Weak equivalence principle and GR

[Q-reeus]

Looking at J.L. Anderson's article http://arxiv.org/abs/gr-qc/9912051, referenced as [16] in http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.2748v2.pdf, sure lends weight in my mind to the belief GR has indeed undergone a radical conceptual transformation:

Does General Relativity Require a Metric?
James L. Anderson
Stevens Institute of Technology
Hoboken, New Jersey 07666, USA
"The nexus between the gravitational field and the spece-time metric was an essential element in Einstein’s development of General Relativity and led him to his discovery of the field equations for the gravitational field/metric. We will argue here that the metric is in fact an inessential element of this theory and can be dispensed with entirely. Its sole function in the theory was to describe the space-time measurements made by ideal clocks and rods. However, the behavior of model clocks and measuring rods can be derived directly from the field equations of general relativity using the
Einstein-Infeld-Hoffmann (EIH) approximation procedure. Therefore one does not need to introduce these ideal clocks and rods and hence has no need of a metric."
EFE's that use curvature tensors but no longer need a curving metric. So this is the modern viewpoint - either accept it or not I guess. Pointless then debating on principles that have just vanished from the scene, and if you do, expect the matter of conceptual ambiguities encountered here to make it all too slippery. Just as long as the sun still comes up each day, well and good. Oh, sorry, that should be 'earth keeps revolving' - just my old pre-Copernican thinking there! :shy:

 

[PAllen]

I'll try to formulate my question more clearly, how can a principle that is limited to massless test point-like bodies, like those of an SR flat Minkowski spacetime have any relevance for GR curved spacetime manifolds?

The idea is that the test body is 'small' compared to the sources of 'background curvature', so it does not substantially perturb those sources. Then one can say (to a greater precision the smaller the test body) that it follows a geodesic of the 'background metric'. Once the test body becomes large enough to significantly perturb the nearby sources, the bodies are mutually interacting - as each moves, its impact on curvature moves with it, propagating at finite speed (c); such effect then moving the other body, whose motion's effect on the metric propagates back at c. Because of finite propagation speed, angular momentum of the bodies themselves cannot be exactly conserved. The Carlip paper I linked demonstrates that it is very nearly conserved because a moving source affects another body as if it's location were quadratically extrapolated - but this is not sufficient for exact conservation (only instant action at a distance, a la Newton, would exactly conserve angular momentum for mutually interacting bodies). The periodic disturbances in curvature propagate away from the system, carrying away the lost angular momentum. If you include the angular momentum of the gravitational radiation, the total system conserves angular momentum.

Since to me this is all pretty obvious, and for similar sized objects I cannot conceive of what might even be meant by each following a geodesic of some fixed background metric, when each is constantly perturbing the metric, I posed what seemed like a completely on topic attempt to ask whether a reasonable generalization the geodesic hypothesis could be true: each body's center of mass following a geodesic of the total spacetime metric including periodic perturbations of it. The answer I got was basically, maybe yes (Physics Monkey), maybe no (Bcrowell), hard to answer because of ambiguities of what is meant by center of mass in GR (though Mentz114 posted a link to a paper that claimed to mostly resolve this by showing the equivalence of several popular coordinate independent formulations of center of mass; but this is a new paper, not necessarily accepted as consensus yet). Instead of off topic, I think this is the only possibly meaningful question that can be asked about geodesic motion of massive bodies.

I feel I have tried hard to constructively contribute, and the a large majority of non-constructive attitude has been yours.

 

[bcrowell]

Instead of off topic, I think this is the only possibly meaningful question that can be asked about geodesic motion of massive bodies.

I considered it on topic and interesting.

 

[TrickyDicky]

Instead of off topic, I think this is the only possibly meaningful question that can be asked about geodesic motion of massive bodies.
I feel I have tried hard to constructively contribute, and the a large majority of non-constructive attitude has been yours.

Hey, if you feel that I think I can feel it too, certainly bcrowell feels it which says a lot in your favour.
By the way have you read the last posts? If you have you will be able to see that your question, that has been answered with maybe yes and maybe no answers, can't be answered within the context of the limited and restricted version of the EP that you are using. And actually the paper "Theory of gravitation theories: A no progress report" gives a lot of hints about why no progress can be achieved with the restricted EP in relation with your possibly meaningful question (but not the only one that can be asked).

