Is GR a wrong apparoach to gravitation?

[Stingray]

Are you asking why light doesn't move on the coordinate lines? Those lines are arbitrary, and are mainly chosen so that the picture looks good. Nothing would naturally move along them.

 

[Anomalous]

Are you asking why light doesn't move on the coordinate lines? Those lines are arbitrary, and are mainly chosen so that the picture looks good. Nothing would naturally move along them.

Now I am more confused. Thoes black lines show that the space is bend towards the BH in the nearby areas , So are not cordinates of space bending towards the BH, How else can space bend ?

 

[Juan R.]

Waiting for Ehlers' paper

Stingray

I solicited a copy of Ehlers J: Examples of Newtonian Limits of Relativistic Spacetimes, Class. Quantum Grav. 14 (1997), A119 but I have not got still.

Still I can do some preliminary comments (remember that I didn’t read still Ehlers work) and all is based in my survey of last days.

Ehlers' work appears to be mainly focused to cosmological models.

It appears that his work has not been very popular for the construction of post-Newtonian models.

When I mean the recovering of Newton gravity from GR, I mean a consistent derivation of the full Newtonian model. Of course, one can obtain the “correct” Newton equation for trajectories in coordinate time

d^2 x / dt^2 = – “time-time connection”

but the physical metric corresponds to curved spacetime g = nu(SR) + gamma.

Often, one takes formally the c--> infinite in the obtaining of coordinate time, but one maintains c finite in the nu metric.

Generally, one argues for the derivation of “correct” Newtonian equation

a = – grad (phy)

from the “geodesic” equation

a + “time-time connection” = 0

and, therefore, the covariant derivatives does not commute, this implies that one cannot use ordinary derivatives in this regime.

If one want that covariant derivative exactly coincides with ordinary derivatives then one obtain that bodies are unaffected by gravity.

If one works all of this in detail for a Schwarzschild metric, one obtains either a pure flat spacetime with zero affine connections and zero Rieman curvature tensor, or usual GR “linear” gravitation on curved spacetime and c finite. Newtonian theory is a theory of gravity in flat spacetime and c --> infinite.

I don’t see how Ehlers’ work can modify this maintaining intact the basic structure of GR.

Please read my reference. Newtonian gravity can be recovered as a formal c->infinity limit without any ad hoc procedures (at least for asymptotically flat spacetimes).

As said I didn’t read paper yet, but I have found others interesting related works. It is interesting that other author refers to same Ehlers’ work like the “c--> infinite” limit and carefully emphasize the “”. This suggests to me that Ehlers is performing not the real limit after of all, only some formal "reparametrization". Of course, i am not sure of thyis because didn't read the article.

Yet Ehlers use at least (I didn’t read the paper) an ad hoc assumption: asymptotically flat spacetime. Not only is ad hoc, moreover, it is unphysical. In the basis of experimental evidence and analysis from Penrose or Misner:

“universe is not an island of matter surrounded by emptiness”.

Perhaps other better work imposing on the curvature tensor an ad hoc condition "prohibiting rotational holonomy" can permit us obtain Newton gravity in a “consistent” manner, but I doubt like one can obtain curved “geodesic” motion with a zero Christoffel. All attempts that I know until now are mixed approaches with flat structure plus a Newtonian potential = non-flat spacetime such as the world lines of test bodies follow the true non-flat metric. If the true metric is flat there is no gravitation, only pure free motion.

I unknown if Ehlers’ paper deals with solar system problem, but all works that I am seeing are focusing to cosmological issues, where one may expect deviations from pure Newtonian gravity and therefore the fact one does not obtain exactly Newtonian theory is not a problem, it is a virtue.

I think that GR clearly states that gravitation is curvature as was said by Einstein. By curvature I do not mean exclusively the Riemann curvature tensor, since that Christoffel symbols are another form of defining curvature.

Your distinction between “curvature” or “connection” regarding the true origin of gravity is not applicable to my non-technical work

http://www.canonicalscience.com/stringcriticism.pdf

Because I mean that in pure Newtonian theory, both vanish.

I think that Carlip know very well that there is no complete derivation of Newtonian theory from GR and that there are problems still unsolved. The cite that I quoted

“general relativity very nearly reproduces the infinite-propagation-speed Newtonian predictions. ”

is best understood in their surrounds

For weak fields, however, one can describe the theory in a sort of Newtonian language. In that case, though, one finds that the "force" in GR is not quite central---it does not point directly towards the source of the gravitational field---and that it depends on velocities as well as positions. The net result is that the effect of propagation delay is almost exactly cancelled; general relativity very nearly reproduces the infinite-propagation-speed Newtonian predictions.

