Conservation of angular momentum in GR

[PAllen]

I haven't got a reference but I'll do some resarch. It was probably proved by Frottmann in 1742.

I have a lot of respect for J. L. Synge, so I'll have to invoke a local frame to define the COM with some normal coordinates. If the extension is smaller than the radius curvature, but not too small, it might work.

I've sketched a proof, but it needs thinking about.

[Edit]After 5 minutes I have found a paper where the conservation of angular momentum is used to define a 'centre-of-mass' line inside the world tube. So I got it backwards.

Schattner, (1978)
http://www.springerlink.com/content/mg846n70582873n8/


This recent survey

http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0456v1.pdf

concludes

Thanks a lot for the references. Very interesting. (I don't think there is any contradiction with Synge; his definitions were different, but presumably, for some reasonable coordinates, his would match the modern definition(s) [several of them shown to be equivalent in the cited paper]. )

 

[Mentz114]

One of the discussions on this thread is whether real massive bodies exactly follow geodesics, in general, if there are no forces on them. While interesting, a finding that idealized massive bodies in one unique state of relative modtion exactly follow geodesics does little to answer the general issue.

Granted.

My original assertion is a cinch to prove in Newtonian gravity with a spherically symmetric field and might extend to a local coordinate system on a geodesic. But I have other things to do. The motion of extended bodies in GR is obviously a big and difficult topic.

 

[bcrowell]

I'm pretty sure that if two masses are free-falling toward one another, they radiate. The quadrupole moment is changing. Actually purely radial motion is one of the easiest cases to use if you want to convince someone without a lot of math that gravitational radiation exists. Taylor and Wheeler have an argument to this effect in Spacetime Physics (where they have Atlas lifting a gigantic weight).

I'm not sure it makes sense to discuss the proper acceleration of a massive self-gravitating body in GR. If you attach an accelerometer to the earth, all you measure is the Earth's field. An accelerometer attached to a radiating body acts like part of that body, radiating coherently along with it.

[EDIT] Deleted my original version of the second paragraph, which I decided was wrong.

 

[TrickyDicky]

This seems to be a case when the principles, the foundations of a theory like GR can not be refuted by supposed consequences of something, like GW that is said to be derived from the theory, without incurring in contradiction or falacy.

It is important to remark that a geodesic itself is inevitably a somewhat idealized concept, like most habitually used concepts in physics without loss of generality and validity of the empirical results obtained from them. Since the trajectory a body describes is also a set of points, necessarily a massive body, no matter its size is idealized to a massive point when speaking about its location in the trajectory it draws, a timelike geodesic path or trajectory is a set of points at which the Chrystophel symbols can be made to vanish, irrespective of the size or mass of the body that describes that path, that path is a one-dimensional curved line in curved spacetime. For reference see:
http://en.wikipedia.org/wiki/Normal_coordinates#Geodesic_normal_coordinates
This is notwithstanding the fact that in a body in motion for instance rotating there will be points that are obviously not describing the same path than the object to which they belong because they have motion relative to the center of mass of the object.


Here is a paragraph taken from the GR textbook of a relevant relativist that summarizes perfectly what I'm trying to get across
So going to the principles of GR one finds that:

"In Einstein’s General Relativity, gravity manifests itself by a tensor field and, in the absence of other forces, the motion of a particle is determined by this tensor field; it does not depend on the mass of the particle. Einstein’s revolutionary idea is that the arena where the gravity tensor lives is itself determined by gravity, both are united in a 4-dimensional Lorentzian manifold (V, g), called the spacetime. The trajectories of particles are geodesics of the metric g. Einstein’s weak equivalence principle corresponds to the fact that in a general Lorentzian spacetime the geodesic equations are formally the same as in Minkowski spacetime with arbitrary coordinates where the non-vanishing of the Christoffel symbols signals the presence of inertial forces due to the noninertialframe of reference.

In a general spacetime (V, g), it is always possible at one given point, to choose local coordinates such that the Christoffel symbols vanish at that point; gravity and relative acceleration are then, at that point, exactly balanced. It is even possible to choose local coordinates such that the Christoffel symbols vanish along one given geodesic; astronauts in spacecraft have made popular the fact that in free fall one feels neither acceleration nor gravity; in a small enough neighbourhood of a geodesic the relative accelerations of objects in free fall are approximately zero. Massive pointlike objects in free fall follow a timelike geodesic"
General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation.


From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.
The consequences this may have for GW are not for me to spell out with authority (I'd rather let everyone draw their own conclusions), since I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.