I'm sure if you are as constructive as you claim you'll easily realize why your post is still off-topic. Edit: on further consideration I declare it on-topic. If bcrowell being such a reasonable guy thinks it is it must be.

 

[TrickyDicky]

I missed this one post.

Even in Newtonian gravity, this is true only when restricted to test particles.

Sure but in Newtonian gravity test particles are point masses that can have any mass, see the comments about the ambiguity of the term "test particle" in GR.

It means the Newtonian physics can also be formulated as curved spacetime. This is called Newton-Cartan theory.

It couldn't mean this because as I say in Newtonian gravity all bodies (in the absence of non-gravitational forces) regardless their mass follow inertial paths, the Newtonian theory fulfills the WEP in the version I'm defending also for GR, not in its restricted version put forth by several posters and authors cited in this thread. That is the reason the Newton-Cartan formulation is possible.

 

[PAllen]

I missed this one post.Sure but in Newtonian gravity test particles are point masses that can have any mass, see the comments about the ambiguity of the term "test particle" in GR.

I think what atty might be referring to is that in neither Newton's gravity nor GR is it true that the trajectory of test particle is independent of mass, no matter what the mass is. Drop a feather in vaccuum, near earth; drop a canonball : same path. Now drop a point mass of mass of Jupiter, while standing on earth. It follows completely different trajectory (much faster 'fall'). In both theories, the test particle concept is restricted to particles small enough to ignore them as a source of gravity. And in both theories, the WEP is true to the same extent, for such test particles.

 

[Q-reeus]

..Drop a feather in vaccuum, near earth; drop a canonball : same path. Now drop a point mass of mass of Jupiter, while standing on earth. It follows completely different trajectory (much faster 'fall')..

Which to my mind merely implies that mutual free-fall between 'point Jupiter' and Earth is larger owing to Jupiter's much larger gravity, but assuming it is indeed free-fall, then surely WEP holds. Strange and troubling that on the matter of self-gravity invalidating WEP in GR there are discordant views from various expert opinions. The view of WEP as limit approximation expressed by some here is found also in the WP article Geodesic (general relativity) at http://en.wikipedia.org/wiki/Geodesic_(general_relativity)#Approximate_geodesic_motion
"Approximate geodesic motion
True geodesic motion is an idealization where one assumes the existence of test particles. Although in many cases real matter and energy can be approximated as test particles, situations arise where their appreciable mass (or equivalent thereof) can affect the background gravitational field in which they reside.
This creates problems when performing an exact theoretical description of a gravitational system (for example, in accurately describing the motion of two stars in a binary star system). This leads one to consider the problem of determining to what extent any situation approximates true geodesic motion. In qualitative terms, the problem is solved: the smaller the gravitational field produced by an object compared to the gravitational field it lives in (for example, the Earth's field is tiny in comparison with the Sun's), the closer this object's motion will be geodesic."

An entirely different picture, by folks specializing in this sort of thing is presented elsewhere:

In #23 the status of SEP (strong equivalence principle = WEP + self-gravitation leaves WEP unaltered) in The Confrontation between General Relativity
and Experiment - Clifford Will was gone through, and the conclusion there clear. SEP is part of and unique to GR, and holds within current experimental limits. Another article backing this crucial point: "New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
"1. Equivalence principles and gravitational self-energy
The principle of equivalence between gravitational force and acceleration is a common feature to all viable theories of gravity. The Strong Equivalence Principle (SEP), however, is unique to Einstein’s general theory of relativity (GR). Unlike the weak equivalence principle (which dates back to Galileo’s demonstration that all matter free falls in the same way) and the Einstein equivalence principle from special relativity (which states that the result of a non-gravitational experiment is independent of rest-frame velocity and location), the SEP states that free fall of a body is completely independent of its gravitational self energy." Goes on to give the new limits which may or may not be considered to have closed the case, but that's not the point here.

