Weak equivalence principle and GR

[TrickyDicky]

In another thread, it's considered a generally accepted fact that the WEP is not valid anymore the way it was initially postulated:

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 equivalence principle is a statement about the limiting case where the mass of the test object is small

I find this surprising, because such a change in the principles of the theory should be more stressed in introductory GR textbooks, and kind of disturbing because precisely the WEP is considered to be a necessary condition for any theory about gravity, if only because the notion of gravitational redshift rests on it.
So it is argued by the previous quoted posters that the WEP is only valid at the limit of vanishing mass test particles. But this seems to make useless the WEP given the fact that the principle is precisely about the equivalence of inertial mass and gravitational mass no matter how big or small is that mass, how can it be valid only at the limit of negligible mass,that would simply make the principle null. Which is probably what the above quoted poster hint at when they cite authors that assert that the WEP is simply false for GR.
Here is Einstein's presentation of the principle:

"A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass)times (Acceleration) = (Intensity of the gravitational field)times (Gravitational mass).

It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body. "

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

Is it really the consensus current view that the WEP is only valid at the limit of negligible mass and therefore it does not apply to physical bodies such as binary pulsars?
Wouldn't this amount to saying the WEP as it was originally stated by Einstein is no longer valid? In this case I believe this very important fact is not sufficiently stressed in GR textbooks.

 

[TrickyDicky]

Another way to see it doesn't make much sense to formulate the WEP in GR in terms of being valid only at the limit of vanishing mass, is that at that limit we are dealing with flat spacetime of SR,with absence of mass, so how could then the WEP say anything about curved spacetimes in GR where bodies are massive.
Obviously the WEP is also valid at that limit, not only valid at that limit. Otherwise the WEP would be just another way to state Lorentz invariance in SR flat spacetime, which is self-evident.
The WEP in its original form is precisely what keeps Lorentz invariance from acting in GR other than at the local limit. Maybe there's too much rush to turn GR into an "effective field theory", a la quantum gravity FT, but that hardly justifies distorting the original principles IMO.

any thoughts or comments?

 

[bcrowell]

Presumably you pulled the Einstein quote out of the WP article, where it is footnoted to this source:
'A. Einstein. “How I Constructed the Theory of Relativity,” Translated by Masahiro Morikawa from the text recorded in Japanese by Jun Ishiwara, Association of Asia Pacific Physical Societies (AAPPS) Bulletin, Vol. 15, No. 2, pp. 17-19 (April 2005). Einstein recalls events of 1907 in talk in Japan on 14 December 1922.'
This isn't Einstein's statement of the e.p. in a scientific paper, it's his attempt to explain it for a nontechnical audience in a talk about the history of the subject's development. Therefore it's not surprising that he leaves out technical conditions like a low mass for the test body.

For a careful discussion, see the following references that were already pointed out to you in a previous thread:

http://arxiv.org/abs/gr-qc/9909087
http://arxiv.org/abs/0806.3293
http://relativity.livingreviews.org/Articles/lrr-2006-3/
http://arxiv.org/abs/gr-qc/0309074v1
http://arxiv.org/abs/grqc/0306052
http://arxiv.org/abs/hep-th/0409156

This may also be helpful:

http://arxiv.org/abs/0707.2748

See p. 4 for a discussion of the condition of low mass.

In all these papers, the use of the word "test" in phrases like "test mass" or "test body" is code for "low mass." I'm sure Einstein understood this thoroughly from the time he first formulated the e.p. ca. 1907, because electromagnetism already had the notion of a test charge, which was a charge small enough that it could be used to measure fields without being strong enough to disturb the sources of the fields.

I find this surprising, because such a change in the principles of the theory should be more stressed in introductory GR textbooks[...]

There hasn't been a change in the principles of the theory. You're comparing a casual statement of the theory with more careful ones, not the original, pure, and unsullied statement with later revisions.

 

[TrickyDicky]

Presumably ...
There hasn't been a change in the principles of the theory. You're comparing a casual statement of the theory with more careful ones, not the original, pure, and unsullied statement with later revisions.