It signifies that is not clear if GR can describe solar system dynamics due to aberration and other issues. There is no consensus if GR is compatible with experimental data or no. I think that no, as said many standard “proofs” and “verifications” are misleading. For example, the famous recent claim of measure of gravity speed may be seen like misleading (was not measure of that). Recent Carlip's paper in aberration of celestial bodies is, unfortunately, full of failures, and finally he agrees that interpretations of others using instantaneous interaction (for example canonical gravitodynamics) are consistent with experimental absence of aberration. But he is not demonstrating that absence of aberration is consistent with GR.

As said I didn’t not post here what are the errors of Carlip’s papers (in fact, one would need several pages in a paper for a detailed following), but I put an "indicator".

One cannot demonstrate a thing if begins assuming that thing in the form of a hidden assumption.



Therefore, one needs a theory of gravity with next requirements:

1) A theory giving exactly the Newtonian limit in a flat Euclidean space and absolute time. “Cartan-like” covariant “reformulations” are not that.

2) A theory for gravity on a flat spacetime. Unless one can measure curved spacetime, all our experimental evidence is for flat space and time.

3) A theory explaining usual Solar system tests: perihelion, radar delay, redshift, etc.

4) A theory explaining other tests, e.g. binary stars, but without appeal to unobserved gravitational waves, etc.

5) A theory where gravity speed is infinite. The model cannot violate SR but may, at the same time, fits experimental orbiting and astronomical data on BH, binary stars, aberration, etc.

6) A theory departing from GR at extragalactic regimes explaining data and empirical laws (e.g. TF one) without ad hoc assumptions like unobserved dark matter and fine tuning with two-three parameters.

7) A theory unified with EM.

8) A theory that can be satisfactorily quantized from first principles.

9) Solving of most hard problems of cosmology: inflation, cosmological dark matter (90%!), cosmological constant, etc.


At least twenty-five alternative theories to Einstein GR have been investigated from the 60’s. I cannot say that I have solved all those problems already (I don’t studied 9 still) but already said that things I have already obtained.

The research is very young but very, very promising.

 

[Stingray]

Now I am more confused. Thoes black lines show that the space is bend towards the BH in the nearby areas , So are not cordinates of space bending towards the BH, How else can space bend ?

If you had a bent rubber sheet in front of you, would you need lines drawn on it to know that it was bent?

Coordinates can be chosen in all kinds of ways that don't necessarily have any physical significance. It's actually very rare to have any good notion of a "preferred" coordinate system in curved spacetimes (like Cartesian coordinates in flat spacetime). Going from some 'random' system of coordinates to something physical or invariant is what a large part of differential geometry is about.

 

[Stingray]

Ehlers' work appears to be mainly focused to cosmological models.

There is no restriction in that article to cosmological models. In fact, it is somewhat more applicable to solar system-type problems to cosmological ones.

I don’t see how Ehlers’ work can modify this maintaining intact the basic structure of GR.

He does not go by the route you described. The structure of GR remains intact. The Newtonian structure is what is different than what you are expecting.

Although Newtonian spacetime is flat in a sense, it is not nearly as simple as Minkowski spacetime. There are two natural metrics, and both are singular. There always exists a coordinate system where one metric takes the form diag(1,0,0,0), and the other diag(0,1,1,1). It is clearly impossible to invert these, so you can't compute connections or curvatures from them.

At a fundamental level, though, connections are not defined by metrics. They have a separate existence describing the notion of parallel transport. This is certainly meaningful in Newtonian gravity, so there is a connection. Restricting the connections to ones which reproduce Newton's concepts does not leave you with something that is necessarily zero. The remaining freedom is actually just enough to allow for a gravitational potential giving you all the motions you'd expect.

This was all worked out back in the 1920's by Cartan. The rumor is that he got annoyed at Einstein's claims that GR was the only theory that remained elegant in generally covariant form, so he went off and showed that Newton's theory was only slightly more complicated!

If you're going to try to find a Newtonian limit of GR, then you really have to write the two in same (generally covariant) language first. This requires adopting Cartan's notation/formalism for Newton's theory. Ehlers shows that Einstein's theory goes over to Newton's in this way.

He writes Einstein's theory in an unusual way with a free parameter equal to 1/c^2. When this parameter is zero, you get something called Newton-Cartan theory (trivially). This isn't quite what I outlined above, but a slight generalization of it. It allows for what might be thought of as an overall rotation of the universe. If you only allow for asymptotically flat solutions, then you recover Newton's theory (in Cartan's notation).

I do not understand your objection to this last step. Newtonian gravity is only used in asymptotically flat problems. You would only encounter possible problems when going to cosmology, and to quote you,

[...] cosmological issues, where one may expect deviations from pure Newtonian gravity and therefore the fact one does not obtain exactly Newtonian theory is not a problem, it is a virtue.