 

[atyy]

Sure but in Newtonian gravity test particles are point masses that can have any mass, see the comments about the ambiguity of the term "test particle" in GR.

Here I was using one of the definitions you brought up, that the trajectory of a particle is independent of its mass and composition. This is not true in Newtonian physics, it is an extremely good approximation if the particles have small mass, and becomes a better and better approximation as their masses get smaller.

It couldn't mean this because as I say in Newtonian gravity all bodies (in the absence of non-gravitational forces) regardless their mass follow inertial paths, the Newtonian theory fulfills the WEP in the version I'm defending also for GR, not in its restricted version put forth by several posters and authors cited in this thread. That is the reason the Newton-Cartan formulation is possible.

OK, so here you use a different definition, that a particle follows an inertial path. I don't know if this is true in Newtonian gravity for non-test particles - I'd be interested to find out. I do know that it is not true in GR. bcrowell once gave an extremely simple argument for this, which I cannot now remember, so let me point you to the complicated:

http://arxiv.org/abs/0907.5197 (short and "sweet")
http://arxiv.org/abs/1102.0529 (extensive background)

 

[PAllen]

Which to my mind merely implies that mutual free-fall between 'point Jupiter' and Earth is larger owing to Jupiter's much larger gravity, but assuming it is indeed free-fall, then surely WEP holds.

Not according the definition referred to, unless limited mass is assumed. This is where the issue of precise definitions is important, and an unfortunate issue is that there is no accepted mathematically precise statement of the equivalence principle. The definition referred to was, in the opening post of this thread, was:

A more modern definition: "The world line of a freely falling test body is independent of its composition or structure"

It was claimed that this should apply independent of mass of test particles. My example demonstrates clearly that this definition fails (in all plausible theories of gravity) if test particles are allowed to be arbitrarily massive. As written, it is cagey: it says test particle, which has implications - that the test particle not perturb the environment being tested. It does not say 'irrespective of mass', only independent of composition and structure. It is, in fact, very carefully written, but has been over-interpreted by some of the posts in this thread.

 

[PAllen]

In #23 the status of SEP (strong equivalence principle = WEP + self-gravitation leaves WEP unaltered) in The Confrontation between General Relativity
and Experiment - Clifford Will was gone through, and the conclusion there clear. SEP is part of and unique to GR, and holds within current experimental limits. Another article backing this crucial point: "New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
"1. Equivalence principles and gravitational self-energy
The principle of equivalence between gravitational force and acceleration is a common feature to all viable theories of gravity. The Strong Equivalence Principle (SEP), however, is unique to Einstein’s general theory of relativity (GR). Unlike the weak equivalence principle (which dates back to Galileo’s demonstration that all matter free falls in the same way) and the Einstein equivalence principle from special relativity (which states that the result of a non-gravitational experiment is independent of rest-frame velocity and location), the SEP states that free fall of a body is completely independent of its gravitational self energy." Goes on to give the new limits which may or may not be considered to have closed the case, but that's not the point here.

If you read the discussion and math, not just verbal summaries, you will see the following points made about SEP:

1) Bodies must be sufficiently far apart that tidal forces are not significant. Thus, mathematically still true only in the limit.

2) It says, then (and only then) gravitational self energy can be ignored and the body treated as determined by mass and angular momentum. (This is the core of the SEP).

3) It is does not say, one way or the other, whether large, mutually interacting masses are following geodesics.

It is trivially obvious that large, mutually interacting bodies cannot follow geodesics of some background geometry derived without considering the dynamics of mutual interaction. The question I raised is the only one so far on this thread that meaningfully poses whether there is any plausible sense in which large, mutually interacting bodies can be said to follow geodesics. Unfortunately, it appears that the answer is not well known. At least, none of the scientific advisers knows of reference that definitively answers this.

 

[Q-reeus]

Not according the definition referred to, unless limited mass is assumed. This is where the issue of precise definitions is important, and an unfortunate issue is that there is no accepted mathematically precise statement of the equivalence principle. The definition referred to was, in the opening post of this thread, was:

A more modern definition: "The world line of a freely falling test body is independent of its composition or structure"

It was claimed that this should apply independent of mass of test particles. My example demonstrates clearly that this definition fails (in all plausible theories of gravity) if test particles are allowed to be arbitrarily massive. As written, it is cagey: it says test particle, which has implications - that the test particle not perturb the environment being tested. It does not say 'irrespective of mass', only independent of composition and structure. It is, in fact, very carefully written, but has been over-interpreted by some of the posts in this thread.