Great, now everybody knows you agree with your own quote.

 

[Physics Monkey]

I find this surprising, because such a change in the principles of the theory should be more stressed in introductory GR textbooks, and kind of disturbing because precisely the WEP is considered to be a necessary condition for any theory about gravity, if only because the notion of gravitational redshift rests on it.

As I said in the other thread, this is a complaint about pedagogy. Perhaps it is valid, but it has nothing to do with what the theory actually predicts.

So it is argued by the previous quoted posters that the WEP is only valid at the limit of vanishing mass test particles. But this seems to make useless the WEP given the fact that the principle is precisely about the equivalence of inertial mass and gravitational mass no matter how big or small is that mass, how can it be valid only at the limit of negligible mass,that would simply make the principle null. Which is probably what the above quoted poster hint at when they cite authors that assert that the WEP is simply false for GR.

First, there is no issue of consensus, GR is a well defined theory and one can simply ask how massive objects move. The answer is that they follow geodesics in the limit of small mass and size, if moving in completely flat spacetime, etc but that in general there are (often small) deviations from geodesic motion in a "background geometry" because the massive object affects the gravitational field.

In the other thread I provided a link to a complete derivation of this effect from first principles using the equations of GR found in any textbook. There is nothing more to say about this point without getting into the technical details of how the calculation proceeds.

Furthermore, I don't understand why this issue keeps being painted as black and white. Like most everything else in physics, rules for the behavior of physical systems are stated making certain physical assumptions. It is obviously true that we move in approximate geodesics when jumping in the air just as it is true that the Earth moves in a geodesic in the sun's background field to a high degree of approximation. I want to emphasize this point. The equivalence principle is obviously a very useful statement about the behavior of a wide variety of objects from people to planets over reasonable time scales. Of course if you consider a sufficiently extreme situation, or look at small enough effects, or measure for a very long time you can see some deviation from the simplified rule, but this hardly invalidates the usefulness of the rule.

Here is Einstein's presentation of the principle:

"A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass)times (Acceleration) = (Intensity of the gravitational field)times (Gravitational mass).

It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body. "

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

Your "modern" definition contains that all important phrase "test body". Just as in electromagnetism where we consider test charges of vanishing charge, test bodies are assumed to have vanishing effect on the gravitational fields they are designed to measure. It is clearly an unbelievably good approximation to say that I would follow a geodesic if I was orbiting the sun. I don't have vanishing mass, but I'm close enough.

Is it really the consensus current view that the WEP is only valid at the limit of negligible mass and therefore it does not apply to physical bodies such as binary pulsars?
Wouldn't this amount to saying the WEP as it was originally stated by Einstein is no longer valid? In this case I believe this very important fact is not sufficiently stressed in GR textbooks.

This is again a complaint about pedagogy, but let me address it. Do you really think we should teach students of GR on day 1 that even though the motion of almost all familiar objects can be very well approximated by geodesic motion in a fixed background geometry, we must actually solve a horribly complicated set of equations that will predict almost exactly the same motion? I'm certainly all for mentioning that the simplified rule has limitations, but do you really want more than that?

 

[Q-reeus]

The OP has raised a very important matter which surely is more than 'merely pedagogical'. We are talking about behavior of matter under the action of purely gravitational interaction. To say that geodesic motion fails for a massive body because it perturbs the gravitational field in which it is immersed sounds a bit like an oxymoron - in old time GR at least gravity is perturbation of spacetime, no? It is still commonly stated that in such a setting the sole possible local experience is tidal forces. For instance newbies often engage in ad nauseam arguments about falling into a notional static BH, and are typically answered along the lines "the infalling observer feels nothing special in crossing the EH, being in free fall there is only tidal forces which can be arbitrarily small for a super-massive BH". But can this be logically correct if WEP fails - especially given the extreme metric curvature (but not necessarily the gradient) involved here? Tiny departures from WEP under 'normal' conditions might become huge, and the standard dictum of free-fall = locally inertial motion would be seriously wrong.
But that raises the matter of what the infalling observer could experience - plummeting in vacuo, in what sense can there be a net force. Are we to imagine that the curved spacetime is acting as it's own air cushion so to speak - the observer 'parachuting in' in some sense? Does it make any sense at all?