I'll continue in another post...

 

[Stingray]

I still think Carlip is saying what I outlined in a previous post. I can't see how you're misinterpreting him. Maybe it's a language problem?

Anyway, his paper isn't meant to be profound. It's basically something to demonstrate to students how things work. There are much more developed post-Newtonian and post-Minkowski formalisms around, and these do match up very well to solar system observations. Yes, you can agree with solar system measurements by starting from some non-GR, non-Newtonian theory having instantaneous interactions, but this is trivial. It's just a statement that you can do a Taylor expansion when everything is moving at speeds much less than c.

1) A theory giving exactly the Newtonian limit in a flat Euclidean space and absolute time. “Cartan-like” covariant “reformulations” are not that.

2) A theory for gravity on a flat spacetime. Unless one can measure curved spacetime, all our experimental evidence is for flat space and time.

To point 1: Cartan's reformulation of Newtonian gravity is exactly equivalent to the original version. There is an absolute time, and the spatial hypersurface defined by a single instant of time is Euclidean.

Point 2: Ok, fine. In some sense, "curved spacetime" is just semantics. It's a very nice mental picture, though, and I don't know why you want to spoil it.

 

[Juan R.]

On the Newtonian limit

Stingray

First an important detail; initially I was a “believer” on GR, but a problem with symmetries in canonical science obligated to me to reconsider the question of gravity. Then I tought that (see page 17 of www.canonicalscience.com/stringcriticism.pdf[/URL]) a canonical force in a flat spacetime could be compatible with curved spacetime GR gravitation, somewhat in the spirit of Lagrangian mech. <=> Hamiltonian mech. A more rigorous analysis, from derivation of Newtonian limit, to unification with EM, quantization requirements passing by some Solar system test, etc. obligated to me to re-thinking about gravitation. A careful discussion will appear in the paper.

Still I didn’t read Ehlers’ paper, but have studied additional stuff on the topic, and think that I already have got the point.

From standard GR (let me call it “metric gravity”) one cannot obtain Newtonian theory. I want to be clear here, the “metric” approach does not obtain the original Newton theory. Therefore, people have searched for alternative ways. Now my comments on “affine gravity” ("Cartan-like" approach).

Newton and Einstein are geometrically different.

The first step consists on “reformulate” Newton theory in a covariant form. First, this is not a simple reformulation (like Hamiltonian-Lagrangian of mech.). From a conceptual point of view, Newton-Cartan is not the same than original Newton approach. [b]The geometric Newton-like theory is not the same that original Newton theory[/b]

The second step consists on reformulating also Einstein GR. In some sense, the method resembles the (3+1) formalism of HGR but one works with powers of parameter 1/cc (Ehlers) or 1/c (others). Both approaches are compatibles in the Newtonian limit. The differences arise in post-Newtonian approaches. The curvature Riemann tensor of the Newtonian hypersurfaces is zero, so spatially is flat. Ok.

The total spacetime is not flat and one introduces a single curved
derivative operator. The operator is splinted into two parts: a flat derivative operator more a scalar field.

Since that this reformulation of GR begins with a curved spacetime, only the single curved operator is physical. Therefore, there is ambiguity since the decomposition will not be unique (this resembles to me the infamous problem of time of quantum HGR). Then one cannot obtain original Newton theory from a reformulation of GR. All that one can obtain is a family of geometric Newton-like theories from a reformulation of GR.

For obtaining a single real geometric Newton-like theory, one needs to introduce some additional [i]ad hoc[/i] condition [b]does not contained in GR[/b]. There are many different covariant Newton-like theories and a great discussion in literature on which is the correct (if any).

Ehlers showed that one of the usual [i]ad hoc[/i] equations for the Riemann tensor (the weak condition?) can be obtained from special boundary conditions: he showed from asymptotic flat spacetime and therefore the adittional [i]ad hoc[/i] equation is not need.

I see two problems with that. The first that assumption is unphysical, the so-called “island assumption” by cosmologists. Cosmological experimental evidence does not support it and people rejects it as I already said. Still Ehlers could claim that unobserved asymptotic condition is valid very far from radio of observable Universe, but we cannot see it with our limited spacetime window. Maybe! But it continue being a hypothesis additional to GR.

But the second problem is much more interesting. Really assuming boundaries at infinitum, Ehlers is assuming instantaneous gravitation. This is difficult to see in static models but, in dynamic models, one can see that the choosing of different boundaries [b]would[/b] leave intact the dynamical properties for example at the Solar system scale. Still, only one boundary leave to the correct Newtonian limit (in Cartan sense of course), the others boundaries offer wrong answers. Ehlers is fixing the “gauge” of the decomposition by means of a large (infinite) correlation that connect local spacetime dynamics with spacetimes regions at infinitum. Really very, very interesting.