Right well given all the problems various definitions seem to be creating, I'm butting out at this point, and just hope everyone in the end has learned something useful!

 

[TrickyDicky]

Not according the definition referred to, unless limited mass is assumed. This is where the issue of precise definitions is important, and an unfortunate issue is that there is no accepted mathematically precise statement of the equivalence principle.

Yes, there is, according to a a well known reference "Introduction to General relativity" by L. Ryder in page 43 it says: "The Principle of General Covariance is a mathematical statement of the Equivalence Principle". Now if that is not a precise statement of the EP, I don't know what precise is, the very reason tensors are the appropriate math objects to use in GR lies on this mathematical statement.
I only ask not to use the argument here that Ryder only says this for pedagogical reasons, and that to dumb down a bit the theory he is allowed this little fib.

The definition referred to was, in the opening post of this thread, was:

A more modern definition: "The world line of a freely falling test body is independent of its composition or structure"

It was claimed that this should apply independent of mass of test particles. My example demonstrates clearly that this definition fails (in all plausible theories of gravity) if test particles are allowed to be arbitrarily massive. As written, it is cagey: it says test particle, which has implications - that the test particle not perturb the environment being tested. It does not say 'irrespective of mass', only independent of composition and structure. It is, in fact, very carefully written, but has been over-interpreted by some of the posts in this thread.

Ok, so please explain to me why the definition specifies independence of structure or composition? what is in your opinion the composition of a massless test particle, or its structure?

 

[TrickyDicky]

Here I was using one of the definitions you brought up, that the trajectory of a particle is independent of its mass and composition. This is not true in Newtonian physics, it is an extremely good approximation if the particles have small mass, and becomes a better and better approximation as their masses get smaller.

It is perfectly true in Newtonian physics, are you acquainted with the experiments of a guy named Galileo?

OK, so here you use a different definition, that a particle follows an inertial path. I don't know if this is true in Newtonian gravity for non-test particles - I'd be interested to find out.

Obviously I meant bodies follow inertial paths in Newtonian physics in the absence of all types of forces, and that the non-inertial acceleration of the paths they follow are independent of their mass.I don't use a different definition, I'm just human too and make mistakes.

 

[PAllen]

Yes, there is, according to a a well known reference "Introduction to General relativity" by L. Ryder in page 43 it says: "The Principle of General Covariance is a mathematical statement of the Equivalence Principle". Now if that is not a precise statement of the EP, I don't know what precise is, the very reason tensors are the appropriate math objects to use in GR lies on this mathematical statement.
I only ask not to use the argument here that Ryder only says this for pedagogical reasons, and that to dumb down a bit the theory he is allowed this little fib.


Ok, so please explain to me why the definition specifies independence of structure or composition? what is in your opinion the composition of a massless test particle, or its structure?

1) Ryder is simply wrong. Ever since 1917, it was noted by Kreschman and confirmed by Einstein (and every significant author since, including several pages of discussion on this in MTW) that the principle of general covariance has no physical content at all. Instead, an ongoing, not yet fully concluded, activity is to try to restore what Einstein seemed to mean by this. MTW takes a crack with 'no prior geometry', but they don't give a formal definition or any proof. James L. Anderson took the approach of requiring the 'symmetry group' of a theory to be the manifold mapping group. There is an unending chain of papers on these themes proposing and attacking the sufficiency of attempted definitions. So far as I see, no final conclusion has been reached. One recent paper purports to prove that GR, indeed, does have a hidden prior geometric object.

2) I cannot do better than Physicsmonkey has in explaining why the 'test particle' concept is useful even though never *mathematically* exact except in the limit of massless particles. All I can suggest is re-read the early posts on this and ponder. I don't think there is anything else that can be said to clarify this more than it has been already. Also, think about my example, showing that it is patently absurd to put 'and any mass' into this WEP definition.