How about some sensible definition, free of self-contradictions, of what departure from geodesic motion means in the local frame of a massive body under only gravitational interaction with another such mass or masses.

 

[TrickyDicky]

Your "modern" definition contains that all important phrase "test body". Just as in electromagnetism where we consider test charges of vanishing charge, test bodies are assumed to have vanishing effect on the gravitational fields they are designed to measure.

Only the analogy with electromagnetism is not completely valid here, since there is a fixed background in electromagnetism that is independent of charges, while in GR gravity is what configurates spacetime, thus the linearity of Maxwell eq. and the non-linearity of the EFE which makes really hard to solve problems with more than one body in GR.

Actually my point is that this goes beyond a pedagogical issue as I imply in my second post

 

[TrickyDicky]

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?

 

[bcrowell]

The OP has raised a very important matter which surely is more than 'merely pedagogical'.

Physics Monkey's #5 did not claim that it was all merely pedagogy, only that specific parts of it were.

We are talking about behavior of matter under the action of purely gravitational interaction. To say that geodesic motion fails for a massive body because it perturbs the gravitational field in which it is immersed sounds a bit like an oxymoron - in old time GR at least gravity is perturbation of spacetime, no?

It's a limiting process. You take the limit of the trajectory as the mass approaches zero.

But can this be logically correct if WEP fails - especially given the extreme metric curvature (but not necessarily the gradient) involved here? Tiny departures from WEP under 'normal' conditions might become huge, and the standard dictum of free-fall = locally inertial motion would be seriously wrong.

It depends on how small the test body is. The Riemann tensor sets a distance scale for curvature. For a solar-mass black hole, this is on the order of kilometers. If the test body is much smaller than a kilometer, and if an energy condition holds, the e.p. will be a good approximation.

But that raises the matter of what the infalling observer could experience - plummeting in vacuo, in what sense can there be a net force. Are we to imagine that the curved spacetime is acting as it's own air cushion so to speak - the observer 'parachuting in' in some sense? Does it make any sense at all?

Sure. It's exactly analogous to radiation resistance: http://en.wikipedia.org/wiki/Radiation_resistance

How about some sensible definition, free of self-contradictions, of what departure from geodesic motion means in the local frame of a massive body under only gravitational interaction with another such mass or masses.

http://arxiv.org/abs/gr-qc/9909087
http://arxiv.org/abs/0806.3293
http://relativity.livingreviews.org/Articles/lrr-2006-3/
http://arxiv.org/abs/gr-qc/0309074v1
http://arxiv.org/abs/grqc/0306052
http://arxiv.org/abs/hep-th/0409156
http://arxiv.org/abs/0707.2748

 

[TrickyDicky]

Or put in more simple terms: how can objects have exactly equivalent inertial and gravitational mass only at the limit of their not having mass at all? This is as profound as saying 0=0. It amounts to say nothing. How could have Einstein been inspired by this trivial nonsense to create a GR?
This equivalence is clearly valid no matter the mass or composition of the bodies as the Eotvos experiments show.

 

[Physics Monkey]

The OP has raised a very important matter which surely is more than 'merely pedagogical'. We are talking about behavior of matter under the action of purely gravitational interaction. To say that geodesic motion fails for a massive body because it perturbs the gravitational field in which it is immersed sounds a bit like an oxymoron - in old time GR at least gravity is perturbation of spacetime, no? It is still commonly stated that in such a setting the sole possible local experience is tidal forces. For instance newbies often engage in ad nauseam arguments about falling into a notional static BH, and are typically answered along the lines "the infalling observer feels nothing special in crossing the EH, being in free fall there is only tidal forces which can be arbitrarily small for a super-massive BH". But can this be logically correct if WEP fails - especially given the extreme metric curvature (but not necessarily the gradient) involved here? Tiny departures from WEP under 'normal' conditions might become huge, and the standard dictum of free-fall = locally inertial motion would be seriously wrong.
But that raises the matter of what the infalling observer could experience - plummeting in vacuo, in what sense can there be a net force. Are we to imagine that the curved spacetime is acting as it's own air cushion so to speak - the observer 'parachuting in' in some sense? Does it make any sense at all?