 

[Physics Monkey]

From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.
The consequences this may have for GW are not for me to spell out with authority (I'd rather let everyone draw their own conclusions), since I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.

I can't agree. Let us take the simple limit of a binary where one star is much larger than the other. This allows us to study the motion of the smaller star in the bigger star's background. Now it simply must be true that the smaller star does not behave exactly as a point mass following a geodesic in the background of the big star because energy, angular momentum, etc. will be radiated in the form of gravitational waves. This conclusion is well supported by analytical, numerical, and experimental evidence.

Confusion may arise because:
1) the effect can be small, thus considerations of astronauts or moons may not be sufficient
2) calculation of the effect requires one to regulate certain divergences, thus it is technically a bit onerous

Nevertheless, the issue is well studied as the many references in the thread indicate. There is no contradiction and no need to assault the foundations of the theory. Perhaps the strongest complaint one might make is that various pedagogical sources obfuscate the physics in the name of simplicity or otherwise make overly general introductory claims. I am very sympathetic to this claim, but I don't think we should let it distract us now.

Here are two more references that I find useful:
1) a comprehensive review http://arxiv.org/abs/grqc/0306052
2) a nice conceptual framework http://arxiv.org/abs/hep-th/0409156

 

[TrickyDicky]

I can't agree. Let us take the simple limit of a binary where one star is much larger than the other. This allows us to study the motion of the smaller star in the bigger star's background. Now it simply must be true that the smaller star does not behave exactly as a point mass following a geodesic in the background of the big star because energy, angular momentum, etc. will be radiated in the form of gravitational waves.

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.
Besides your example happens to go against your thesis, even in your own view, the bigger the difference in mass between two bodies in a binary orbit the less GW radiation. In fact for systems like the sun and planets the radiation is usually neglected for all practical calculations.

This conclusion is well supported by analytical, numerical, and experimental evidence.

To this date no direct experimental evidence of GW has been shown.

...various pedagogical sources obfuscate the physics in the name of simplicity or otherwise make overly general introductory claims. I am very sympathetic to this claim, but I don't think we should let it distract us now.

I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

 

[PAllen]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.

WEP was a motivating principle for the theory. It is not an axiom. Some respected authors (e.g. J. L. Synge strongly argue that it shouldn't even be taught anymore because, mathematically speaking, it is simply false for GR. The more consensus view is that it is valid heuristically, and can be made true in the limit, though there are numerous papers (Bcrowell has provided links) that show it is basically impossible to formulate fully precise, mathematically true, formulation of it).

The geodesic hypothesis was initially introduced as a separate element of GR from the field equations. However, after overcoming his flip flopping on GW, Einstein (with Infeld and Hoffman) became a strong proponent of the idea that the geodesic hypothesis should be deleted from the theory as if it never exised; that all motion follows from the field equations - which directly lead to tiny deviations from geodesic motion, while also showing (in the limit) that geodesic motion follows from the field equations.

To this date no direct experimental evidence of GW has been shown.

True, only strong indirect evidence. However, there are other predictions of GR that have also not been verified yet. One can even say length contraction in SR has not been directly verified. So what?

I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

Your reference is propounding the pedogogical value of the equivalence principle and the geodesic hypothesis. I am sure, if you asked the author, he/she would agree that they were glossing over the details.

 

[bcrowell]

From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.

Nope, you're wrong. The quote you gave from the textbook doesn't explicitly say so, but it's implicitly assuming that the objects have low enough masses so that radiation is negligible. This has been well understood and noncontroversial since ca. 1940.

I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.

Let's get straight what would really be meant by an appeal to authority versus an appeal to logic. An appeal to authority would be if A invokes a statement by an authority figure B without fully understanding the evidence and logic behind that statement. This is what is happening when you quote from the text by "General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation." An appeal to logic would be what I did by linking to my exposition, which you didn't seem to understand, of the ideas in the Geroch proof.

The reason we keep telling you that there has been a universal consensus about this since 1940 is not because we are trying to win an argument by an appeal to authority. It's in response to your repeated erroneous claims that it is controversial or unknown, based on your failure to understand the physics.

 

[bcrowell]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.

No, nobody is reasoning backward. We are starting from the Einstein field equations, and as a result we find that trajectories are not geodesic and gravitational waves exist. The equivalence principle is a statement about the limiting case where the mass of the test object is small, so there is no contradiction.

 

[Physics Monkey]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.