Summary:

1º) [b]“[/b]Reformulation[b]” [/b] of Newton

2º) Reformulation of GR

3º) [b]Additional assumptions[/b] (equations) for the correct splitting of the defined single curved derivative.

4º) If one want “eliminate” the ad hoc equatios one [b]may assume an ad hoc unphysical boundary[/b] for the geometry that, moreover, is introducing instantaneous gravity.

A note, Ehlers unphysical boundary => instantaneous gravity, but a violation of that boundary (our universe is not of “island” type) =/=> that gravity is delayed because above there is a “=>” and not a “<=>”

And finally one (is exhaust :-) obtains a [i]Newton-like theory[/i] in the limit of c--> infinite, probably full compatible with original Newton theory in an empirical sense

a = -grad(phy)

[b]but incompatible in a conceptual sense[/b]. That is, there are two “phys” numerically agreeing but conceptually (theoretically) different: one is Newton real potential in flat space defining a real force, other is a scalar field that arise in the decomposition and that is related via connections with a physical curved spacetime where test bodies move in a “geodesic manner”.


From [b]canonical gravitodynamics[/b], one obtains the full Newtonian theory without modification of conceptual or theoretical issues simply taking the [b]limit[/b] c--> infinite (without mathematical ambiguities nor singular points) in the expression for canonical force [b]or[/b] applying it to a stationary case because canonical gravitostatics = Newton gravitation

[b]A single well-defined mathematical step and need for zero assumptions outside of the canonical theory[/b].

Canonical => original Newton theory

I call this a rigorous derivation.

Newton-Cartan-like approaches are summarized in

GR reformulation + ad hoc equations => Newton-like theories =/= original Newton theory

There is not derivation because the ad hoc equations (or boundary assumptions) are not derivable from GR alone. Moreover, with each assumption one obtains different “Newtonian” theories: Neo-Newt NG, Max NG, Weak NCG, etc.

 

[Juan R.]

On Carlip’s ideas

Stingray

I disagree, Carlip’s paper is meant to be profound and fixes the beliefs of one of the schools of gravity in dispute. He published this paper in PLA for a rebuttal of others’ ideas. Interestingly, in the web he maintained that instantaneous gravity was impossible, whereas in the final published paper Carlip recognizes that instantaneous explaining of experimental data is also possible. Great!

It is also interesting that Carlip agrees that there is “absence of direct
measurements of propagation speed”. Therefore the GR interpretation v = c is simply a theoretical interpretation like the assumption of curved spacetimes and the belief on gravitational radiation.

Problems with Carlip’s paper already begin with its EM review. He carefully chooses a specific model (velocity constant) with the aim of eliminate accelerating terms in the Electric field. It is interesting that also omit the discussion of magnetic fields (the movement of test charge is affected by both). Summing all physical terms, the force does not appoint toward the “instantaneous” position, contrary to his claim. This is natural and, in fact, is one of greatest problems in numerical Maxwell EM, the instability of computed trajectories due to time-delay. Some authors claim for solving this problem using preMaxwell fields in 5D (in the spirit of Kaluza-Klein :-) but a carefull analysys demonstrate that are using a concept of instantaneus interaction in 5D for coupling the preMaxwell fields.

Curiously, Carlip carefully chooses (he is astute) the models and equations just for eliminating the most part of aberration effects. If you are eliminating it from the beginning, it is very difficult that you obtain it at final.

Carlip (great GR specialist in the words of others) says,

“One could, of course, try to formulate an alternative model in which the Coulomb field acted instantaneously, but only at the expense of ‘deunifying’ Maxwell’s equations and breaking the connection between electric fields and electromagnetic radiation.”

It is completely wrong. Precisely advanced mathematical analysis of Maxwell EM show that there is an implicit “deunifying” of EM into transverse and longitudinal effects, and a “deunifying” on “pure” particles and “pure” fields contributions in the standard approach. From canonical EM a single unified formula can explain those topics.

Curiously, astronomers compute orbits, without retarded positions (violating GR). Only perihelion and light deflection are computed. Time-delay in gravitatory orbits is ignored. However, EM time-delay is always used! Full GR is not used as already said in previous posts.

The effect of a gravitational time-delay destroys the orbit. The effect is very small, probably undetectable in a direct measurement, but it is accumulative, and after of several miles of rotations, the usual orbit is destroyed. Astronomers’ chronology shows no signs of that.

Carlip continues

“If gravity could be described exactly as an instantaneous, central interaction, the mechanical energy and angular momentum of a system such as a binary pulsar would be exactly conserved, and orbits could not decay.