How about some sensible definition, free of self-contradictions, of what departure from geodesic motion means in the local frame of a massive body under only gravitational interaction with another such mass or masses.

First, I don't think I ever used the phrase "merely". That is your own invention. It is especially disingenuous given that I have repeatedly expressed sympathy for possible confusion caused by poor pedagogy.

Second, there is no "old time GR" or new time GR, there is simply GR. Here is the action:
$$S = \frac{1}{16 \pi G} \int d^4 x \sqrt{-\det{(g)}} \,R[g] + S_{\mbox{matter}}$$
where the matter action can be point particles
$$S = - m \int d\tau[g]$$
or some fluid or scalar fields or whatever you want.

All you need to do in principle is analyze the equations following from this action to deduce the motion of objects. The main subtlety is the presence of infinities when considering point particles that must be dealt with carefully. Fortunately, the analysis has already been done many times, and you may read about it in the review articles that I and others have linked to. A result of said analysis is that the motion of a small object around a much bigger object may be well approximated by a geodesic in the background geometry produced by the larger object. Furthermore, this approximation becomes better and better as the mass of the smaller object goes to zero.

Now perhaps you don't understand the answer, there's certainly nothing wrong with that, but we should stop pretending that there are multiple versions of the theory or confusion within the theory about what should happen and so on. If you want to dispute that GR is relevant for the real world, fine, but it would really help the discussion if we could stop pretending as if GR itself failed to give a definite answer or was somehow on shaky ground on this point.

 

[Physics Monkey]

Or put in more simple terms: how can objects have exactly equivalent inertial and gravitational mass only at the limit of their not having mass at all? This is as profound as saying 0=0. It amounts to say nothing. How could have Einstein been inspired by this trivial nonsense to create a GR?
This equivalence is clearly valid no matter the mass or composition of the bodies as the Eotvos experiments show.

Pretending that we can extract a universal like "equivalence is clearly valid no matter the mass or composition of the bodies" from an experiment with errors which considered a limited range of object masses and sizes is bad science.

Once again, no one disputes that for small enough objects the motion is very well approximated by a geodesic in a given background geometry. Since this is true for any small object whatsover it is an extremely general and useful statement which follows from GR.

To say two things agree in a certain limit and are very close with calculable deviations away from that limit is a very powerful statement. To say that anything from a single neutron to an object the size of the Earth moves roughly in a geodesic in background geometry generated by the sun is an astounding statement covering some 50 orders of magnitude in mass. I hope casual observers will agree that this is one hell of a principle.

 

[TrickyDicky]

All you need to do in principle is analyze the equations following from this action to deduce the motion of objects. The main subtlety is the presence of infinities when considering point particles that must be dealt with carefully. Fortunately, the analysis has already been done many times, and you may read about it in the review articles that I and others have linked to. A result of said analysis is that the motion of a small object around a much bigger object may be well approximated by a geodesic in the background geometry produced by the larger object. Furthermore, this approximation becomes better and better as the mass of the smaller object goes to zero.

You may call the appearance of infinities a subtlety, other people may consider them a sign of warning something very wrong happens with that action, the trivial answer being if you treat massive bodies as massless points you're bound to get infinities. Obviously that happens because you try to fit GR scenarios in SR principles. No wonder then that the approximations become better as masses tend to zero.

 

[TrickyDicky]

Pretending that we can extract a universal like "equivalence is clearly valid no matter the mass or composition of the bodies" from an experiment with errors which considered a limited range of object masses and sizes is bad science.