 

[PAllen]

It is perfectly true in Newtonian physics, are you acquainted with the experiments of a guy named Galileo?

A Jupiter mass of neutron star material (very small in size) will fall to Earth same as a cannonball? Think again: it will 'fall' *much* faster as it pulls the Earth towards it. All statements of this principle had the implicit caveat that the test particle couldn't be so massive as to be a major source of gravity on its own.

AND that means, *mathematically* no two bodies of different mass fall the same. For realizable precision, it is a different story, which is why the principle is actually extremely useful, in practice.

 

[TrickyDicky]

A Jupiter mass of neutron star material (very small in size) will fall to Earth same as a cannonball? Think again: it will 'fall' *much* faster as it pulls the Earth towards it. All statements of this principle had the implicit caveat that the test particle couldn't be so massive as to be a major source of gravity on its own.

AND that means, *mathematically* no two bodies of different mass fall the same. For realizable precision, it is a different story, which is why the principle is actually extremely useful, in practice.

You shot down the universality of freefall just like that, I hope some casual observer at this point says something for the sake of the scientific rigor of this forums. This is getting ridiculous.

 

[TrickyDicky]

1) Ryder is simply wrong. Ever since 1917, it was noted by Kreschman and confirmed by Einstein (and every significant author since, including several pages of discussion on this in MTW) that the principle of general covariance has no physical content at all. Instead, an ongoing, not yet fully concluded, activity is to try to restore what Einstein seemed to mean by this. MTW takes a crack with 'no prior geometry', but they don't give a formal definition or any proof. James L. Anderson took the approach of requiring the 'symmetry group' of a theory to be the manifold mapping group. There is an unending chain of papers on these themes proposing and attacking the sufficiency of attempted definitions. So far as I see, no final conclusion has been reached. One recent paper purports to prove that GR, indeed, does have a hidden prior geometric object.

2) I cannot do better than Physicsmonkey has in explaining why the 'test particle' concept is useful even though never *mathematically* exact except in the limit of massless particles. All I can suggest is re-read the early posts on this and ponder. I don't think there is anything else that can be said to clarify this more than it has been already. Also, think about my example, showing that it is patently absurd to put 'and any mass' into this WEP definition.

You admitted a few weeks ago you were new to GR but as I see you are ready to write a new GR textbook rectifying reknown authors. Way to go.

 

[DrGreg]

A Jupiter mass of neutron star material (very small in size) will fall to Earth same as a cannonball? Think again: it will 'fall' *much* faster as it pulls the Earth towards it. All statements of this principle had the implicit caveat that the test particle couldn't be so massive as to be a major source of gravity on its own.

AND that means, *mathematically* no two bodies of different mass fall the same. For realizable precision, it is a different story, which is why the principle is actually extremely useful, in practice.

You shot down the universality of freefall just like that, I hope some casual observer at this point says something for the sake of the scientific rigor of this forums. This is getting ridiculous.

PAllen is correct. In Newtonian theory, the "universality of freefall" applies only relative to the centre-of-gravity of the Earth+object system.

The acceleration of the object relative to the C-of-G is independent of the object's mass, but does depend on the mass of the Earth.

The acceleration of the Earth relative to the C-of-G is independent of the Earth's mass, but does depend on the mass of the object.

The relative acceleration between Earth and object is the sum of both of the above accelerations, but for small masses (relative to the Earth's mass), the second of those is negligible compared with the first. For Jupiter masses, the second is much larger than the first.

$$g = \frac{G(M + m)}{r^2}$$​

 

[atyy]

It is possible that the Principle of General Covariance is a statement of the equivalence principle. It is also possible that it is not. Weinberg notes two different definitions of the principle of general covariance in his text, one is not the same as the EP, the other is. With the definition PAllen uses, he is right - possibly except for the part about Ryder being wrong, since I don't know which definition Ryder uses.

 

[TurtleMeister]

As a casual observer I was going to post in answer to TrickyDicky's invitation. But I see that DrGreg has beat me to it. One point I would like to make however is the use of the term "gravitational mass" that has been used in this tread. In most cases refers to passive gravitational mass. But there were a few posts that seemed to refer to active gravitational mass. It may help the "casual observer" if you specified which one you're talking about.