In general relativity, the gravitational radiation reaction appears as a slight mismatch between the effects of aberration and the extra noncentral terms in the equations of motion.

One could again try to formulate an alternative theory in which gravity propagated instantaneously, but, as in electromagnetism, only at the expense of “deunifying” the field equations and treating gravity and gravitational radiation as independent phenomena. ”

The first part is, of coourse, completely wrong. I believe that Carlip misunderstands the concept of “central” force. On the second part (GR) I agree. On the third part again incorrect, see my previous words on EM.

Carlip continues (in the next section) with “The naïve choice for a retarded Newtonian potential would be phy = m/R, where R is the propagation delayed distance”

This is childish, if phy is a number, a simple number, one could be tempted to follow that strange suggestion. But phy is not a number, it is a potential and therefore it has a well defined sense: like a measure of instantaneous correlations at one specific instant. The attempt to relativize that, one may substitute not R by retarded R, if not the delta(t), implicit in the nonrelativistic Hamiltonian formalism due to “collapse” of light cones, by a relativistic delta(tau) for the cones surfaces.

Moreover, Carlip posterior suggestion for phy = m/r, is not complete. The carefull discussion of those and others errors is outside of this forum, but is will be done in the paper in preparation.

 

[Juan R.]

On canonical gravitodynamics vs GR

Stingray

Yes, you can agree with solar system measurements by starting from some non-GR, non-Newtonian theory having instantaneous interactions, but this is trivial. It's just a statement that you can do a Taylor expansion when everything is moving at speeds much less than c.

It is a bit more complex. From a conceptual point of view, the theories are very different. From an empirical point, instantaneous gravity is not equivalent to a Taylor series expansion of a delayed formula in powers of (1/cc). Only first terms are equivalent, to higher orders there exist differences between both approaches. I have no computed still the different terms for gravitation, for solar system test that i did recently i needed just the first expansion. But, in a future, there is possibility for an experimental confrontation.

Now we can compare preliminary canonical with GR

1) A theory giving exactly the Newtonian limit in a flat Euclidean space and absolute time. “Cartan-like” covariant “reformulations” are not that.

2) A theory for gravity on a flat spacetime. Unless one can measure curved spacetime, all our experimental evidence is for flat space and time.

3) A theory explaining usual Solar system tests: perihelion, radar delay, redshift, etc.

4) A theory explaining other tests, e.g. binary stars, but without appeal to unobserved gravitational waves, etc.

5) A theory where gravity speed is infinite. The model cannot violate SR but may, at the same time, fits experimental orbiting and astronomical data on BH, binary stars, aberration, etc.

6) A theory departing from GR at extragalactic regimes explaining data and empirical laws (e.g. TF one) without ad hoc assumptions like unobserved dark matter and fine tuning with two-three parameters.

7) A theory unified with EM.

8) A theory that can be satisfactorily quantized from first principles.

9) Solving of most hard problems of cosmology: inflation, cosmological dark matter (90%!), cosmological constant, etc.

I agree with you in that point 2 is fine, curvature is just semantics. My reply to your question is

A) because there is no experimental evidence of them and i follow Bohr advice of that a physicist may be the most conservator possible. I will choose curved spacetimes if i) they are measured and/or ii) someone shows that from flat spacetime theories one cannot explain all available data.

B) there are problems with the geometrical view, for example regarding the choosing of correct boundaries, the problem of how Earth knows what is the curvature of spacetime (GR proposes no mechanism), energy conservation, etc.

C) Because, that “deunify” physics, precisely it is the problem with quantum gravity and the rest of interactions considered forces. I see very logical to modify current gravity for adapting it to the other three.

D) From a computational point of view, GR is difficult and most of difficulties are unnecessaries for real computations due to weak character of corrections. By this reason, there is so much practical interest in computational models based in direct post-Newtonian approaches.


Corresponding GR points vs canonical ones

1) One cannot obtain exact original Newton theory in either “metric” or “affine” gravity. In the latter (more recent) one needs reformulate GR and add ad hoc assumptions for obtaining not he original theory, just a theory that look like.

2) GR is based in unobserved curved spacetime that enters like a “mathematical tool” in the theory.

3) GR fits with usual Solar system tests very well. I refer to perihelion, light deflection, etc. Others GR effects have not been carefully studied still!

4) GR explains other tests, e.g. binary stars but appealing to unobserved gravitational waves for maintaining energy conservation, etc.

5) GR claims gravity speed is c. There is serious mathematical and empiricial tests on contrary. E.g. absence of orbit aberration is not explained (at contrary of common claims).