I guess it is the same bad science everyone else uses. No experiment is without errors. All experiments have an accuracy level, are you not aware of this? Actually all I said is that experiments so far show the WEP to be valid to the level of accuracy of those experiments, are you denying this too? is saying this really bad science in your opinion?

To say two things agree in a certain limit and are very close with calculable deviations away from that limit is a very powerful statement. To say that anything from a single neutron to an object the size of the Earth moves roughly in a geodesic in background geometry generated by the sun is an astounding statement covering some 50 orders of magnitude in mass. I hope casual observers will agree that this is one hell of a principle.

Casual observers might also agree that the principle says more than you make it to say. In fact the principle says, as shown in my references, that the motion is not roughly a geodesic but geodesic, making it really one hell of a principle. By the way, if the roughness of the geodesic approximates a "pure" geodesic as the mass approaches the limit as mass tends to zero, how can bodies with 50 orders of difference in mass approximate both roughly geodesic motion?

 

[PAllen]

I have a question suggested by theme running through some of Q-reeus comments on this and the conservation-angular-momentum thread, that I find really quite interesting. Hopefully I can ask it in a sufficiently clear way. I present the question in two takes, since the first runs into an obvious failure (or so it seems to me).

Start with the idea of inpiralling black holes radiating away mass as gravitational radiation. Let's be strictly classical, so there is no Hawking radiation. Let the black holes start out non-rotating and be as close to Schwarzschild as possible in some initial hypersurface. However, difficult to actually compute or specify it, there is some unique, complicated, 4-manifold with geometry representing the complete time evolution of this system (under GR field equations), including of course the GW. I want to ask whether the black hole centers might be following geodesics of the total geometry (nothing about background geometry here; total geometry includes all effects of GW and mass loss and inspiral). But woops! I'm asking whether a locus of points not included in the manifold is a geodesic of the manifold. No go, so far as I see.

Take two. Start with two massive balls of perfect fluid, but nowhere near graviational instability - no chance of collapse. Again have them mutually orbit. They radiate, and orbit degenerates (and balls deform, have fluid waves, and other such messy complications, but so what, this is a conceptual argument). As they inspiral, lose mass, shrink, we have spiral paths (in spacetime) approaching but never reaching each other. Assuming such a solution is possible in principle (where due to shrinkage, the balls forever spiral closer into asymptotic nothingness), is it possible that some plausible definition of center of mass for each ball forever follows a geodesic of the complete, incredibly complex, 4-manifold?

(Note, that since we are not talking about background geometry, the following type of argument Bcrowell raised is inapplicable: two different masses cannot both follow an exact geodesic because, if they radiate differently, you would have two different paths with the same starting point and direction; you can't have two such geodesics. In this case, each different mass would be comletely different total geometry, with different goedesics).

 

[Physics Monkey]

You may call the appearance of infinities a subtlety, other people may consider them a sign of warning something very wrong happens with that action, the trivial answer being if you treat massive bodies as massless points you're bound to get infinities. Obviously that happens because you try to fit GR scenarios in SR principles. No wonder then that the approximations become better as masses tend to zero.

It simply means one must exercise some caution and physical sense. But one doesn't need to have infinities, simply consider continuous fluid droplets held together by surface tension or something analogous and kept from collapsing by some pressure. As a mathematical problem in GR there are no infinities in sight, and while the equations will be more complicated (corresponding to a more complicated model for the material objects) the conclusions will be the same. It's just a choice of where you want to put your complications. There is nothing profound here since one doesn't take either model too seriously as an ultimate description.

 

[TrickyDicky]

Pallen, I consider your post as off topic, and ask you to start your own thread, the OP is about the WEP and you can try and answer any of the questions I've put forward.

 

[Physics Monkey]

I guess it is the same bad science everyone else uses. No experiment is without errors. All experiments have an accuracy level, are you not aware of this?

