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Please do not cite Cartan-like derivation, because that one obtains there is a modified (geomtrized) version of Newtonian mechanics after using additional asumptions like the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Please do not cite usual textbooks.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Please do not cite Cartan-like derivation, because that one obtains there is a modified (geomtrized) version of Newtonian mechanics after using additional asumptions like the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Please do not cite usual textbooks.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. :tongue2: 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. :tongue2: 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. :rofl:

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. :rofl:

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.