6) GR cannot explaining data at extragalactic regimes and empirical laws (e.g. TF one) without ad hoc assumptions like unobserved dark matter or the appeal to fine tuning with two-three parameters or statistical ad hoc asumptions on galactic formation.

7) GR is not unified with EM. Einstein’s search for unified field theory failed.

8) GR is incompatible with QM. All attempts (dozens and dozens) of quantize it have failed. Recent non-commutative program is “stopped”. String theory is at one very bad stage with many recent publications in the form of “no-go theorems”, and the number of publications down this year. LQG continues with its very fundamental problems, in the limits of my knowledge still nobody demonstrated any classical limit, the problem of time remain unsolved, the interpretation of cosmological wavefunctions, etc.

9) GR introduces really difficult problems in the cosmological scale: singularities, need for hypothetical inflation for explaining large scale structure (interestingly this is related with the “local” character of GR), ad hoc assumption of a 90%! of dark matter, the old crux of cosmological constant, etc.

I sincerely think that in an average view of all points canonical gravitodynamics looks very promising :!), especially seing that Einstein developed three or four previous versions of GR before the definitive, and needed of 10-15 intense years, when canonical gravitodynamics is at a very early stage, it born this year, and still has been not published. Once, the first draft manuscript is ready, i will send to several especialists for comment/review and correct possible errors.

Like a scientist, I think that i would follow this way of research closely and verify what we could obtain, even if relativists would prefer the closing of this "dangerous" new posibility, that GR was a kind of Dirac hole theory.

 

[Stingray]

The first step consists on “reformulate” Newton theory in a covariant form. First, this is not a simple reformulation (like Hamiltonian-Lagrangian of mech.). From a conceptual point of view, Newton-Cartan is not the same than original Newton approach. The geometric Newton-like theory is not the same that original Newton theory

What is the difference? All predictions are the same. "Conceptual points of view" are not really important. Any theory can be equivalently reformulated in an infinite number of different ways. Each of these might suggest different underlying concepts. Take, for example the Ashtekar formulation of GR. It bears no resemblance to textbook GR, but it's the same thing.

Since that this reformulation of GR begins with a curved spacetime, only the single curved operator is physical. Therefore, there is ambiguity since the decomposition will not be unique (this resembles to me the infamous problem of time of quantum HGR).

This is not true. Any well-defined derivative operator is "physical." In both GR and (standard old-fashioned) Newtonian theory, you normally use derivative operators adapted to whichever coordinate system is most useful for the problem at hand. Covariant derivatives are rarely used in 'real' problems. Anyway, preferred coordinate systems (basically inertial frames) can be defined as ones which diagonalize the Newton-Cartan metrics. This is invariant, intuitive, and gives a unique split

As for your objection to asymptotic flatness, can you point to any non-asymptotically flat system that should have a Newtonian limit? Newton's theory does not apply to cosmology, and that's the only place where this assumption could be problematic. The solar system, for example, can be assumed asymptotically flat. Also, almost all non-cosmological work in GR makes this same assumption.

Really assuming boundaries at infinitum, Ehlers is assuming instantaneous gravitation. This is difficult to see in static models but, in dynamic models, one can see that the choosing of different boundaries would leave intact the dynamical properties for example at the Solar system scale.

I'm not sure what you're saying here. There is no assumption of instantaneous gravitation. Of course boundary conditions do matter, but this is also true in Newtonian gravity (or any field theory). You have to make sure that the system isn't being strongly influenced by things extremely far away (either through Coulomb-type interactions, or gravitational radiation). You're right that this assumption is technically separate from GR, but GR doesn't say c->infinity either. It is quite reasonable to assume that the appropriate limiting process makes both of these assumptions.

Now that we've beaten this topic to death, how about appropriate limits in other physical theories. Does non-relativistic quantum mechanics go over to classical mechanics in an appropriate sense? Does quantum field theory go over non-relativistic QM? As far as I know, nobody has shown either of these things to the degree of rigor that NG follows from GR.

 

[Stingray]

I agree with your point that Carlip only looks at very special cases. That was why I said before that the paper was just meant to illustrate a point. Don't take it too seriously.

I was not aware of any instabilities in numerical EM. Can you elaborate?

Going back to gravity, Post-Newtonian expansions of GR now exist to very high order. Everything is stable. If you still claim otherwise, cite a source. Self-force effects are negligible if that's what you're talking about.

 

[Juan R.]

What is the difference? All predictions are the same. "Conceptual points of view" are not really important.

Of course that are important! Physics is not engineering. Numbers alone are not sufficient: empirical models are not theoretical models, that is the reason you claim that gravity cannot be instantaneous, because you are inferring from a previous theoretical framework called GR where vG = c. But c in my theory has the same valor but is not the velocity of gravitation.