I will simply note that you are the one who made universal statements based on the results of some experiment. Hence my comment about bad science. I suppose I should also point out that experiments to test the predictions of GR are ongoing precisely because other scientists agree with me that universal statements cannot be made. I have repeatedly emphasized the role of approximation, and I'm confident that reasonable observers will see that my position is perfectly in accord with the presence of experimental error, your personal attacks notwithstanding.

By the way, if the roughness of the geodesic approximates a "pure" geodesic as the mass approaches the limit as mass tends to zero, how can bodies with 50 orders of difference in mass approximate both roughly geodesic motion?

Very easily. Suppose for the sake of argument that the expansion is in powers of $$m/m_{\mbox{Sun}}$$. Then we would have a good approximation for the Earth, a great approximation for people, and a truly ridiculous level of approximation for a neutron.

 

[TrickyDicky]

It simply means one must exercise some caution and physical sense. But one doesn't need to have infinities, simply consider continuous fluid droplets held together by surface tension or something analogous and kept from collapsing by some pressure. As a mathematical problem in GR there are no infinities in sight, and while the equations will be more complicated (corresponding to a more complicated model for the material objects) the conclusions will be the same. It's just a choice of where you want to put your complications. There is nothing profound here since one doesn't take either model too seriously as an ultimate description.

No idea of what you mean with this disconnected sentences.

 

[PAllen]

Pallen, I consider your post as off topic, and ask you to start your own thread, the OP is about the WEP and you can try and answer any of the questions I've put forward.

I think it is on topic. If true, it would be a sense in which WEP could remain 'exactly' true in the presence gravitational radiation, which is something Q-reeus sort of asked in the other thread.

 

[TrickyDicky]

I will simply note that you are the one who made universal statements based on the results of some experiment. Hence my comment about bad science. I suppose I should also point out that experiments to test the predictions of GR are ongoing precisely because other scientists agree with me that universal statements cannot be made. I have repeatedly emphasized the role of approximation, and I'm confident that reasonable observers will see that my position is perfectly in accord with the presence of experimental error, your personal attacks notwithstanding.

I never made any universal statement,I just mentioned that experiments back the WEP, anything wrong with that too?

What personal attacks are you talking about?

Very easily. Suppose for the sake of argument that the expansion is in powers of $$m/m_{\mbox{Sun}}$$. Then we would have a good approximation for the Earth, a great approximation for people, and a truly ridiculous level of approximation for a neutron.

Right, but you said all of them would be roughly geodesic, not great, good and ridiculous.

 

[Physics Monkey]

Start with the idea of inpiralling black holes radiating away mass as gravitational radiation. Let's be strictly classical, so there is no Hawking radiation. Let the black holes start out non-rotating and be as close to Schwarzschild as possible in some initial hypersurface. However, difficult to actually compute or specify it, there is some unique, complicated, 4-manifold with geometry representing the complete time evolution of this system (under GR field equations), including of course the GW. I want to ask whether the black hole centers might be following geodesics of the total geometry (nothing about background geometry here; total geometry includes all effects of GW and mass loss and inspiral). But woops! I'm asking whether a locus of points not included in the manifold is a geodesic of the manifold. No go, so far as I see.

Take two. Start with two massive balls of perfect fluid, but nowhere near graviational instability - no chance of collapse. Again have them mutually orbit. They radiate, and orbit degenerates (and balls deform, have fluid waves, and other such messy complications, but so what, this is a conceptual argument). As they inspiral, lose mass, shrink, we have spiral paths (in spacetime) approaching but never reaching each other. Assuming such a solution is possible in principle (where due to shrinkage, the balls forever spiral closer into asymptotic nothingness), is it possible that some plausible definition of center of mass for each ball forever follows a geodesic of the complete, incredibly complex, 4-manifold?

This is an interesting question. Rather than get into details, let me merely point out a discussion of interest in http://arxiv.org/abs/grqc/0306052 starting on p 15. After equation 1.9.3 Poisson describes obtaining the equations of motion by formally demanding that the point particle move in a geodesic in the perturbed spacetime. Of course, as you said this is only formal precisely because there is a singularity that must be resolved. The subsequent discussion describes the physical resolution of the singularity culminating in equation 1.9.7 for the motion of the particle. See also the subsequent discussion for additional points of view about the black hole case you mentioned and some further subtleties about gauge invariance.

