# Advanced books/papers on derivation of Newtonian mechanics from GR

1. Sep 15, 2005

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Please do not cite Cartan-like derivation, because that one obtains there is a modified (geomtrized) version of Newtonian mechanics after using additional asumptions like the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Please do not cite usual textbooks.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

2. Sep 15, 2005

### robphy

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).

3. Sep 15, 2005

### Juan R.

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

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

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]

4. Sep 15, 2005

### robphy

What is your issue with this boundary condition?

5. Sep 15, 2005

### DrChinese

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?

6. Sep 15, 2005

### Berislav

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.

7. Sep 15, 2005

### 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.

8. Sep 16, 2005

### pervect

Staff Emeritus
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.

9. Sep 16, 2005

### 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.

10. Sep 16, 2005

### Juan R.

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.

Last edited: Sep 16, 2005
11. Sep 16, 2005

### Juan R.

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.

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?

Last edited: Sep 16, 2005
12. Sep 16, 2005

### Juan R.

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.

13. Sep 16, 2005

### Juan R.

Nobody obligate to you

Arguing =/= demostration

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, wich 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.

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

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

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

Last edited: Sep 16, 2005
14. Sep 16, 2005

### Juan R.

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.

15. Sep 16, 2005

### robphy

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

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.

16. Sep 16, 2005

### Juan R.

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 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".

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 [Broken] is the same

The author says

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 [Broken]

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 [Broken]

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?

Last edited by a moderator: May 2, 2017
17. Sep 16, 2005

### 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]?

Last edited: Sep 16, 2005
18. Sep 16, 2005

### Berislav

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.

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

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.

Last edited: Sep 16, 2005
19. Sep 16, 2005

### Juan R.

What? Bosonic string theory?

20. Sep 16, 2005

### Juan R.

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