Phy in Newton theory is not the same that Phy in Cartan-Newton-Ehlers theory even if numerically both agree in the “Newtonian” limit. This is the reason of that the asymptotic condition

lim R--> inf; {Phy = (1/R)} = 0 (*)

is valid in the first one approach (in fact is the well proven and famous principle of decomposition of clusters), but totally unphysical (the so-called island assumption by cosmologists and rejected because violates direct observation) in the second. Why is (*) accepted by all people in the first case but neglected by many people in the second? Because the two phy are not the same. Newton-Cartan theory is not Newton original theory

This is not true. Any well-defined derivative operator is "physical." In both GR and (standard old-fashioned) Newtonian theory, you normally use derivative operators adapted to whichever coordinate system is most useful for the problem at hand. Covariant derivatives are rarely used in 'real' problems. Anyway, preferred coordinate systems (basically inertial frames) can be defined as ones which diagonalize the Newton-Cartan metrics. This is invariant, intuitive, and gives a unique split

The only well defined (physical) derivative operator is the corresponding to curved spacetime. This is splinted into two terms, but each term is not well-defined (only the total sum) by this reason each term need to be fixed with an additional physical equation does not contained in GR and that needs to be invoked for pure consistency with experimental data. This is not pure math, the failure for spliting adequately the curve operator is related to well-known problem of inertial and gravitatory masses in gravitation. From a conceptual point of view we fix the splitting for fixing the relation between inertial mass and gravitatory mass.

Reformulated GR + ad hoc equation/assumption => NC gravity =/= Original Newton gravity.

With each ad hoc equation/assumption introduced in the formalism, different spacetime structures and different Newtonian-like theories arise. I summarized some of them in #77 post. If am not wrong (i don't read his paper still), Ehlers’ formalism may be of the weak NCG type.

Canonical gravitodynamics is clearly superior here. There is no ambiguity and there is full consistency with Newton original theory.

As for your objection to asymptotic flatness, can you point to any non-asymptotically flat system that should have a Newtonian limit? Newton's theory does not apply to cosmology, and that's the only place where this assumption could be problematic. The solar system, for example, can be assumed asymptotically flat. Also, almost all non-cosmological work in GR makes this same assumption.

It is not my objection, as said cosmologists, relativists, say,

“universe is not an island of matter surrounded by emptiness”.

Ehlers uses ad hoc asymptotic flatness for ignoring the additional equation needed for fixing the splitting of the total derivative operator. As said i) is empirically unphysical, ii) is introducing an instantaneous component for gravitation. The same ii) question arises if one admits asymptotic flatness in solar or other GR models. Some of geometric models of Newtonian gravity that said above claim to do not use directly that boundary and use others ad hoc equations, but I don’t know if it is “still here” (hidden) because I have not checked the formulas.

I'm not sure what you're saying here. There is no assumption of instantaneous gravitation. Of course boundary conditions do matter, but this is also true in Newtonian gravity (or any field theory). You have to make sure that the system isn't being strongly influenced by things extremely far away (either through Coulomb-type interactions, or gravitational radiation). You're right that this assumption is technically separate from GR, but GR doesn't say c->infinity either.

There is not explicit assumption of instantaneous gravitation but there is implicit one. Basically, you are connecting two infinitely separated regions of spacetime in a pure geometric manner, breaking the causality connection that would correspond to a dynamical approach where c is finite. You are “taking” the “group of word lines” outside of the light cone.

As said above (*) in Newtonian gravity the limit at infinite does not imply an anticausal link of spacetimes (matter densities), it has other interpretation because the theory is completely different. The (*) is not a boundary conditions in Newtonian gravity, just reflects the famous principle of decomposition of clusters and is totally physical, in fact is perfectly compatible with cosmological data, whereas Ehlers’ boundary not. By this reason nobody reject the principle of decomposition of clusters, but most cosmologists reject Ehlers boundary condition like unphysical. I am repeating because I think that are not fixing the point here.

Now that we've beaten this topic to death, how about appropriate limits in other physical theories. Does non-relativistic quantum mechanics go over to classical mechanics in an appropriate sense? Does quantum field theory go over non-relativistic QM? As far as I know, nobody has shown either of these things to the degree of rigor that NG follows from GR.

NG does not follow from GR I did extensive comments on that above and in #77. Resume:

Canonical => original Newton theory

I call this a rigorous derivation.

Newton-Cartan-like approaches are summarized in

GR reformulation + ad hoc equations => Newton-like theories =/= original Newton theory

There is not derivation because the ad hoc equations (or boundary assumptions) are not derivable from GR alone.