 

[Q-reeus]

[pre Edit: this was composed in response to #9 - a lot has happened since but oh well done now so tossing this in anyway]
In #9 bcrowell made some interesting comments and gave some useful links (thanks). Seems this can be split into two distinctinctly different claims re non-geodesic motion (neglecting Lens-Thirring type of spin couplings):
1: Self-gravitation of a mass violates WEP a la Eötvös style (in the non-dynamic limit).
2: GW radiation damping means non-geodesic motion.
A look at what Clifford Will has to say about 1:
From one reference http://relativity.livingreviews.org/Articles/lrr-2006-3/
3.1 Metric theories of gravity and the strong equivalence principle:
"...These ideas can be summarized in the strong equivalence principle (SEP), which states that:

WEP is valid for self-gravitating bodies as well as for test bodies.
The outcome of any local test experiment is independent of the velocity of the (freely falling) apparatus.
The outcome of any local test experiment is independent of where and when in the universe it is performed.

The distinction between SEP and EEP is the inclusion of bodies with self-gravitational interactions (planets, stars) and of experiments involving gravitational forces (Cavendish experiments, gravimeter measurements). Note that SEP contains EEP as the special case in which local gravitational forces are ignored.

The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is strictly valid, there must be one and only one gravitational field in the universe, the metric g. These arguments are only suggestive however, and no rigorous proof of this statement is available at present. Empirically it has been found that almost every metric theory other than GR introduces auxiliary gravitational fields, either dynamical or prior geometric, and thus predicts violations of SEP at some level...General relativity seems to be the only viable metric theory that embodies SEP completely. In Section 3.6, we shall discuss experimental evidence for the validity of SEP."

3.6 Tests of the strong equivalence principle
The next class of solar-system experiments that test relativistic gravitational effects may be called tests of the strong equivalence principle (SEP). In Section 3.1.2 we pointed out that many metric theories of gravity (perhaps all except GR) can be expected to violate one or more aspects of SEP.
3.6.1 The Nordtvedt effect and the lunar Eötvös experiment
...This violation of the massive-body equivalence principle is known as the “Nordtvedt effect”. The effect is absent in GR (η = 0 N) but present in scalar-tensor theory...The resulting bound of 1.4 parts in 10¹³ [19, 2] from composition effects reduces the ambiguity in the LLR bound, and establishes the firm SEP test at the level of about 2 parts in 10¹³...

From the above I conclude that both theoretically and up to current experimental limits, in GR 'massive' test particles will follow geodesics - sans GW considerations.

So what of GW damping? Made the point elsewhere that the perturbations representing GW emission are purely metric in nature. Logically then, a test mass is always responding locally to metric curvature only - there is never any non-gravitational interaction (assuming a 'stiff', compact mass and negligible tidal deformations). So the basic question remains - given SEP findings above, how can there ever be other than local free fall, and thus geodesic motion by that definition? Now if the definition of geodesic specifies a path notionally free of GW emissions, and that any deviation (in-spiral) is non-geodesic, that seems to me to be one of definition only. [but I see PAllen is tackling that - I'm off to bed so sorry will see how it all goes much later] :zzz:

 

[bcrowell]

Or put in more simple terms: how can objects have exactly equivalent inertial and gravitational mass only at the limit of their not having mass at all? This is as profound as saying 0=0. It amounts to say nothing. How could have Einstein been inspired by this trivial nonsense to create a GR?

Oh, please. In other words, how can $\Delta y$ and $\Delta x$ have a definite ratio in the limit of their not having any finite value at all? This is as profound as saying 0=0. It amounts to saying nothing. How could Newton and Leibniz have been inspired by this trivial nonsense to create the differential calculus?

 

[atyy]

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

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

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?