I merely stated that assuming gravitational waves are emitted then the orbit must not be a geodesic. Gravitational waves are derived from the theory and this derivation in no way conflicts with the basic premise that particles follow geodesics in the limit of vanishing size and mass. Are you challenging the claim the a small body orbiting a larger body in GR will emit gravitational waves?

Also, you are conflating two issues. Issue 1 is whether GR is a consistent theory that predicts gravitational waves and deviations from geodesic motion for bodies of finite size and mass. Both of these facts are well established consequences of GR. You need only go through the derivations yourself (I provided a comprehensive reference). Note that observations in our universe have nothing directly to do with issue 1. Issue 2 is whether GR is a good description for the low energy physics of gravity in our universe. This is obviously an experimental/observational question. The title of this thread and the subsequent discussion suggest to me that we are primarily discussing issue 1.

Besides your example happens to go against your thesis, even in your own view, the bigger the difference in mass between two bodies in a binary orbit the less GW radiation. In fact for systems like the sun and planets the radiation is usually neglected for all practical calculations.

I fail to see how this follows. Whether the radiation output is large or small by some measure is irrelevant for the conceptual question. I used the large mass difference example because it is a clean theoretical laboratory, but I explicitly said that the effect may turn out to be quite small. Certainly the effect exists in GR and can be made large by varying parameters. In fact, the system I described may be applicable for direct detection efforts e.g. for neutron stars orbiting supermassive black holes.

To this date no direct experimental evidence of GW has been shown.

By direct presumably you mean no one has seen a mirror wobble due to a gravitational wave. Of course I agree. However, as you surely know the effects of gravitational waves have been indirectly shown in orbital decay of binary pulsars. The observed decay agrees well with theoretical calculations using GR. Given the many other tests of GR, I'd be happy to bet a large sum that gravitational waves are a part of the physical universe. The real question now is not do gravitational waves exist (they do), but instead what we can learn from them about cosmology, black holes, etc.

Of course, this is again irrelevant for the question of geodesic motion within the theory of GR. As I said, there is convincing analytic and numerical evidence that gravitational waves exist in GR, that they are emitted in the system I described, and that relatedly bodies of finite mass do not exactly follow geodesics.

I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

You misunderstand me. I was not attempting to impugn your references but rather to encourage you by noting that this would not be first time simplified statements lead to confusion.

 

[Q-reeus]

Admittedly as a GR layman seems to me TrtickyDicky (and I assume still Mentz114) have logic on their side here. Are not the EFE's themselves simply a mathematical expression of 'matter tells spacetime how to curve; spacetime tells matter how to move'? So given that in vacuuo two orbiting masses can only experience metric curvature effects, what principle in the EFE's allows effective COM motion other than geodesic? My own simple reasoning here is that finite propagation delay guarantees each co-orbiting mass will 'fall slightly behind' the other - but only as determined by a distant observer. And that this is quite consistent with the COM of each mass locally following a perfect geodesic - but such perfect geodesic motion is an inspiralling non-conservative path. Maybe that's where Loinger is both right and wrong. Or is it a matter of definition that a pure geodesic path must be completely conservative in nature (again - as determined by a distant reference frame)? If so that seems imho to be a sure recipe for paradox.
EDIT: BTW the first bit above is saying I'm the laymen - not TrickyDicky - sorry for any ambiguity there.

 

[bcrowell]

But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

Here are six previous posts linking to six papers giving detailed discussions of this topic:

https://www.physicsforums.com/showpost.php?p=3261324&postcount=11
https://www.physicsforums.com/showpost.php?p=3266404&postcount=40
https://www.physicsforums.com/showpost.php?p=3266574&postcount=42
https://www.physicsforums.com/showpost.php?p=3266793&postcount=44
https://www.physicsforums.com/showpost.php?p=3266989&postcount=46
https://www.physicsforums.com/showpost.php?p=3270075&postcount=75

These are not vague non-arguments.

If you want to, you can make the effort to understand all the detailed mathematical arguments in these papers. If you don't want to do that, then it's ridiculous that you're still complaining that the rest of us are arguing based on appeals to authority, engaging in circular reasoning, violating widely accepted principles, or arriving at contradictions. If, for example, you think there is circular reasoning going on, please point out where that is happening in the Ehlers and Geroch paper.

 

[Mentz114]

Admittedly as a GR layman seems to me TrickyDicky (and I assume still Mentz114) have logic on their side here...

I'm agnostic, and I hope still learning about angular momentum in GR. My views can change hourly as I discover more. Questions I've asked here have been answered sufficiently to make me think, so ascribing any position to me on these issues would be a risk.

 

[Q-reeus]

I'm agnostic, and I hope still learning about these things. My views can change hourly as I discover more.