 

[TrickyDicky]

In Newtonian theory, the "universality of freefall" applies only relative to the centre-of-gravity of the Earth+object system.

The acceleration of the object relative to the C-of-G is independent of the object's mass, but does depend on the mass of the Earth.

The acceleration of the Earth relative to the C-of-G is independent of the Earth's mass, but does depend on the mass of the object.

So far so good, this basically agrees with what I've been saying.

The relative acceleration between Earth and object is the sum of both of the above accelerations, but for small masses (relative to the Earth's mass), the second of those is negligible compared with the first.

For Jupiter masses, the second is much larger than the first.

$$g = \frac{G(M + m)}{r^2}$$​

Nothing wrong with this either, but you should be aware that relative acceleration between different bodies is not what we are discussing here, the WEP is concerned with the body's own mass (gravitational and inertial, which are equated by this principle), not with the relative accelerations various bodies have towards each other or a a third, which are tidal effects problems referred to the sources of curvature, usually not solvable within GR due to the non-linear nature of the EFE but solvable within Newtonian theory. Precisely the EP is what allows bodies to respond to background curvature regardless of their own mass and describe geodesic motion that responds to the global curvature that includes them as a source of curvature, thus the non-linearity. If the WEP didn't permit ignore the mass of the body, it couldn't be treated as a test particle (neglecting its mass) to begin with and the Newtonian limit of GR couldn't be recovered, but that is different than saying WEP only applies to bodies with negligible mass. It's the other way around, because of the WEP, massive bodies masses can be neglected to solve problems in the Newtonian limit such as Mercury's perihelion advance.

 

[atyy]

It is perfectly true in Newtonian physics, are you acquainted with the experiments of a guy named Galileo?

Galileo never used Jupiter as a test particle.

 

[TurtleMeister]

TrickyDicky, I read PAllen's post as an argument against the universality of free fall also. However, after re-reading the posts I think I just misinterpreted it. Whatever, I think everyone can agree with DrGreg's post.

Galileo never used Jupiter as a test particle.

I think the point is that it doesn't have to be a test particle. It can be a body of any mass.

 

[atyy]

So it looks as if Tricky Dicky has been using this definition:

In Newtonian theory, the "universality of freefall" applies only relative to the centre-of-gravity of the Earth+object system.

The acceleration of the object relative to the C-of-G is independent of the object's mass, but does depend on the mass of the Earth.

The acceleration of the Earth relative to the C-of-G is independent of the Earth's mass, but does depend on the mass of the object.

Does this apply for arbitrary shapes of the earth?

Eg. Consider 3 collinear massive particles which initially are spatially separated from each other. 2 form "the earth" and one is the "test particle".

(Sorry, I can work this out for myself, but am lazy.)

Edit: Actually, even in the 2 body case, is it true for finite time?

 

[PAllen]

You admitted a few weeks ago you were new to GR but as I see you are ready to write a new GR textbook rectifying reknown authors. Way to go.

Actually, if you read 'about me' in my profile, you will see the story. I learned a decent amount about GR in the 1960s, but have only been an avid physics 'fan' since 1973. What I said in thread you refer to was that I was not then familiar with conformal flatness and conformal definitions of asymptotic flatness because they were not included in GR books in the 1960s. So, I proceeded to read about them from the numerous university relativity websites available now. I know quite a bit about history of relativity and have a collection of books on it going back to 1921.

Oh, and though I have MTW, I bought after leaving academia, and have only read random sections of it, referring to it as needed. The last book I read through was published in the 1960s.

 

[atyy]

In Newtonian theory, the "universality of freefall" applies only relative to the centre-of-gravity of the Earth+object system.

The acceleration of the object relative to the C-of-G is independent of the object's mass, but does depend on the mass of the Earth.

The acceleration of the Earth relative to the C-of-G is independent of the Earth's mass, but does depend on the mass of the object.

So far so good, this basically agrees with what I've been saying.

I suspect this holds only for infinitesimal time if both masses are finite.