It means the Newtonian physics can also be formulated as curved spacetime. This is called Newton-Cartan theory. It also means that GR may not be the only relativistic theory consistent with classical gravity. This is the case, eg. Nordstrom's second theory, which obeys some form of the EP.

It is true that many textbooks don't stress the limitations of the EP, but many do, PAllen pointed out Synge's, and I know J L Martin's and Rindler's do too. An online one that does is http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html , in many places, but especially in section 24.7.

 

[bcrowell]

Take two. Start with two massive balls of perfect fluid, but nowhere near graviational instability - no chance of collapse. Again have them mutually orbit. They radiate, and orbit degenerates (and balls deform, have fluid waves, and other such messy complications, but so what, this is a conceptual argument). As they inspiral, lose mass, shrink, we have spiral paths (in spacetime) approaching but never reaching each other. Assuming such a solution is possible in principle (where due to shrinkage, the balls forever spiral closer into asymptotic nothingness), is it possible that some plausible definition of center of mass for each ball forever follows a geodesic of the complete, incredibly complex, 4-manifold?

This is an interesting question. I think the answer is no, because I think there are similar systems that act like counterexamples:
"Swimming in Spacetime: Motion in Space by Cyclic Changes in Body Shape" Jack Wisdom 2003, Science , 299 , 1865. http://groups.csail.mit.edu/mac/users/wisdom/
"The relativistic glider," Eduardo Gueron and Ricardo A. Mosna, Phys.Rev.D75:081501,2007. http://arxiv.org/abs/gr-qc/0612131
"'Swimming' versus 'swinging' in spacetime", Gueron, Maia, and Matsas, http://arxiv.org/abs/gr-qc/0510054

There is a nice popularization of this in Scientific American, Eduardo Gueron, Aug 2009, which I think gets at exactly the relevant point. In GR, vector addition is nonassociative when the vectors are located in different places. As an example, take 3 points A, B, and C on a single line of latitude on the Earth's surface. Their c.m. could be close to the north pole if you average all 3 at once, but if you find the c.m. of 2 of them and then average in the third, it can be much closer to the equator. This means that there's an ambiguity in conservation of momentum. This is why an isolated body can exploit this by doing a repetitive, asymmetric motion of its parts.

I think there are really two separate questions here:
(1) Do these relativistic gliders conserve momentum?
(2) Can the center of mass be well defined?

#1 is one that I've thought about and haven't come to a definite conclusion about. Of course there is no reason to expect momentum to be conserved in GR except locally, and it's not clear whether the gliders are local enough to qualify. I've spent some time trying to work out scaling laws, estimate radiation reaction forces and reactions with the body creating the field, etc. I didn't come to any conclusions I was certain of.

However, I think #2 is easier. It's pretty clear that there is an ambiguity in the definition of the center of mass in curved spacetime, and this is exactly the mechanism that Gueron invokes to explain the gliders.

 

[TrickyDicky]

Oh, please. In other words, how can $\Delta y$ and $\Delta x$ have a definite ratio in the limit of their not having any finite value at all? This is as profound as saying 0=0. It amounts to saying nothing. How could Newton and Leibniz have been inspired by this trivial nonsense to create the differential calculus?

Oh, I see, really brilliant contribution, now you say your version of the WEP is actually defining differential calculus, too bad Leibniz and Newton didn't go a little further and develope the full GR theory.

 

[TrickyDicky]

So far no one has even attempted to answer posts #2 and #8, it is significant that precisely those have been ignored while some have tried to distract the main questions with all kind of sideshow remarks.

 

[WannabeNewton]

I don't know what books you have been looking at but I am staring at General Relativity - Wald and Spacetime and Geometry - Carroll right now and both state (and I paraphrase) that geodesics are the paths followed by unaccelerated TEST particles.

 

[TrickyDicky]

I don't know what books you have been looking at but I am staring at General Relativity - Wald and Spacetime and Geometry - Carroll right now and both state (and I paraphrase) that geodesics are the paths followed by unaccelerated TEST particles.

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.

 

[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!