Fair comment - and sorry if I have made you out to be an antagonist of one color here - did hesitate slightly before typing that bit!

 

[bcrowell]

So given that in vacuuo two orbiting masses can only experience metric curvature effects, what principle in the EFE's allows effective COM motion other than geodesic?

This is a good question. One way to see that this argument doesn't quite work is to see that it can be adapted to make it into an argument about electromagnetism, which then predicts the wrong thing about electromagnetism. Suppose that we have an electromagnetic field in a certain region. We have two test particles. Particle 1 has mass m and charge q. Particle 2 has mass 2m and charge 2q. Based on Newton's laws, we expect that 1 and 2 should follow identical trajectories if they are given the same initial motion. But in fact they both radiate electromagnetic waves, and the Larmour formula shows that the radiated power is (initially) equal for 1 and 2. Since the radiated power is equal, the decelerations of the particles are unequal, so they follow different trajectories. Now we could ask, "what principle in Maxwell's equations allows motion other than that predicted by Newton's laws?" The thing to realize here is that Maxwell's equations are fundamental, and Newton's laws are not. Newton's laws don't apply to electromagnetic waves. Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation.

Completing the analogy with GR, what is fundamental is the Einstein field equations(~Maxwell's equations), and what is not fundamental is geodesic motion(~Newton's laws). Newton's laws are not a good approximation for an electromagnetically radiating system. Geodesic motion is not a good approximation for a gravitationally radiating system.

My own simple reasoning here is that finite propagation delay guarantees each co-orbiting mass will 'fall slightly behind' the other - but only as determined by a distant observer. And that this is quite consistent with the COM of each mass locally following a perfect geodesic - but such perfect geodesic motion is an inspiralling non-conservative path.

To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1, even if their initial conditions are identical. Since geodesics that start out parallel at the same point are the same geodesic, it follows that the trajectories of 1 and 2 cannot both be geodesic -- in fact, neither is.

 

[TrickyDicky]

Let's get straight what would really be meant by an appeal to authority versus an appeal to logic. An appeal to authority would be if A invokes a statement by an authority figure B without fully understanding the evidence and logic behind that statement. This is what is happening when you quote from the text by "General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation." An appeal to logic would be what I did by linking to my exposition, which you didn't seem to understand, of the ideas in the Geroch proof.

What exposition?, all you did was link papers, but I guess when you do it is not an appeal to authority, that only happens when I quote pertinent paragraphs because obviously I'm very confused and wrong about all I write, I don't think I'll ever achieve your perfect understanding about GR.
You certainly don't seem to be a teacher, kind of feel sorry for your students.

 

[Q-reeus]

This is a good question. One way to see that this argument doesn't quite work is to see that it can be adapted to make it into an argument about electromagnetism, which then predicts the wrong thing about electromagnetism. Suppose that we have an electromagnetic field in a certain region. We have two test particles. Particle 1 has mass m and charge q. Particle 2 has mass 2m and charge 2q. Based on Newton's laws, we expect that 1 and 2 should follow identical trajectories if they are given the same initial motion. But in fact they both radiate electromagnetic waves, the Larmour formula shows that the radiated power is (initially) equal for 1 and 2. Since the radiated power is equal, the decelerations of the particles are unequal, so they follow different trajectories. Now we could ask, "what principle in Maxwell's equations allows motion other than that predicted by Newton's laws?" The thing to realize here is that Maxwell's equations are fundamental, and Newton's laws are not. Newton's laws don't apply to electromagnetic waves. Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation.

Completing the analogy with GR, what is fundamental is the Einstein field equations(~Maxwell's equations), and what is not fundamental is geodesic motion(~Newton's laws). Newton's laws are not a good approximation for an electromagnetically radiating system. Geodesic motion is not a good approximation for a gravitationally radiating system.

Thanks for your input here, but some aspects are leaving me baffled. The analogy with Newton's laws vs Maxwell's equations is apt to a point but of course represents the way Newton would have understood it without knowing about radiation, but I think we agree including radiation ('photons flitting off') Newton's laws retain their validity in a more generalized setting - all played out on a flat Minkowski backdrop.
My understanding has been that GR being a metric theory expresses all gravitationally induced effects as owing to metric curvature - including any 'radiation reaction' resulting in in-spiral. Now if that has become a bit old-hat all I can think is GR has quietly evolved into a kind of field theory. If that's even remotely correct then my earlier position clearly becomes no longer relevant. But such a transformation of the very foundations of GR would be big news to me and I think many others here, even if not to GR insiders. If not a now de facto field theory, the rationale for non-geodesic behavior still eludes me. I don't consider it fundamental in that respect that unequal masses would follow different geodesic paths - only that each mass COM is locally geodesic - ie perfectly free-fall. Again, maybe a matter of defining the term appropriately.