Therefore your phrase “NG follows from GR” would rigorously read like “NCG follows from GR more ad hoc equations or assumptions”

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Effectively, quantum mechanics does not go over to classical mechanics. The derivation that appears in textbooks (derivation of Newton law) is completely false. This is the reason of that many groups around the world are studying the topic seriously since 70 years ago. There exist several levels of mathematical conceptual rigor (from less rigor to more rigor).

Textbook derivation, multiple-worlds, etc. <= decoherence <= Gell-Mann histories <= generalizations of QM.

In generalizations of QM, there is again different levels of rigor/sophistication, again from few to high

Direct modifications of Schrödinger <= spacetime foam, non-critical strings, etc. <= Brussels School <= Thermomaster (from canonical science).

Of course, it is rigorously impossible to derivate classical physics from QM, therefore you will newer see such one derivation.

The same comments apply to relativistic quantum field theory. The best comment is from Dirac.

“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, in which one neglects only quantities that are small. One is not allowed to neglect infinitely large quantities. The renormalization idea would be sensible only if it was applied with finite renormalization factors, not infinite ones. For these reasons I find the present quantum electrodynamics quite unsatisfactory. One ought not to be complacent about its faults. 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. Quantum electrodynamics ... was built up from physical ideas that were not correctly incorporated into the theory and it has no sound mathematical foundation. One must seek a new relativistic quantum mechanics and one’s prime concern must be to base it on sound mathematics.”

He is correct. Specially in the “radical” idea of that agreement with observation is presumably a “coincidence”. This is rigorously demonstrated from quantum part of canonical science!

I was not aware of any instabilities in numerical EM. Can you elaborate?

The usual Maxwell formulation requires knowledge of one of the world lines in order to compute the electromagnetic field acting on the other everywhere in spacetime. One may then compute the trajectory of the other, and given this, the effect of its field on the first. The resulting motion of the first particle may then not be consistent with the original assumption, and the process of trial and error, or iteration, often is unstable.

F. Rohrlich already pointed clearly that the N-body problem is intrinsically unstable in the standard Maxwell theory.

I think that the same problem would arise in gravitatory bodies. I think that nobody has rigorously solved the 2 and 3-body problem in gravitation. Right?

Those computational problems are easily solved in canonical electro- or gravitodynamics. My objective is not make a rigorous and elegant theory, it may be ueful also.

Everything is stable.

I think that none relativist has seriously studied the effect of instabilities due to time delay in gravitatory bodies still.

Self-force effects are negligible.

What do you mean by “negligible”?

 

[tdunc]

I would seriously rethink your belief that gravity has an instantaneous effect regardless of distance if you are thinking what I think you are thinking. This completely lacks logic and common sense, I don't care where you get it from, I'm telling you right now your wrong. What you really want to say is that the force of gravity from a distance object is present in the time of a second local body that for example is about to travel over an area being affected by the gravity of the distance body, only in that sense is it being affected by the gravity of the distance object in real time. Know that the force of gravity had to travel over space-time (borrowing from GR) to affect the local area. Certainly we will not want to call this instantaneous though because people will misunderstand what it implies. (If that didnt make sense, its just like the travel of light.)



You say that gravity cannot be explained by a curvature of space. It COULD BE and it IS explained, in GR that is.., however your right, gravity is NOT caused by a curvature of space. GR is false.

You say gravity is not governed by C, correct. I'm surprised you know this.


"Newton equation permits us only compute the force. In the same way, Einstein field equations permit just compute the curvature without an underlying mechanism for this curvature effect, and therefore, you are just substituting a mystery by other: force by curvature."

The above paragraph I will take it to mean that you realize that there is no underlieing force to cause a curvature in the first place. Clever, your correct. Forget the math. Most people do not get this Very simple crucial concept. In other words, you need gravity to pull down on an object to make a curve in space, the object having mass & weight alone will not curve space! (obviously to have mass requires gravity too, sort of like a chicken and egg thing going on) And Yes GR regards a physical curve of space as in the context of a "fabric of space", it is derived from relating his math into a physical model or interpretation of the real world. No other interpretation of GR is true, he really DID believed in a physical curvature.

Which brings me to another point, space in all relateable contexts of the word cannot be curved, it just... cant. Its a non fixed medium of free moving particles such that of a gas or liquid. It has no semi-solid structure about it. Whatever you move ("curve") will move around you, such that you can never push against it and it will stop you. (There is however concievable ways to create an artificial boundry of particles involving the suspention of particles using magnetics but that's as far as it goes, and not even related)

There are quite a few other obvious conceptual disproofs of GR that you are missing... I happen to know them, but I would have to look them up in my journal because they are not coming to mind at the moment.

Hey good luck with that, I'm certainly not against you, I didnt really read to much into it but looks like you got a good start. You make no mistake arguing String theory that's for sure.