Theoretical Astrophysics of Caltech, in his talk "Ephemeral Ephemerides? From General Covariance to Relativistic Geodesy and Astrometry" on Parma, June 22, 2001). Emphasis mine.

- We use GR as a theory of gravitation to study astrophysics and cosmology, to formulate astrometry and geodesy, to help navigation and time-keeping

- This is done with a collection of tools, widely different and often inconsistent

*** Geometrodynamics

*** Linearized theory

*** Nonlinear CFT on flat bkg

*** Global methods

*** Post—Newtonian, Post—Minkowskian

*** Quantum gravity, superstrings, ...

- From this pragmatic viewpoint, we

*** Use eclectic methods to study the sector of GR that applies to
our universe

*** Exploit the very special structure of our universe to confer a
special status to particular coordinate systems

i think that GR and NM are inconsistent. My claim is (copying Dirac) What is the correct relativistic generalization of NM if GR is not?

Where i could find (if exists) the rigorous derivation of NM from GR?

 

[Juan R.]

You couldn't derive Newtonian physics even from the Minkowski metric. At least not directly by setting c to be infinite. Time is not a dimension in Newtonian physics, it's a parameter.

Therefore, doing c--> infinite one does not obtain Newtonian gravity where effectively c is infinite. That is correct? Then Diagram of theories that Penrose uses in his last book (i read time ago but I do not remember title sorry. Help!) where Quantum gravity is reduced to GR when h-->0 and this to Newtonian mechanics when c-->infinite, or where Quantum gravity is reduced to QFT when G --> 0 and this to NQM when c--> infinite is pure fantasy.

 

[Juan R.]

Not this again ...

Nobody obligate to you

There was already a thread where I spent way too much time arguing with Juan on this

Arguing =/= demostration

To reiterate without going into detail again, there's no good reason to have a problem with the (specialized) Newton-Cartan theory. It provides the same predictions as Newton's original theory. That's all you can possibly ask for.

1) I do not know derivation of Newtonian effects from GR theory. All i know is a reformulation of GR for "weak fields" called Newton-Cartan that need of aditional asumptions, E.g. Ehlers obtain the Newtonian limit using asymptotic flatness, which is empirically false. Others simply add a new equation to GR as vanishing of divergence of theoretical Newtonian connection. But that vanishing does nof follow from field equations of GR. It is invoked ad hoc by consistency with NG, which may be previously known.

Previous knowledge + ad hoc equations (or empirically false boundaries) = theory that is not original Newton theory =/= Popular claim "NG is derived from GR".

If you know a paper or book where the derivation was detailed please cite it

2) I was not claiming for the derivation of a theory "covariant NG" that "may" offer the same results that original NG (nobody has shown to me that both formulations are equivalent). I am claiming for derivation of original NG. It is a basic principle of epistemology that any new theory may reduce to previous theory in the limit where previous theory already worked.

Now some philosophers like to say that no two theories using different languages can ever be equivalent because a perfect translation is impossible. In my opinion, this is a ridiculously pedantic and useless point of view that would (apparently) invalidate most of science. If this is the point you want to argue, however, be clear about it.

Philosophers? Is this a ridiculously pedantic and useless point of view? I already cited to sufficiently important physicist, Dirac, who maintained similar points regarding the imposibility of reduction of RQFT to NRQM.

Dirac, P.A.M. Mathematical Foundations of Quantum Theory. (Academic Press, Inc., 1978) emphasis mine

The appearance of this [Dirac] equation did not solve the general problem of making quantum mechanics relativistic... When one tried to solve it, one always obtained divergent integrals... Rules for discarding the infinities
[(renormalization) have been developed]. Most physicists are very satisfied with this situation. They argue that if one has rules for doing calculations and the results agree with observation, that is all that one requires. But it is not all that one requires. One requires a single comprehensive theory applying to all physical phenomena. Not one theory for dealing with non-relativistic effects and a separate disjoint theory for dealing with certain relativistic effects. Furthermore, the theory has to be based on
sound mathematics... For these reasons, I find the present quantum electrodynamics quite unsatisfactory... The agreement with observation is presumably a coincidence, just like the original calculation of the hydrogen spectrum with Bohr orbits. Such coincidences are no reason for turning a blind eye to the faults of a theory.