..To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1.

Much the same comments as above. Not trying to be picky here, but wouldn't the smaller mass, having to move that much further and faster (almost perfectly Newton's laws in action!, ha ha) be experiencing the greater power loss? EDIT: Was thinking the two masses were meant to be a binary system - but you were meanng each are separately gravitationally coupled to a much larger mass (planet or whatever)?
FURTHER EDIT: Some more thought on your last point (taking the last viewpoint is correct). I agree that is a strong consideration. Would be convinced if the sums have been done showing the net curvature of 'planet' + test mass is definitely not locally geodesic - something needing a proper explanation surely!

 

[TrickyDicky]

WEP was a motivating principle for the theory. It is not an axiom. Some respected authors (e.g. J. L. Synge strongly argue that it shouldn't even be taught anymore because, mathematically speaking, it is simply false for GR. The more consensus view is that it is valid heuristically, and can be made true in the limit, though there are numerous papers (Bcrowell has provided links) that show it is basically impossible to formulate fully precise, mathematically true, formulation of it).

The geodesic hypothesis was initially introduced as a separate element of GR from the field equations. However, after overcoming his flip flopping on GW, Einstein (with Infeld and Hoffman) became a strong proponent of the idea that the geodesic hypothesis should be deleted from the theory as if it never exised; that all motion follows from the field equations - which directly lead to tiny deviations from geodesic motion, while also showing (in the limit) that geodesic motion follows from the field equations.

I have to disagree with your view on relativity history,but then this is a highly subjective and opinable matter.

True, only strong indirect evidence. However, there are other predictions of GR that have also not been verified yet. One can even say length contraction in SR has not been directly verified. So what?

So what? My phrase was merely answering a previous statement saying otherwise. This question is gratuitously argumentative, don't you think?

Your reference is propounding the pedogogical value of the equivalence principle and the geodesic hypothesis. I am sure, if you asked the author, he/she would agree that they were glossing over the details.

The wikipedia article linked had all needed details. But if there is any specific detail you may wan to discuss just ask.

 

[bcrowell]

Thanks for your input here, but some aspects are leaving me baffled. The analogy with Newton's laws vs Maxwell's equations is apt to a point but of course represents the way Newton would have understood it without knowing about radiation, but I think we agree including radiation ('photons flitting off') Newton's laws retain their validity in a more generalized setting - all played out on a flat Minkowski backdrop.

No, there is no way to patch up Newton's laws to handle photons.

Now if that has become a bit old-hat all I can think is GR has quietly evolved into a kind of field theory.

GR is and always has been a classical field theory.

..To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1.

you were meanng each are separately gravitationally coupled to a much larger mass (planet or whatever)?

Yes.

FURTHER EDIT: Some more thought on your last point (taking the last viewpoint is correct). I agree that is a strong consideration. Would be convinced if the sums have been done showing the net curvature of 'planet' + test mass is definitely not locally geodesic - something needing a proper explanation surely!

Sorry, I don't follow you here.

 

[Q-reeus]

No, there is no way to patch up Newton's laws to handle photons.

But as you have stated earlier "Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation."
Is not conservaton of momentum by inclusion of radiation a manifestation of Newton's Third Law - equal and opposite reaction? EM radiation may be massless but there is a well defined momentum density.

GR is and always has been a classical field theory.

OK then what I meant was having taken on at least some non-metric character (seemingly implied if as has been stated WEP and geodesic motion are no longer considered fundamental to GR) - where the notion of field directly acting on matter generates a force. Without that, I cannot understand the concept of non-geodesic motion being possible, GW's or not. There is imo either free-fall, or force, or both. Can you explain then how departure from free-fall in GR is possible without introducing the notion of force. If one says GW back-reaction provides such, this to my mind only begs the question - what part of a GW is non-metric!? If one admits it's all metric, how, in a conceptual way is free-fall condition violated - assuming for simplicity an orbiting 'point' test mass?

Sorry, I don't follow you here.

Was referring to your argument about two test masses starting from the same point and following different geodesics. This indeed seems to prove non-geodesic motion in general. But I conjecture this may be untrue once the combined curvature owing to both central mass plus test mass are fully accounted for. Hence the call for some literature having proven otherwise. If you know of any, would appreciate a link. :zzz: