Weak equivalence principle and GR

[TrickyDicky]

One of the papers cited in the this thread is actually quite ad hoc to this discussion as it refers to definitions of WEP and the confusion about what TEST particles that I mentioned in my previous post.
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.2748v2.pdf

When it says:
" It is important to stress that the WEP only says that there exist some preferred trajectories, the free fall trajectories, that test particles will follow and these curves are the same independently of the mass and internal composition of the particles that follow them (universality of free fall). WEP does not imply, by itself, that there exist a metric, geodesics, etc. — this comes about only through the EEP by combining the WEP with requirements (ii) and (iii)."

And later it refers to the subtleties of the definitions of the EP:
"The second subtle point is the reference to test particles in all the EP formulations. Apart from the obvious limitation of restricting attention to particles and ignoring classical fields
(such as, e.g., the electromagnetic one), apparently no true test particles exist, hence the question is how do we know how “small” a particle should be in order to be considered a test particle (i.e., its gravitational field can be neglected)? The answer is likely to be theory-dependent"
There seems to be some degree of contradiction or confusion even in this comments that are allegedly meant to clarify when the authors don't make up their minds as to whether the trajectories of the test bodies will be the same independently of their mass or else later referring to test particles only in relation to their size instead of their mass, when it is obvious that the principle refers to the trajectories which are simply lines without any width nor depth, just the flow of a point in one dimension.
So it is plain to see that the concept of "test" body or particle can be used in a deliberately confusing way (in a theory-dependent way at the least), so that it can be made to mean different things for different authors as it most convenes to their purposes. And while it is often well used to simplify certain problems, this doesn't seem to be the case here as the authors of this paper admit that it rather confuses than simplifies.
Precisely what the WEP (and the EEP) assert is that the gravitational field of a body can be neglected for its own motion in the absence of non-gravitational forces, how can then the same principle imply that self- gravitation alters that motion?
Hopefully some GR expert will clarify this important issues.

 

[Q-reeus]

There seems to be confusion about the meaning of the term TEST particle in this context, it is an idealized concept, in nature there is no such thing as TEST bodies, it can only be used as approximation,and as such is perfectly valid in many physis contexts, but if the WEP is only valid as an approximative principle as it is claimed it defeats itself completely as a principle for GR, after all the WEP is a restatement of the equality of gravitational and inertial mass which was postulated without specifying at any moment from its formulation that it referred to the vanishing limit of the mass but precisely that the equality holds for any mass:feathers, trucks or neutron stars. The uniqueness of free fall states that they must fall with the same acceleration in a given external gravitational field. Consider for a moment the difference of stating that only idealized massless bodies have inertial paths, which is self evident from Special relativity and adds nothing to explain dynamics of massive bodies of GR, from stating that this extends to massive bodies of curved spacetime.

First an apology TrickyDicky for having muddied things with introducing GW considerations - will henceforth refrain from responding to others input on that matter, respecting your remarks in #17 & #28. So in regards to the SEP/WEP issue as originally intended, agree with your thrust here, and would like to add the following:

In #23 Clifford Will's summary of whether massive, significantly self-gravitating bodies follow geodesic intervals no different from a small 'test mass' is surely unequivocal - they do in GR, full stop. If non-geodesic motion is inferred, this is therefore tantamount to saying we are dealing with a different theory of gravity. I suppose some might say Clifford Will has now become 'out of touch with consensus opinion', but that would need justification. To my mind it all gets back to clear definitions - what exactly in a purely single-metric theory like GR is a geodesic if not identical in meaning to local free-fall of an object's COM? And what exactly could define departure from geodesic motion in that setting? Take a specific scenario: Two identical mass, non-spinning neutron stars are co-orbiting at sufficient separation that tidal deformation and GW's are a negligible. We assume the backdrop is a patch of flat background metric - after all most cosmologist's tell us the universe is overall flat or very nearly so. In this setting talk of the limiting case of a tiny test mass negligibly perturbing a background curvature has it all backwards. Here, local curvature owing to each mass is far greater than that induced from the other. Let's say Will has it wrong and the motion is non-geodesic owing to effects of strong self-gravity for each neutron star. What, physically speaking, is the effect? Do we still have exact free-fall of each star's COM, or not. If not, how does a fully metric theory explain this departure of worldline from geodesic? I think it all gets down to defining things in certain not necessarily self-consistent ways. Remember - in-spiral is a zero or negligible consideration here.
As a footnote, managed to look at the interesting article http://groups.csail.mit.edu/mac/users/wisdom/ referenced in #26, but while genuinely fascinating. has no bearing here on entirely 'non-swimming' entities.

 

[TrickyDicky]

First an apology TrickyDicky for having muddied things with introducing GW considerations - will henceforth refrain from responding to others input on that matter, respecting your remarks in #17 & #28.

I actually had in mind in #28 the Physics Monkey allusion to "personal attacks"(sic) directed to him from my part and similarly off-topic manouvers by PAllen and bcrowell.

 

[Q-reeus]

I actually had in mind in #28 the Physics Monkey allusion to "personal attacks"(sic) directed to him from my part and similarly off-topic manouvers by PAllen and bcrowell.

Relieved personally on that score, but nonetheless best to keep it as stated - just looking at self-gravity 'vs' WEP makes it a cleaner issue to thrash out.

 

[Q-reeus]

One of the papers cited in the this thread is actually quite ad hoc to this discussion as it refers to definitions of WEP and the confusion about what TEST particles that I mentioned in my previous post.
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.2748v2.pdf...

And it just gets better and better. Some more choice selections from that article:

3. Metric Postulates
...Appealing as they may seem, however, the metric postulates lack clarity. As pointed out also by the very authors of the paper introducing them14, any metric theory can perfectly well be given a representation that appears to violate the metric postulates (recall, for instance, that g_µν is a member of a family of conformal metrics and that there is no a priori reason why this metric should be used to write down the field equations). See also Anderson16 for an earlier criticism of the need for a metric and, indirectly, of the metric postulates. One of the goals of this paper is to elaborate on the problems mentioned above, as well as on other prominent ambiguities stated below and trace their roots...

3.2. What does “non-gravitational fields” mean?
There is no precise definition of “gravitational” and “non-gravitational” field. One could say that a field non-minimally coupled to the metric is gravitational whereas the rest are matter fields. This definition does not appear to be rigorous or sufficient and it is shown in the following that it strongly depends on the perspective and the terminology one chooses...

5.1. Alternative theories and alternative representations: Jordan and Einstein frames
...The moral is that one can find quantities that indeed formally satisfy the metric postulates but these quantities are not necessarily physically meaningful. There are great ambiguities as mentioned before, in defining the stress-energy tensor or in judging whether a field is gravitational or just a matter field that practically makes the metric postulates useless outside of a specific representation (and how does one know, in general, when given an action, if it is in this representation, i.e., if the quantities of this representation are the ones to be used directly to check the validity of the metric postulates or a representation change is due before doing so?)...

Honestly this is looking more like a free-for-all theorist's playground where definitions themselves are up for grabs. I might just go fishing!

 

[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.