Your issues with asymptotic flatness are also unfounded.

I already cited several authors, including Penrose, who claim that asymptotic flatness does not characterize our universe. Moreover there are more difficulties with that and you simply ignore them...

The objective of a physicists is to obtain a satisfactory model of reality. It is not develop a model of how "i want universe was" ignoring both difficulties and inconsistencies.

I already explained to you in the past that even ignoring experimental data, asymptotic flatness is theoretically imposible inside GR teoretical framework, because basically one is assuming that one may ignore light cones and causality.

I do not understand your last part

It is clear that all predictions of Newton's theory which have ever been tested are derived assuming that there is no significant amount of matter "at infinity."

Do you refer that Newtonian potential may verify Phy --> 0 when R --> infinite?

 

[Juan R.]

pervect, I think you might have misunderstood me. I was not saying the (full) Newton-Cartan theory is equivalent to GR, and I'm pretty sure the original poster wasn't either. The argument that I had with him was over the reduction of GR to a special case of the Newton-Cartan theory in a particular limit (basically 1/c \rightarrow 0). It is known that this special case is basically Newton's original theory (as any such limit should be), although Juan disagrees because it uses a notation that looks very different from Newton's.

You interpretation of my post is very superfitial. You are claiming that i disagree with that Newtonian limit because of notation issues! Please read again my posts. I do not think that Penrose is talking of notation issues and i do not think that people that does not follow Ehlers derivation is talking of notational issues. The violation of causality is not about notation

Moreover i have a question for you. What is the curvature of spacetime on the particular limit (basically 1/c \rightarrow 0) according to GR.

If i take that particular limit on a GR Schwarzild metric, i obtain, for example

g₀₀ = 1

which is the same value that SR FLAT metric.

 

[robphy]

Not this again ... There was already a thread where I spent way too much time arguing with Juan on this. It was eventually moved to theory development, and then locked.

(snipped good points)

Stingray, can you direct me to that thread? I'd be curious to see how you argued those points.

During many time i have searched a complete and rigorous derivation of Newtonian limit from GR but i found none. I suspect that it does not exist!

Since you say you have "searched", may I ask for a list of journal references that you have found?

It may be the case that such a "complete and rigorous derivation" does not exist [at this time] and that what we have heard could be called a "folklore theorem" up to this time. So, finding such a "derivation" [to your satisfaction] may be of interest (Research problem!)... although I don't think that the failure to find one implies (say) that GR is wrong... it may simply be that your problem (formulated as a mathematical theorem which would presumably state precisely how the limit is being taken) is somehow not well posed or well formulated. If there are assumptions that you don't like in the various attempts to such a theorem, you might try to weaken the assumptions or find alternate ones before completely abandoning their approaches.

Since I am away from my usual desk [my city is closed off right now], I can only offer these references (as an answer to your original post) as a starting point to the literature that I am aware of (although I have not completely read them):

http://www.arxiv.org/abs/gr-qc/0506065
also at http://www.lps.uci.edu/home/fac-staff/faculty/malament/papers/GRSurvey.pdf
http://www.lps.uci.edu/home/fac-staff/faculty/malament/papers/GravityandSpatialGeometry.pdf
http://www.lps.uci.edu/home/fac-staff/faculty/malament/papers/NewtCosm.pdf

Perhaps to your disappointment, these do use a Newton-Cartan type formulation. I just can't see any other way.

Please don't forget to provide your references.

 

[Juan R.]

Since you say you have "searched", may I ask for a list of journal references that you have found?

Thanks by your list!

An incomplete list is

arXiv:gr-qc/0304014 v1 2 Apr 2003

Commun. Math. Phys. 166, 221-235 (1994).

arXiv:gr-qc/9610036 v1 16 Oct 96

arXiv:gr-qc/9806108 v1 27 Jun 1998

arXiv:gr-qc/0506123 v1 27 Jun 2005

arXiv:gr-qc/9604054 v1 29 Apr 96

arXiv:gr-qc/0004037 v2 21 Jul 2000

Also studied several books. Wald, MOller, etc.

Also several courses and lecture notes, e.g that of Sean M. Carroll available online, etc.