 

[TrickyDicky]

Galileo never used Jupiter as a test particle.

Good one-liner.
I guess he would have had some trouble putting it at the top of the Pisa tower.

 

[TrickyDicky]

I think the point is that it doesn't have to be a test particle. It can be a body of any mass.

Exactly.

 

[TrickyDicky]

I suspect this holds only for infinitesimal time if both masses are finite.

Nope, to describe a trajectory you need finite time.

 

[TrickyDicky]

PAllen is correct.

By the way DrGreg , in this post you seem to disagree with what PAllen is saying now:
https://www.physicsforums.com/showpost.php?p=2444886&postcount=7

Have you changed your mind?

 

[TrickyDicky]

$$g = \frac{G(M + m)}{r^2}$$​

I forgot to mention that formula is of course referring to relative acceleration.
All this thread has dealt with absolute acceleration, in lay terms what a body "feels" or what an accelerometer in the COM of the body measures, and what the EP states, and was the insight from which Einstein developed GR is precisely that this absolute acceleration is canceled in freefall so that the subject doesn't feel its own weight when describing geodesic motion in curved spacetime. This is so basic that I'm amazed that almost everybody in this forum seems to disagree with it. It is obvious that objects in free fall without the influence of non-gravitational forces have exact geodesic motion no matter their own mass.
Maybe what confuses people is that it is also obvious that since the object is also acting as a source in the spacetime curvature, the geodesic it draws is different depending on its mass and the mass of the other objects that act as sources of curvature, that is where the non-linearity comes in and what makes so hard to deal with 2-body or n-bodies problems in GR and the reason that in Newtonian gravity it can be done since the background is Euclidean space and the masses of the bodies can be treated linearly plus the fact that in Newtonian physics the WEP also applies so that the mass of one of the bodies wrt the other can always be neglected.

Let's remember some textbooks (I believe one is the Eyvind Gron one but not sure so don't quote me on this) outline GR by saying it is basically the sum of SR+WEP and the only way to reconcile SR with WEP and recover the results at the Newtonian limit is a curved spacetime with general covariance.

 

[atyy]

Good one-liner.
I guess he would have had some trouble putting it at the top of the Pisa tower.

Well, he tried to, but changed his mind once it started leaning.

 

[atyy]

In Newtonian gravity,
1) gravitational mass=inertial mass (always true)
2) two objects of different mass and composition dropped from the same height will take the same time to reach the ground (approximately true for small masses)
3) an object in a two body system in which both objects have finite mass has the same acceleration relative to the centre of gravity of the system, regardless of its mass, but is dependent on the mass of the second object (I'm not sure, but I think it's true only for an infinitesimal time)

 

[TrickyDicky]

In Newtonian gravity,
1) gravitational mass=inertial mass (always true)
2) two objects of different mass and composition dropped from the same height will take the same time to reach the ground (approximately true for small masses)
3) an object in a two body system in which both objects have finite mass has the same acceleration relative to the centre of gravity of the system, regardless of its mass, but is dependent on the mass of the second object (I'm not sure, but I think it's true only for an infinitesimal time)

I agree except with what is between parenthesis in your points 2 and 3.
Unless what you mean is that if we throw a trailer and a marble, due to the much bigger size of the trailer it will touch the ground before the marble, that is trivially true, but remember that I've been alway referring to geodesic paths and the COM of bodies of any mass will pass an arbitrary point at the same time exactly, not just approximately true for small masses.

Here is a post where you quote as valid what I'm considering the WEP and that is in the definition as the first requisite of the strong equivalence principle. BTW if you agree with PAllen I guess you don't accept the SEP either, which is normally accepted in mainstream GR. (not by all authors tha's true, but the ones that don't are usually considered crackpot by mainstreamers which IMO doesn't follow necessarily).

https://www.physicsforums.com/showpost.php?p=2492126&postcount=3

"These ideas can be summarized in the strong equivalence principle (SEP), which states that:
1. WEP is valid for self-gravitating bodies as well as for test bodies.
2...
3..."

 

[TrickyDicky]

Well, he tried to, but changed his mind once it started leaning.

...lol