It may be the case that such a "complete and rigorous derivation" does not exist [at this time] and that what we have heard could be called a "folklore theorem" up to this time. So, finding such a "derivation" [to your satisfaction] may be of interest (Research problem!)... although I don't think that the failure to find one implies (say) that GR is wrong... it may simply be that your problem (formulated as a mathematical theorem which would presumably state precisely how the limit is being taken) is somehow not well posed or well formulated. If there are assumptions that you don't like in the various attempts to such a theorem, you might try to weaken the assumptions or find alternate ones before completely abandoning their approaches.

It is more simply that all that. I did a well defined question in PF.

I simply read on textbooks (or even in original Einstein writtings) that GR reduces to NG in the appropiate limit. I simply want that anybody indicates to me a reference where i can learn the derivation, since all i have revised is not a "derivation" is just a "this looks like".

I already knew your

http://www.arxiv.org/abs/gr-qc/0506065

Effectively, it is based in NC, which is not Newton theory. It clearly assumes that decompositon into flat derivative more potential is far from unique. He claim that one may check that from boundaries, but does not specify what boundaries are correct. Ehlers ones? Are unphysical like already said.

Others authors fix the split of derivative operator adding new equations that are not contained in GR. At the best one is deriving and theory that is not NG, from a theory that is not GR (because one add new equations).

http://www.lps.uci.edu/home/fac-sta...rs/GRSurvey.pdf is the same

The author says

It is significant for several reasons. (1) It shows that several features of relativity theory once thought to be uniquely characteristic of it do not distinguish it from (a suitably reformulated version of) Newtonian gravitation theory. The latter too can be cast as a “generally covariant” theory in which (a) gravity emerges as a manifestation of spacetime curvature, and (b) spacetime structure is “dynamical”, i.e., participates in the unfolding of physics rather than being a fixed backdrop against which it unfolds.

Point (1) is false and nobody has proved this (I think), in fact the author does not show (1). And on (b) is gravity in the limit c-->infinite described via curved spacetime? In standard GR the metric becomes flat on that limit.

GR may explain all gravitational phenomena, not only a part of phenomena and NG other part. Here my emphasis on where NG is derived from GR, (not if some papers or books or course claim that one "could" derive it)

http://www.lps.uci.edu/home/fac-sta...ialGeometry.pdf

I do not understand the metric (1,0,0,0) that obtains in page 407. I do not see derivation of Newton law. i do not see how fixes the "gauge" of the curved derivative (via boundaries?), etc.

Moreover, i think that it is imposible that author is deriving Newtonian Poisson equation when in page 410 is assuming a Poisson like equation for the Riemann tensor (which is logical if one begins grom GR).

In fact, the author does not prove like one can obtain a function with implicit time dependence "Newtonian Phy", from a equation with explicit time dependence Rab. I think that he simply obtain

nabla (Phy) = 4 pi rho

and after he believe that like Newtonian equation is

nabla (Phy) = 4 pi rho

then both are equal because "look equal" (this is also one of problems of textbooks, Newtonian law that appears in GR textbooks is not the Newtonian equation).

Phy in Newton equation is not the same that Phy derived from Rab (as said above) because very different funcional forms.

http://www.lps.uci.edu/home/fac-sta...rs/NewtCosm.pdf

here assumes that "boundary conditions are needed" for fixing the gauge but said not what one would use or if those solve the problem.

does not obtain Newtonian law. Does not obtain Poisson-Newtonian law (only work with some that looks like by using the same notation). Does not explain like one could do the transition from D'Alembert to Poisson regimes (it is imposible from GR), etc.

Again, are there some strict and real derivation in literature or only is a "myth" due fact that people use the same notation?

 

[Juan R.]

For example in the Wald, one finds

a = - GRAD (Phy) [GR]

and in Newtonian mechanics (which is experimentally verified)

one finds

a = - GRAD (Phy) [NG]

they look equal because Wald use the same notation, but are not equal!

t in [GR] is not the t in [NG]. In fact, are equal only when there is not gravitation.

Above [GR] is defined only for c finite (due to curvature of spacetime). [NG] works with c--> infinite. If c--> infinite curvature --> 0 and according to geometric approach of GR a=0 but [NG] which is defined for c--> infinite says that a =/= 0.

Spacetime in [GR] is (ct, x) and, therefore, one cannot apply Galilean Transformation. There is not that in [NG] where one always applies GT between frames.

In [GR] Phy = Phy(x,t) but in [NG] Phy = Phy(R(t)).

In practice, astronomers work with [NG] and its Phy = Phy(R(t)) and after add numerically some of [GR] effects (e.g. perihelion anomaly) to computation of orbits.

In [NG] does not appears c, in [GR] appears. In fact, in [GR]

Phy = -GM / {R-vR/c}

Etc.

I ask again, are there two theories, one [NG] used in some problems and other [GR] used in others?

Where can I find derivation of [NG] from [GR]?

 

[Berislav]

Then Diagram of theories that Penrose uses in his last book (i read time ago but I do not remember title sorry. Help!) where Quantum gravity is reduced to GR when h-->0

There is no problem with that in theories dealing only with bosons. If one has fermions as well, the quantum numbers would go to infinity. There exist, however, variables of anticommuting c(lassical)-numbers which can be thought of as the classical limit of a Fermi quantum system. They are a part of the Grassman algebra, which is used in string theory and other theories of quantum gravity. But it is unphysical.

or where Quantum gravity is reduced to QFT when G --> 0 and this to NQM when c--> infinite is pure fantasy.

I don't know. I see no reason for this to be a problem.

Spacetime in [GR] is (ct, x) and, therefore, one cannot apply GT. There is not that in [NG] where one applies GT between frames.

Yes, because there is no spacetime in Newtonian physics. If there was a spacetime then it would be SR. The Lorentz transformation can be derived from the Minkowski metric. From the LT it is clear that Newtonian physics is the limit of SR, as is it clear from the limit of the equations for geodesic motion that Newtonian gravity is the limit of GR.

 

[Juan R.]

There is no problem with that in theories dealing only with bosons.

What? Bosonic string theory?

I don't know. I see no reason for this to be a problem.

Read above Dirac quote!

 

[Juan R.]

From the LT it is clear that Newtonian physics is the limit of SR, as is it clear from the limit of the equations for geodesic motion that Newtonian gravity is the limit of GR.

Where, that limit of the geodesic motion is computed please. i already explained that equation that appears in textbooks is not the Newton equation. Please read my posts. I detailed above why

a = -GRAD (Phy) on Wald textbook is not the Newton law of motion. You are simply ignoring.

For example in Wald Phy = Phy(x, t) In Newton law Phy = Phy(R(t)), but do not explain like one funtional form is derived from the other appealing to a magical limit that nobody has done still.

People do is work directly with [NG] newer with the [GR] that looks equal but is not equal. Still if you take c--> infinite on [GR] curvature of spacetime tends to zero and then the geodesic interpretation break. When c--> infinite the geodesic equation of motion is a = 0 but [NG] is a =/= 0

 

[Berislav]

There is no problem with that in theories dealing only with bosons.

What? Bosonic string theory?

All Bose quantum systems. Not just bosonic string theory.

Read above Dirac quote!

Sorry. I can't find a quote by Dirac.

a = -GRAD (Phy) on Wald textbook is not the Newton law of motion. You are simply ignoring.

I'm sorry if it seemed that way.

You said:

...
a = - GRAD (Phy) [NG]

they look equal because Wald use the same notation, but are not equal!

t in [GR] is not the t in [NG]. In fact, are equal only when there is not gravitation.

And I said:

Time is not a dimension in Newtonian physics, it's a parameter.

Yes, because there is no spacetime in Newtonian physics. If there was a spacetime then it would be SR.

 

[Juan R.]

All Bose quantum systems. Not just bosonic string theory.

Then are you claiming that any bosonic theory is a quantum gravity and reduce to GR when h --> 0

Sorry. I can't find a quote by Dirac.

post #13

You said:

a = - GRAD (Phy) [NG]

they look equal because Wald use the same notation, but are not equal!

t in [GR] is not the t in [NG]. In fact, are equal only when there is not gravitation.

And I said:

Time is not a dimension in Newtonian physics, it's a parameter.

Yes, the concept of time is different. I rectify above phrase.

the clock rate t in [GR] is not the the clock rate t in [NG]. In fact, are equal only when there is not gravitation.

Where is then [NG] obtained from [GR]?

Still more simple (we would begin from a more simplistic question)

Where can I find the derivation of Newton second law from GR motion law?

 

[Berislav]

Then are you claiming that any bosonic theory is quantum gravity and reduce to GR when h --> 0

No. I'm claiming that that limit is not a problem in any bosonic theory.

Where is then [NG] obtained from [GR]?

It's there. But the mathematical apparatus is different - time has no geometric properties in Newtonian physics and hence it makes no sense to compare the time coordinate in relativity to the classical notion of time as just a variable.

post #13

Oh, right. I don't think Dirac was talking about this, though.

 

[JesseM]

Several!

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



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

The asymptotic flatness assumption isn't really supposed to be an assumption about cosmology, is it? Isn't it just supposed to represent something like the idea that you have a system far enough from other sources of gravity that you don't have to worry about their effects? And if you object to such an assumption in GR, how come you don't object to it in Newtonian mechanics? After all, Newton's derivation of why planetary orbits are elliptical assumes the only object that has any non-negligible gravitational effects on the planet is the sun, would you treat this as a cosmological assumption too and say it's unphysical since there are actually a lot of other stars in the universe besides the sun? If not, what's the difference?

 

[Juan R.]

No. I'm claiming that that limit is not a problem in any bosonic theory.

Ahh! now you are claiming that ANY bosonic theory in the limit h --> 0 coincides with classical physics. Sure? Also a boson described via KG equation reduces to correct classical limit when h --> 0.?

It's there.

Where?

But the mathematical apparatus is different - time has no geometric properties in Newtonian physics and hence it makes no sense to compare the time coordinate in relativity to the classical notion of time as just a variable.

Nooo! It is the physics what is different! This is the reason that

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

but

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

when mathematically both limits are defined in functions Phy or [GR] and Phy of [NG]

Again, i ask to you if t in [GR] equation a = - GRAD (Phy) is different of t in [NG], what t may i take the Newtonian clock the Einstein clock?

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

i am computing the trajectory of Earth around Sun using both Newton equation and Wald equation (which is calimed to be the Newton equation but is not). Now Sun explode and his mass is (1/2) Sun mass. How would i compute the trajectory of Earth 3 seconds after of Sun explosion if i use above [GR] equation and if i use [NG].

Of course, if it is imposible to derive NG from GR via standard geodesic motion, one would follow Cartan-like via -which is, curiously, done by specialists working in the topic-. How i obtain the "4D Newtonian gravity" from GR? Using that?

Unphysical boundaries of any universe that is not our universe?

Using ad hoc equations does not contained in GR?



If i am using ad hoc equations, does this indicate that GR alone is not sufficient?

Oh, right. I don't think Dirac was talking about this, though.

I remember that you replied to my

or where Quantum gravity is reduced to QFT when G --> 0 and this to NQM when c--> infinite is pure fantasy.

with

I don't know. I see no reason for this to be a problem.

I cited Dirac talking about the limit c --> infinite, and now you claim that Dirac was not talking about that

I read again and i see Dirac critizing QFT because does not reduce to NRQM when c --> infinite. The clearly says that one work with two inconsistent theories. Curiously, one also work with two inconsistent theories in gravitation.

As said, Michele Vallisneri, Theoretical Astrophysics of Caltech, also note that in practical problems we are using inconsistent theories...

Adapted to gravitation Dirac thoughts on RQFT would read

They argue that if one has rules for doing [astrophysical] calculations and the results agree with observation, that is all that one requires. But it is not all that one requires. One requires a single comprehensive theory applying to all [gravitational] phenomena. Not one theory for dealing with non-relativistic [gravitational] effects and a separate disjoint theory for dealing with certain relativistic [gravitational] effects.

 

[Juan R.]

The asymptotic flatness assumption isn't really supposed to be an assumption about cosmology, is it?

Irrelevant!

If you are working in solar system tests using Cartan like formulation you may fix the gauge. Ehlers does via

limit R --> infinite Phy = 0

that is, asymptotic flatness.

Isn't it just supposed to represent something like the idea that you have a system far enough from other sources of gravity that you don't have to worry about their effects?

No! you are mixing asymptotic flatness with the principle of decomposition of clusters.

And if you object to such an assumption in GR, how come you don't object to it in Newtonian mechanics?

A detail, i am not Penrose.

No, in Newtonian mechanics you are applying decomposition of clusters which is experimentally verified.

After all, Newton's derivation of why planetary orbits are elliptical assumes the only object that has any non-negligible gravitational effects on the planet is the sun, would you treat this as a cosmological assumption too and say it's unphysical since there are actually a lot of other stars in the universe besides the sun? If not, what's the difference?

No! Newtonian gravity does not assumes asymptotic flatness. This is reason that Penrose, I, Christian, and others have no problem with Newtonian gravity.

Penrose says that our universe is not of island type. He of course does not claim that Newtonian gravity was unphysical. In fact he work with it and with quantum generalizations of it.

I see many confusion here on the topic of gravitation and i am wasting my time with irrelevant replies that either simply are ignoring details i am saying or do not really know physics under both [GR] and [NG].

After of 26 unuseful replies Perhaps i would post my question on sci.physics.research

 

[JesseM]

Irrelevant!

If you are working in solar system tests using Cartan like formulation you may fix the gauge. Ehlers does via

limit R --> infinite Phy = 0

that is, asymptotic flatness.

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?

A detail, i am not Penrose.

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).

 

[chronon]

Newton believed that a homogeneous infinite universe could be static. Einstein found that under GR such a universe could not be static. So it looks like there isn't going to be any way to derive Newtonian gravity as a limiting case of GR.

My feeling is that Newton's intuition was wrong here, and that the universe can't be static with Newtonian gravity any more than with (zero cosmological constant) GR.

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.

 

[pervect]

The asymptotic flatness assumption isn't really supposed to be an assumption about cosmology, is it? Isn't it just supposed to represent something like the idea that you have a system far enough from other sources of gravity that you don't have to worry about their effects? And if you object to such an assumption in GR, how come you don't object to it in Newtonian mechanics? After all, Newton's derivation of why planetary orbits are elliptical assumes the only object that has any non-negligible gravitational effects on the planet is the sun, would you treat this as a cosmological assumption too and say it's unphysical since there are actually a lot of other stars in the universe besides the sun? If not, what's the difference?

An expanding universe (FRW space-time) won't strictly conserve energy, so it can't be made rigorously equivalent to an asymptotically flat space-time. You *can* keep energy conserved for that "matter" part of the universe, which consists of particles nearly at rest in the isotropic-CMB frame, via a suitable definition of "energy". However, you cannot simultaneously do this and keep the energy in the "radiation" terms conserved. The "radiation" terms include the energy in the CMB itself and any other source of energy which contributes to "pressure" of the cosmological stress-energy tensor.

Since our universe is matter-dominated, conserving the energy in the matter terms is the right thing to do as far as approximations go. Because our universe is matter dominated, the non-conservation of energy is small even over cosmological distance scales (using the approrpriate aproximate defintion of "energy" discussed above) and is totally undetectable for a solar-system sized experiment.

 

[robphy]

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.]

After of 26 unuseful replies Perhaps i would post my question on sci.physics.research

Good luck.

 

[pervect]

pervect, I think you might have misunderstood me. I was not saying the (full) Newton-Cartan theory is equivalent to GR, and I'm pretty sure the original poster wasn't either. The argument that I had with him was over the reduction of GR to a special case of the Newton-Cartan theory in a particular limit (basically 1/c \rightarrow 0). It is known that this special case is basically Newton's original theory (as any such limit should be), although Juan disagrees because it uses a notation that looks very different from Newton's.

OK, that makes a little more sense. I tend to think in the limit as v->0 rather than in the limit as c->infinity, but it's the same thing.

You might want to look over my response to Jesse and see if you have any comments about it. Basically I argue that because we only have an apprxomiate notion of energy conservation in an expanding universe (FRW cosmology), without asymptotic flatness we cannot rigorously get Newtonian physics which does strictly conserves energy. However, this isn't particularly scary, because with an appropriate approximate defintion of "energy", the non-consevation due to the universal expansion is small even on a cosmological scale, and is totally undetectable in a solar-system sized experiment.

Reading this over, it's a bit hand-wavy, so I'll refer to MTW page 705 for a further clarification of how to go about doing definiing a suitable measure of "energy", though I won't go through the detailed calculations to show just how small the non-conservation is.

 

[Berislav]

Ahh! now you are claiming that ANY bosonic theory in the limit h --> 0 coincides with classical physics. Sure? Also a boson described via KG equation reduces to correct classical limit when h --> 0.?

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

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

Potential is not physical, it's gauge.

i am computing the trajectory of Earth around Sun using both Newton equation and Wald equation (which is calimed to be the Newton equation but is not). Now Sun explode and his mass is (1/2) Sun mass. How would i compute the trajectory of Earth 3 seconds after of Sun explosion if i use above [GR] equation and if i use [NG].

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.

Unphysical boundaries of any universe that is not our universe?

Using ad hoc equations does not contained in GR?



If i am using ad hoc equations, does this indicate that GR alone is not sufficient?

I really don't understand.

 

[Stingray]

Stingray, can you direct me to that thread? I'd be curious to see how you argued those points.

Here you go: https://www.physicsforums.com/showthread.php?t=75197

Anyway, I agree with pervect. If you restrict yourself to a non-asymptotically flat spacetime (and I agree this is realistic), an exact Newtonian limit is not possible. 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.

Another question would be to ask whether non-asymptotically flat solutions can look like Newtonian gravity when certain quantities are small. That can't be done as elegantly because it is not exact (the matter distributions are necessarily different). Still, you can look at the NC limit before the asymptotic flatness assumption is used, and show that the corrections are negligible in practical situations. There are also (for example) generalizations of Schwarzschild with cosmological boundary conditions that you can play around with.

 

[JesseM]

An expanding universe (FRW space-time) won't strictly conserve energy, so it can't be made rigorously equivalent to an asymptotically flat space-time.

Doesn't the expansion rate approach zero as time approaches infinity for a flat universe with no cosmological constant? What prevents you from looking at how a given system will behave in the limit as cosmological time approaches infinity?

 

[pervect]

Doesn't the expansion rate approach zero as time approaches infinity for a flat universe with no cosmological constant? What prevents you from looking at how a given system will behave in the limit as cosmological time approaches infinity?

I think that's a reasonable approach, though one does have to assume that the cosmological constant is zero to make it work. Current observational evidence combined with standard GR suggests that the cosmological constant is non-zero.

This is only true with standard GR, though - non-standard theories such as Garth's SCC have a rather different slant on the whole affair, both as far as the existence of a cosmological constant goes, and the definition of energy.

As far as SCC itself goes, we'll have an experimental test of it from the gravity probe B results in about a year. At that point, one of GR or SCC is going to be falsified. If I had to bet, I'd go with GR as the main candidate, based on the number of experimental tests it's already gone through. But we won't know if it will pass the latest until the GPB results are in.

 

[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 (\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. <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Dirac is saying that in the limit c--&gt; 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 &quot;cristal clear&quot;. Is correct this expresion? </div> </div> </blockquote> I meant that I don&#039;t know what problem, exactly, is he refereing to. Where is the problem in QFT when c---&gt; infinity. <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 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 </div> </div> </blockquote> No, because in that limit the two metrics aren&#039;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 (\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.

<br /> <br /> 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 &quot;Unphysical&quot; 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.<br /> <br /> That is unphysical, that is experimentally false is the limit in GR but is experimentally correct in NG.<br /> <br /> <blockquote data-attributes="" data-quote="Berislav" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Berislav said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> I meant that I don&#039;t know what problem, exactly, is he refereing to. Where is the problem in QFT when c---&gt; infinity. </div> </div> </blockquote><br /> Dirac is cristal clear. There exit two inconsistent theories: one for non relativistic phenomena, other for certain relativistic phenomena.<br /> <br /> <blockquote data-attributes="" data-quote="Berislav" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> Berislav said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> No, because in that limit the two metrics aren&#039;t equivalent. </div> </div> </blockquote><br /> In the limit c ---&gt; infinite<br /> <br /> g00 = 1 and gRR = -1<br /> <br /> curvature R_ab = 0<br /> <br /> Therefore, according to geometrical interpretation of gravity in GR, cannot exist gravitational &quot;force&quot; a = 0<br /> <br /> From NG, however, a =/= 0