Weak equivalence principle and GR

[Q-reeus]

This thread has greater longevity than anticipated so am butting in again. Physics Monkey presented an argument in #72 (mentioned again in #84, and I see just now in #99) that seems open and shut case. For an extended body sampling a non-uniform field it is only to be expected net motion is not generally the result of assuming a COM applicable for a perfectly rigid mass immersed in a perfectly uniform field. But is this truly pointing to the limited validity of WEP, or rather the limited validity of a particularly simple definition of COM? Why wouldn't one define an effective COM that took proper and sensible account of things like tidal deformation, non-uniform 'sampling effects in a tidal field, non-uniformity of energy density owing to gravitational interaction, and non-uniformity of the metric defining COM? In short, COM in the general setting is properly a dynamical quantity. So are we to believe that when all of the above is correctly incorporated, path of free-fall of effective COM still follows a non-geodesic? Depends on convention here surely - what is to be the yardstick for defining what. And I note this extended body matter is departing from the OP's query which centers around mass independence of free-fall, not spatial extent as factor.
[EDIT: Darn it - on second thoughts one will always find with extended rigid-body systems that inertial and passive gravitational COM will generally differ (as per Physics Monkey's extreme example). But in this setting is a 'pathology' of an extended composite entity. So I will stick to the matter of mass as determining factor, and thus below remarks.]

On a similar vein: PAllen in responding to the example of two co-orbiting neutron stars in an otherwise flat background metric, admitted there was no generally agreed on position as to whether a geodesic made sense or could be well defined. But a read of the article raised in #42
"New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
makes it clear there are dynamical consequences if WEP/SEP fails that cut right through any ambiguities about defining geodesic motion. Namely that the combined system will move in ways not consistent with the momentum conservation principle - and that would unambiguously show up on the canvas of a flat background metric - ie astronomical observations. There is no such observed effect. My conclusion: mass-independent free-fall consistent with WEP/SEP is fact, and 'departures' from that are artifacts of adopting simplifying definitions (eg rigid, invariant COM, excising contribution of test mass from total metric curvature). Now you fellas can run rings around me as far as mathematical grasp of GR goes. But looked at just as matter of logical consistency of founding principles (elevator in free-fall etc), seems clear the OP's premise necessarily holds once proper (as opposed to what looks to be purely 'consensus') definitions are made and adhered to. If this is all wrong-headed then please explain where and how exactly! :zzz:

 

[TrickyDicky]

How does the motion of the balls+rod system have little to do with your claim when you claim all bodies move on geodesics? Of course, I'd be happy to hear if this is not your claim, but if not, can you please state your claim clearly and precisely once and for all. Also, you didn't "explain" that we can't talk about geodesics in Newtonian space, you simply declared it.

From the WP:"In mathematics, a geodesic (pronounced /ˌdʒiːɵˈdiːzɨk/, /ˌdʒiːɵˈdɛsɨk/ JEE-o-DEE-zik, JEE-o-DES-ik) is a generalization of the notion of a "straight line" to "curved spaces"
Excuses for not presenting the definition before, I was under the belief that it was a moreless known concept for the participants in this thread.

My claim has been stated many times and actually is not my claim but the definition of the WEP.
Q-reeus has answered any remaining doubts about the specific problem at hand.

I'm starting to suspect the cause of your misunderstanding is 2) in the above post, in which case I would recommend you to read some basic text on GR (Ryder's is a good intro), being aware of the limitations of the concept of "test body" mentioned in this thread.

 

[PAllen]

This thread has greater longevity than anticipated so am butting in again. Physics Monkey presented an argument in #72 (mentioned again in #84, and I see just now in #99) that seems open and shut case. For an extended body sampling a non-uniform field it is only to be expected net motion is not generally the result of assuming a COM applicable for a perfectly rigid mass immersed in a perfectly uniform field. But is this truly pointing to the limited validity of WEP, or rather the limited validity of a particularly simple definition of COM? Why wouldn't one define an effective COM that took proper and sensible account of things like tidal deformation, non-uniform 'sampling effects in a tidal field, non-uniformity of energy density owing to gravitational interaction, and non-uniformity of the metric defining COM? In short, COM in the general setting is properly a dynamical quantity. So are we to believe that when all of the above is correctly incorporated, path of free-fall of effective COM still follows a non-geodesic? Depends on convention here surely - what is to be the yardstick for defining what. And I note this extended body matter is departing from the OP's query which centers around mass independence of free-fall, not spatial extent as factor.
[EDIT: Darn it - on second thoughts one will always find with extended rigid-body systems that inertial and passive gravitational COM will generally differ (as per Physics Monkey's extreme example). But in this setting is a 'pathology' of an extended composite entity. So I will stick to the matter of mass as determining factor, and thus below remarks.]

On a similar vein: PAllen in responding to the example of two co-orbiting neutron stars in an otherwise flat background metric, admitted there was no generally agreed on position as to whether a geodesic made sense or could be well defined. But a read of the article raised in #42
"New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
makes it clear there are dynamical consequences if WEP/SEP fails that cut right through any ambiguities about defining geodesic motion. Namely that the combined system will move in ways not consistent with the momentum conservation principle - and that would unambiguously show up on the canvas of a flat background metric - ie astronomical observations. There is no such observed effect. My conclusion: mass-independent free-fall consistent with WEP/SEP is fact, and 'departures' from that are artifacts of adopting simplifying definitions (eg rigid, invariant COM, excising contribution of test mass from total metric curvature). Now you fellas can run rings around me as far as mathematical grasp of GR goes. But looked at just as matter of logical consistency of founding principles (elevator in free-fall etc), seems clear the OP's premise necessarily holds once proper (as opposed to what looks to be purely 'consensus') definitions are made and adhered to. If this is all wrong-headed then please explain where and how exactly! :zzz:

I will comment only on a few aspects of this.

The reference paper makes no statement as to whether the pulsars may be treated as following geodesics of the complete, dynamic, spacetime. This interesting question is complicated by several factors: lack of universally accepted definition of COM in GR context (though this is not very significant for the case of compact, nearly sperical objects); and the fact that all solutions to the two body problem are numeric, making it hard to accurately decide if some world line is precisely following a geodesic. It definitely appears that the answer to this question is not well known (maybe known by some experts, but not well known; I have no idea of the answer).

Note, also, a fact not mentioned in the paper because it is 'obvious background understanding': If the assemblage of matter into a compact object releases energy (which, of course, it does), the resultant mass of the object declines, the difference representing the gravitational binding energy of the object. In this sense, self gravitation clearly affects the mass of an object. However, the point of the paper and relevant experiments is that to the extent 'finite size' can be ignored, self gravitation has no other impact beyond its affect on mass (specifically, the Nortveldt effect has never been observed).

None of this is really relevant to the WEP, as most commonly stated. Its most common statement is *specifically* to highlight the fact that for test bodies small enough in mass enough not to perturb other 'sources' of gravity, and small enough in extent for finite size effects to be insignificant, that the the trajectory is independent of mass, composition, internal structure. It is is not trying to probe the most general conditions under which an object follows a spacetime geodesic.

 

[Physics Monkey]

From the WP:"In mathematics, a geodesic (pronounced /ˌdʒiːɵˈdiːzɨk/, /ˌdʒiːɵˈdɛsɨk/ JEE-o-DEE-zik, JEE-o-DES-ik) is a generalization of the notion of a "straight line" to "curved spaces"
Excuses for not presenting the definition before, I was under the belief that it was a moreless known concept for the participants in this thread.

My claim has been stated many times and actually is not my claim but the definition of the WEP.
Q-reeus has answered any remaining doubts about the specific problem at hand.

I'm starting to suspect the cause of your misunderstanding is 2) in the above post, in which case I would recommend you to read some basic text on GR (Ryder's is a good intro), being aware of the limitations of the concept of "test body" mentioned in this thread.

Thanks for this, I laughed out loud when I saw that you had included pronunciations in your response.

I guess you aren't aware that the geodesics of the metric I wrote above are, in the Newtonian limit, simply identical to solutions of Newton's 2nd law with potential \phi.

In any event, since it's clear you are unwilling and unable to seriously discuss the issues, I shall not waste anymore time here.

 

[TrickyDicky]

Thanks for this, I laughed out loud when I saw that you had included pronunciations in your response.

I'm glad you did, that was the purpose of including it, to keep a relaxed and humorous tone when treating these sometimes dry issues.

I guess you aren't aware that the geodesics of the metric I wrote above are, in the Newtonian limit, simply identical to solutions of Newton's 2nd law with potential \phi.

Sure, I am aware of that, and guess what makes possible that identity: the WEP in the way I'm formulating it.
The fact remains that the Newtonian limit is the metric of a space at the limit of being flat, and geodesics in strict sense apply to curved spaces.
I've already explained how the WEP allows recovering the Newtonian limit in GR, and linear approximations for calculations such as the precession of Mercury and deflection of light. But IMO it doesn't allow to generalize features intrinsic to the linear solutions to the non-linear theory. (see doubt about GW thread).

In any event, since it's clear you are unwilling and unable to seriously discuss the issues, I shall not waste anymore time here.

I regret you get that impression, I can only assure it doesn't correspond with reality.
Don't consider it a total waste of time though: without any sarcasm, I really think you might learn something. I certainly have.

 

[atyy]

Now you fellas can run rings around me as far as mathematical grasp of GR goes. But looked at just as matter of logical consistency of founding principles (elevator in free-fall etc), seems clear the OP's premise necessarily holds once proper (as opposed to what looks to be purely 'consensus') definitions are made and adhered to. If this is all wrong-headed then please explain where and how exactly! :zzz:

The usual formulation is a freely falling test particle follows a geodesic of the background spacetime.

If the particle is massive, since mass generates curvature, the particle will generate additional curvature in addition to that of the background spacetime. So the full spacetime is not the background spacetime, and the particle should not move on a geodesic of the background spacetime. Presumably it should move on something like a geodesic of the full spacetime? That's tricky to check, because each the the particle moves, the spacetime changes, and you have to compute a new geodesic.

But apparently, it has been calculated. I am not sure I am reading it correctly, but it seems that the result is that up to first order, the particle moves on a geodesic of the full spacetime. See the comments on the generalized equivalence principle in http://arxiv.org/abs/1102.0529 , p143, the section on the Detweiler-Whiting Axiom.

As far as I know, the EP is always "local". What does local mean? It means up to first order in derivatives. What has locality to do with a derivative? Well, a derivative involves taking the difference of values at two spacetime points, so it is "non-local" in that sense (mathematically, it is local, since the limit of that exists at each point). So the EP is never exact in the sense of to all orders. It is exact in the sense of a limit, and provide that limit doesn't include second derivatives ("local"). However if we use the mathematical meaning of "local" and include second derivatives, then the EP is always an approximation. So whether the EP is exact or not depends on your definition of "local".

Now, how did we know that the "local" in the EP meant "up to first derivatives" (as opposed to zero or third order derivatives)? We didn't. The EP was a imprecise rule of thumb until we had GR, in which an EP can be precisely defined.

But more generally, Einstein's EP isn't a "principle of GR". Rather, the EP was one of the many "rules" of thumb Einstein used to construct GR. Those "rules" could have given other consistent theories. These theories are ruled out by experiment, not pure thought. We take GR as the principle, not the EP or some other rule of thumb, because GR has passed experimental tests, not because of pure thought. So GR will survive as the principle, until experiments say otherwise.

 

[PAllen]

The usual formulation is a freely falling test particle follows a geodesic of the background spacetime.

I have always seen this referred to as, e.g. the geodesic hypothesis. It is specific to GR. Meanwhile, the WEP, EEP, and SEP are meant to classify gravitational theories, and make no statements about geodesics (because they are potentially meant to apply to non metric theories, at least at the outset). From what I've seen the following review presents the most generally accepted formulations of the various EPs (and similar wording was used in the OP of this thread):

http://relativity.livingreviews.org/Articles/lrr-2006-3/

If the particle is massive, since mass generates curvature, the particle will generate additional curvature in addition to that of the background spacetime. So the full spacetime is not the background spacetime, and the particle should not move on a geodesic of the background spacetime. Presumably it should move on something like a geodesic of the full spacetime? That's tricky to check, because each the the particle moves, the spacetime changes, and you have to compute a new geodesic.

But apparently, it has been calculated. I am not sure I am reading it correctly, but it seems that the result is that up to first order, the particle moves on a geodesic of the full spacetime. See the comments on the generalized equivalence principle in http://arxiv.org/abs/1102.0529 , p143, the section on the Detweiler-Whiting Axiom.

This whole discussion does not cover the case I was worried about: inspiralling, similarly massive bodies (which in reality are spinning). This discussion is, instead focused (if you read through it) on the case extreme mass ratio system, where the central body is, e.g. a million times the mass of the orbiting body, which allow for the perturbative approaches used. Note specifically, the following summary:

"In the gravitational case the Detweiler-Whiting axiom produces a generalized equivalence
principle (c.f. Ref. [153]): up to order "2 errors, a point mass m moves on a geodesic of the spacetime with
metric g + hR
, which is nonsingular and a solution to the vacuum eld equations. This is a conceptually
powerful, and elegant, formulation of the MiSaTaQuWa equations of motion. And it remains valid for
(non-spinning) small bodies."

I'm sure I could easily have missed something but I did a lot of searches to see if there was paper or expert statement specifiying whether, in a binary pulsar system, the binary pulsars could be said to follow geodesics of the total spacetime (including the radiation). I could find none. Note also, besides being similarly massive, a binary pulsar system (by definition) involves rapid spinning.

But more generally, the EP isn't a "principle of GR". Rather, the EP was one of the many "rules" of thumb Einstein used to construct GR. Those "rules" could have given other consistent theories. These theories are ruled out by experiment, not pure thought. We take GR as the principle, not the EP or some other rule of thumb, because GR has passed experimental tests, not because of pure thought. So GR will survive as the principle, until experiments say otherwise.

I definitely agree with this.

 

[atyy]

This whole discussion does not cover the case I was worried about: inspiralling, similarly massive bodies (which in reality are spinning). This discussion is, instead focused (if you read through it) on the case extreme mass ratio system, where the central body is, e.g. a million times the mass of the orbiting body, which allow for the perturbative approaches used. Note specifically, the following summary:

"In the gravitational case the Detweiler-Whiting axiom produces a generalized equivalence
principle (c.f. Ref. [153]): up to order "2 errors, a point mass m moves on a geodesic of the spacetime with
metric g + hR
, which is nonsingular and a solution to the vacuum eld equations. This is a conceptually
powerful, and elegant, formulation of the MiSaTaQuWa equations of motion. And it remains valid for
(non-spinning) small bodies."

I'm sure I could easily have missed something but I did a lot of searches to see if there was paper or expert statement specifiying whether, in a binary pulsar system, the binary pulsars could be said to follow geodesics of the total spacetime (including the radiation). I could find none. Note also, besides being similarly massive, a binary pulsar system (by definition) involves rapid spinning.

Yes, I wasn't making a point different from yours (ie. the geodesic equation is only exact for test particles).

 

[PAllen]

I want to add the my 'intuition' suggests there is some sense in which mutually orbiting, similar mass, spinning bodies, of large mass (though compact enough relative to their separation to be treated as pointlike) do follow geodesics of the total spacetime (that includes their mutually perturbative effects, finite propagation time, GW, etc). However, I am totally incapable of demonstrating this, and, so far as I can tell, it has not been successfully investigated. Unlike some, I would not declare that my intuition must be true.

 

[TrickyDicky]

Unlike some, I would not declare that my intuition must be true.

I suspect by some you mean me. In that case, I must say that I've been trying to present "my intuitions" in the form of questions and some long explanatory posts, with and without quoting references. Much of what I have asked has remained unanswered, now that doesn't mean my way of seeing things is the right one. I have also said that I don't pretend to hold the absolute truth about anything.
Perhaps from my tone you can derive what you say in the quotation, I've been told before that I show sometimes a certain intellectual haughtiness. If that were true that is not easy to correct since I do it unwittingly.

 

[TrickyDicky]

Since I consider the possibility to be wrong about this as a real one, I'll just describe my questions in the form of perplexity or confusion that might be derived just from ignorance.

But then I go to the WP page on geodesics, and read:
"Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles."
"In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time."
And add it to the WEP definition from Carroll (aimed to college students):"the behavior of freely-falling test particles is universal, independent of their mass (or any other qualities they may have)"

All I can say is that maybe my perplexity with the difference between what is stated in this thread by some and what I read is justified.
It could be blamed on my reading superficially or that all the sources I read are incorrect due to pedagogical reasons, but even in that case it should be explained in what way are those clear statements wrong.

If I may point to a source of misunderstanding, it seems some consider the background geometry in GR like that in Maxwell field theory, to be fixed, and therefore think that the freefalling body's mass should act as a correction of the geodesic it would otherwise describe if it was a particle in the low mass limit described by GR's linear approximation. However, I think GR is non-linear, and further its geometry defines the motion equations, unlike Maxwell theory, that means the mass of the freefalling body is already integrated in its geodesic path. It wouldn't be a correction of the geodesic motion it would have if it was almost massless, but the geodesic motion dictated by the geometric nature of GR non-linear dynamic background that integrates all sources of curvature (including the body that is freefalling unperturbed by other forces): that is the purpose of general covariance and the tensorial form in which the EFE must be formulated.
Looks as if the fact that we can only tackle GR with approximation methods has led some to forget that the theory is not linear and treat it like classic linear field theory.

If something here is wrong I would like it to be specifically corrected by someone more knowledgeable about GR than me (most people around here).

 

[PAllen]

Since I consider the possibility to be wrong about this as a real one, I'll just describe my questions in the form of perplexity or confusion that might be derived just from ignorance.

But then I go to the WP page on geodesics, and read:
"Geodesics are of particular importance in general relativity, as they describe the motion of inertial test particles."
"In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time."

These may seem like very broad statements, but all of these are cases where the object in question is tiny in mass compared to a huge mass gravitational source, and effectively pointlike compared to the gravitational gradient (thus finite size effects are insignificant), and not rapidly spinning.

And add it to the WEP definition from Carroll (aimed to college students):"the behavior of freely-falling test particles is universal, independent of their mass (or any other qualities they may have)"

I don't know why you are having so much difficulty with this. This statement uses the code word test particle. This has established connotations that have been explained by several of the physicists here many times. I also read over the equivalence principle section of the t'Hooft notes you linked. He does not specifically mention test particles (or any concise formulation), but his mathematics makes very clear that he is assuming a large source of gravity is not perturbed by the body under investigation. He assumes the reader is able to see this on their own.

All I can say is that maybe my perplexity with the difference between what is stated in this thread by some and what I read is justified.
It could be blamed on my reading superficially or that all the sources I read are incorrect due to pedagogical reasons, but even in that case it should be explained in what way are those clear statements wrong.

These statements are not wrong. It seems you want to read them differently than the way were intended. This does suggest there is pedagogical weakness in some of these presentations.

If I may point to a source of misunderstanding, it seems some consider the background geometry in GR like that in Maxwell field theory, to be fixed, and therefore think that the freefalling body's mass should act as a correction of the geodesic it would otherwise describe if it was a particle in the low mass limit described by GR's linear approximation. However, I think GR is non-linear, and further its geometry defines the motion equations, unlike Maxwell theory, that means the mass of the freefalling body is already integrated in its geodesic path. It wouldn't be a correction of the geodesic motion it would have if it was almost massless, but the geodesic motion dictated by the geometric nature of GR non-linear dynamic background that integrates all sources of curvature (including the body that is freefalling unperturbed by other forces): that is the purpose of general covariance and the tensorial form in which the EFE must be formulated.
Looks as if the fact that we can only tackle GR with approximation methods has led some to forget that the theory is not linear and treat it like classic linear field theory.

Yes, I think some of this is a source of confusion. One can talk about a geodesic of test particle in a background geometry (excluding the body), versus a complete solution of GR encoding motion of a compact body (otherwise COM difficulties in GR arise) that may turn out to be a geodesic of the complete solution (including other bodies). The former is what is normally done because it is enormously easier to compute, and is sufficient for 'almost all' uses of GR. In particular, note that there is no exact, non-static, two body solution known, so even the simplest case of treating both bodies on the same footing requires approximation. Only in this latter, un-achieved sense, could one talk about "that means the mass of the freefalling body is already integrated in its geodesic path", as you put it. When you compute geodesics in e.g. a Kerr geometry, they have validity as paths only of 'test particles' that may be assumed not to perturb the source of the Kerr geometry; otherwise you would need the non-existent two body solution, or you must accept perturbative approximation methods.

If something here is wrong I would like it to be specifically corrected by someone more knowledgeable about GR than me (most people around here).

I believe I am knowlegeable enough to make these statements. It would be helpful if others more knowledgeable commented on my answers.

 

[TrickyDicky]

Yes, I think some of this is a source of confusion. One can talk about a geodesic of test particle in a background geometry (excluding the body), versus a complete solution of GR encoding motion of a compact body (otherwise COM difficulties in GR arise) that may turn out to be a geodesic of the complete solution. The former is what is normally done because it is enormously easier to compute, and is sufficient for 'almost all' uses of GR. In particular, note that there is no exact, non-static, two body solution known, so even the simplest case of treating both bodies on the same footing requires approximation. Only in this latter, un-achieved sense, could one talk about "that means the mass of the freefalling body is already integrated in its geodesic path", as you put it.

But it is precisely to this last sense only that I've been referring all the time in this thread when bringing up these WEP definitions! I thought it was clear that the EP as an axiom valid for the full non-linear GR had to be expressed that way. Of course that doesn't apply to the linearized version of GR. I really think this was clear from the start, if that is what people has been arguing against, this seems to be a case of prejudiced answering or regrettable misunderstanding.

 

[PAllen]

But it is precisely to this last sense only that I've been referring all the time in this thread when bringing up these WEP definitions! I thought it was clear that the EP as an axiom valid for the full non-linear GR had to be expressed that way. Of course that doesn't apply to the linearized version of GR. I really think this was clear from the start, if that is what people has been arguing against, this seems to be a case of prejudiced answering or regrettable misunderstanding.

I guess both mis-understanding and disagreement. EP definitions are not axioms of GR, but general principles for comparing and motivating theories of gravity, and they make no statements about geodesics (because, among other things, they are meant to be usable to characterize non-metric theories). The geodesic equation of motion is meant to apply only to 'test bodies'. The only axiom of GR is its field equation (equivalently, its action principle). The full field equations allow derivation geodesic motion of test bodies against background geometry, with the normal understanding of test bodies (so this need not be a separate axiom).

For the case of a binary star system, it is obviously meaningless to talk about geodesics of a backgrouond geometry - what would it be? Because there is no known exact, non-static, two body solution, the only thing known about this case at all is numerical approximations from the full equations (which have greatly advanced over the years). It is somewhat surprising to me that the question of whether star's motions are (very nearly) geodesics of the complete (perturbative) solution is not known (for the similar mass case), but that appears to be the case. It may be that no one has been sufficiently interested in this question.

Finally, note that any time you want to make a statement about motion of an arbitrarily large mass in GR, you have to deal with the two body (or N-body) problem, for which ... see last paragrapgh.

In my view, this has all been said, and for whatever reason, this discussion proceeds in circles without progress toward mutual understanding.

 

[Q-reeus]

The heavy going paper "The motion of point particles in curved spacetime" atyy linked to in #106 is while impressing as a masterpiece of technical excellence, also a reminder of the fragmentation in philosophy (definitions, methodology etc) amongst GR specialists on this issue(s). My impression is everyone participating here could find something in that article to vindicate their own position. Used to think GR was the cut-and-dried classical theory where the only real difficulty was in finding solutions to horrendously difficult non-linear equations. But evidently there are numbers of subtle issues still unresolved. I see an unexpected parallel with the situation in QM where numerous interpretations abound and the dictum is "We have Schrodinger's equation - shut up and calculate". Just replace 'Schrodinger's equation' with 'EFE's' and it seems one has GR. That's my way of easing out of this long running thread.
Finally, TrickyDicky, a word of advice. In championing the Equivalence Principle, General Covariance Principle etc, there is one principle you have failed miserably to uphold. What is that you may ask? The Don't-Open-A-Can-Of-Worms-But-If-You-Do-Put-The-Lid-Back-On-It-Quick Principle!

 

[PAllen]

On the question of motion equal mass binaries, the state of the art analyzing the motion is impressive, but the question asked here (is a geodesic of the total geometry followed) was not even asked in e.g. the following:

http://arxiv.org/abs/0904.4551 (Equal mass Neutron star case)

http://arxiv.org/abs/0804.4184 (Equal mass black holes)

 

[TrickyDicky]

The only axiom of GR is its field equation (equivalently, its action principle). The full field equations allow derivation geodesic motion of test bodies against background geometry, with the normal understanding of test bodies (so this need not be a separate axiom).

This doesn't seem right. How can the EFE by themselves be axiomatic when so many different solutions, many of them unphysical can be derived from them depending on the boundary conditions- symmetries applied?

 

[Ben Niehoff]

Tricky asked me via PM to comment on this thread. I'm not sure I have much to add. I am not a GR expert; I just know a lot of differential geometry, of which GR is a special case.

Is the "weak equivalence principle" specifically defined in this thread or in another thread? I couldn't find a definition after a brief skimming.

As for motion on a spacetime manifold, it is certainly true that "test masses follow geodesics". However, remember there is no such thing as a test mass.

If you try to look at point masses, then you have tiny black holes, and then you are asking what "path" a singularity takes. Since the geometry is singular at the singularity, this can't really be formulated as a local law of motion.

If you instead look at extended masses, then you no longer have a single "path". If each part of the extended mass follows a local geodesic, then you have a geodesic spray. But this assumes that the mass has no cohesive forces to hold it together; i.e., it is a dust. These sorts of things don't actually exist either, but they can be good approximations.

I haven't had a chance to read the long review paper on motion in GR, but I probably will later.

 

[TrickyDicky]

Tricky asked me via PM to comment on this thread. I'm not sure I have much to add. I am not a GR expert; I just know a lot of differential geometry, of which GR is a special case.

Thanks for joining.

Is the "weak equivalence principle" specifically defined in this thread or in another thread? I couldn't find a definition after a brief skimming.

One definition that has been used in the thread is:"The world line of a freely falling test body is independent of its composition or structure". By which I understand that they also mean to be independent of mass.

As for motion on a spacetime manifold, it is certainly true that "test masses follow geodesics". However, remember there is no such thing as a test mass.

If you try to look at point masses, then you have tiny black holes, and then you are asking what "path" a singularity takes. Since the geometry is singular at the singularity, this can't really be formulated as a local law of motion.

If you instead look at extended masses, then you no longer have a single "path". If each part of the extended mass follows a local geodesic, then you have a geodesic spray. But this assumes that the mass has no cohesive forces to hold it together; i.e., it is a dust. These sorts of things don't actually exist either, but they can be good approximations.

All this is true. So what it was proposed is to use the path followed by the center of mass of the massive object as the one representing the body's geodesic motion.

 

[PAllen]

This doesn't seem right. How can the EFE by themselves be axiomatic when so many different solutions, many of them unphysical can be derived from them depending on the boundary conditions- symmetries applied?

I don't understand this question. Given a theory, you go in with initial conditions, boundary conditions, and possibly symmetry condtions that corrrespond to reasonable hypotheses about the system you want to study (up to the whole observable universe). Then you look for a solution. There is only a problem if reasonable conditions lead to ambiguous or unreasonable solutions. I am not aware of this being the case for GR. Instead, the unreasonable solutions are associated with initial and boundary conditions that are believed to be unreasonable and certainly not resembling known portions of the observable universe.

 

[PAllen]

Thanks for joining. One definition that has been used in the thread is:"The world line of a freely falling test body is independent of its composition or structure". By which I understand that they also mean to be independent of mass.

I agree with mass also, up to some upper limit where the mass is large enough to perturb other sources. For example, Jupiter orbiting the sun, Jupiter can be considered a test particle for reasonable precision. However, it is not intended to say anything about, e.g. a binary system of comparable masses.

All this is true. So what it was proposed is to use the path followed by the center of mass of the massive object as the one representing the body's geodesic motion.

And a problem discussed in this thread is that COM in general in GR is ill defined. I think, for a compact, nearly spherical object, it may be taken to reasonably well defined.

 

[PAllen]

I haven't had a chance to read the long review paper on motion in GR, but I probably will later.

FYI: That review paper covers motion of a body big enough to have have back reaction from GW (gravitational wave) emission, and perturb the metric in its vicinity, but still small enough compared to a massive central body to allow specialized methods to be used. Under these conditions, a nice geodesic result follows (the small body follows a geodesic of the background metric plus the perturbation (including GW) of its own motion). I am very curious to know if anything similar can be said for binary system of comparable masses.

 

[TrickyDicky]

I don't understand this question. Given a theory, you go in with initial conditions, boundary conditions, and possibly symmetry condtions that corrrespond to reasonable hypotheses about the system you want to study (up to the whole observable universe). Then you look for a solution. There is only a problem if reasonable conditions lead to ambiguous or unreasonable solutions. I am not aware of this being the case for GR.

Yes, but for instance you have several diffrent solutions of the EFE that describe different spacetimes but are all considered part of the theory in practice and used for different purposes or problems. From the static Schwarzschild solution for vacuum, to the FRW non-static cosmological solution, to Kerr's geometry or the linearized equations, etc, they are different solutions with some of their symmetry conditions incompatible with each other, that respond to different hypothesis about the systems they deal with, with totally different outcomes and consequences. But their common denominator is they all use the same field equations, I just don't know how something axiomatic can give rise to so many different solutions, an axiom is a principle of universal application, I don't think the field equations by themselves can qualify as an axiom.

 

[PAllen]

Yes, but for instance you have several diffrent solutions of the EFE that describe different spacetimes but are all considered part of the theory in practice and used for different purposes or problems. From the static Schwarzschild solution for vacuum, to the FRW non-static cosmological solution, to Kerr's geometry or the linearized equations, etc, they are different solutions with some of their symmetry conditions incompatible with each other, that respond to different hypothesis about the systems they deal with, with totally different outcomes and consequences. But their common denominator is they all use the same field equations, I just don't know how something axiomatic can give rise to so many different solutions, an axiom is a principle of universal application, I don't think the field equations by themselves can qualify as an axiom.

Maybe the word axiom bothers you (don't know why). So say, instead, the field equations are whole content of the theory. Nothing else is needed besides differential geometry and what I call correspondence rules: how to relate mathematical objects to natural objects. Each of these solutions covers different problems. I don't see any ambiguity about which corresponds to given situation in nature, e.g. the final collapsed state of a rotating star, use Kerr; non (or minimally) rotating, use Schwarzschild; overall evolution of the universe, FRW (depending on your assumptions about what the universe is like).

The linearized equations are (I'm sure you agree) are not a solution but crude approximation method. There are now high order post-Newtonian equations, as well as direct numerical solutions available.

 

[TrickyDicky]

I agree with mass also, up to some upper limit where the mass is large enough to perturb other sources. For example, Jupiter orbiting the sun, Jupiter can be considered a test particle for reasonable precision. However, it is not intended to say anything about, e.g. a binary system of comparable masses.

It might be like you say, the problem is that this is implicitly assumed, as an assumed code, or so it has been called in this thread.
The thing is if that was the case, that the mass independence is always referred to the body's mass compared with other sources, why isn't that included in the Principle explicitly?
Certainly the low-mass code is true in linearized GR, could it be that it has become a habit to think of the EP and geodesic motion in the terms of linearized GR, since most problems are tackled in this regime, and from the habit of thinking in these kind of setting, this low mass code has become the most prevalent interpretation?

 

[TrickyDicky]

The linearized equations are (I'm sure you agree) are not a solution but crude approximation method. There are now high order post-Newtonian equations, as well as direct numerical solutions available.

Yes, I meant they are derived from the EFE under certain conditions that might be different from the conditions assumed for the other situations mentioned.

 

[PAllen]

It might be like you say, the problem is that this is implicitly assumed, as an assumed code, or so it has been called in this thread.
The thing is if that was the case, that the mass independence is always referred to the body's mass compared with other sources, why isn't that included in the Principle explicitly?
Certainly the low-mass code is true in linearized GR, could it be that it has become a habit to think of the EP and geodesic motion in the terms of linearized GR, since most problems are tackled in this regime, and from the habit of thinking in these kind of setting, this low mass code has become the most prevalent interpretation?

The wording as stated is obviously false for high enough mass (we are repeating ourselves; another star introduced in place of Jupiter will not follow the same world line as Jupiter). There is nothing hidden about this.

 

[TrickyDicky]

The wording as stated is obviously false for high enough mass

I know we get stuck here, still to me is not obvious, at least from the wording.

(another star introduced in place of Jupiter will not follow the same world line as Jupiter).

I agree with this, but I think the wording refers to a single body, it doesn't mean that all bodies no matter their mass have the same worldline, since the worldline depends on the total curvature of the manifold at that point and that is determined by all the sources, not just the body in question. I think there is too much ambiguity in the definition anyway, so we might go on forever in this loop. Hopefully not.

 

[PAllen]

I know we get stuck here, still to me is not obvious, at least from the wording.


I agree with this, but I think the wording refers to a single body, it doesn't mean that all bodies no matter their mass have the same worldline, since the worldline depends on the total curvature of the manifold at that point and that is determined by all the sources, not just the body in question. I think there is too much ambiguity in the definition anyway, so we might go on forever in this loop. Hopefully not.

Yes, this is one of the places we keep getting stuck. Almost everyone except you sees 'test body' and says oh, I know that code word. Note that an atom, me, the moon, earth, and Jupiter introduced at the same place and velocity relative to the sun will fill follow the same world line (to pretty high precision). This gives a huge constraint on how gravity couples to matter. However, another star will not follow the same world line. Almost everyone else is happy to say, oh fine, another star is too massive to be a test body for the Sun's gravity.

 

[atyy]

The EP in full GR is usually thought to be:
1) Lorentzian signature of the metric
2) Minimal coupling between non-metric fields and the metric
3) Ability to state fundamental laws of non-metric fields using first derivatives

1) means that locally Minkowskian coordinates exist
2) & 3) mean that the "fundamental" laws of physics reduce to those of special relativity at a point, and don't probe the curvature of spacetime.

The curvature of spacetime can still be probed because the "derived" laws of physics.

 

[TrickyDicky]

Yes, this is one of the places we keep getting stuck. Almost everyone except you sees 'test body' and says oh, I know that code word. Note that an atom, me, the moon, earth, and Jupiter introduced at the same place and velocity relative to the sun will fill follow the same world line (to pretty high precision). This gives a huge constraint on how gravity couples to matter. However, another star will not follow the same world line. Almost everyone else is happy to say, oh fine, another star is too massive to be a test body for the Sun's gravity.

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

 

[PAllen]

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

And I can't understand what this means. None of the discussion about the WEP we've been having has any connection at all to linearized GR, that I can see.

The WEP, as I understand it, is limiting principle true to exceeding precision for applicable situations in the real world, and is also consistent with several theories of gravity, including full GR. Where does the linearized approximation come into this?

 

[PAllen]

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

I'm wondering, are you thinking that my example of a star not following the same trajectory as Jupiter is related to gravitational waves? That's not what I'm referring to at all. I simply mean, literally, a star would follow a radically different trajectory because it and the sun would mutually orbit; and it doesn't matter whether you compare Jupiter and a star in a center of mass frame or sun centered frame, the trajectories would be completely different. This is true in reality, in Newton, and in any plausible theory of gravity. Thus, the literal wording of this variant of WEP would simply be false (for all theories of gravity). This simply means that this situation is not intended be covered because a star cannot be test body for the sun's gravity.

And we've been here before... and I suspect the circle will not end. Not only has everything been said on this thread, it has been said too many times.

 

[TrickyDicky]

And I can't understand what this means.

Maybe this quote from Ryder's book on relativity helps to see what I'm referring to:
"So in the linearised theory the gravitational field has no influence on the motion of matter that produces the field...
...It is therefore possible in principle, as pointed out by Stephani,1 that an exact solution, provided it could be found, could differ appreciably from the linearised solution. So we must beware, especially since the linear approximation may be used in cases where an exact solution is not known; and therefore the conclusions drawn may not be reliable."

None of the discussion about the WEP we've been having has any connection at all to linearized GR, that I can see.

From the first moment you have been claiming that the WEP, and test partcles must be understood in the weak field limit and as an approximative approach (low mass code) and now you say none of the discussion has any connection with linearized GR, I truly find hard to understand you too.
All the examples used thru the thread refer to Newtonian limit, weak field, linearized approximation.

 

[PAllen]

Maybe this quote from Ryder's book on relativity helps to see what I'm referring to:
"So in the linearised theory the gravitational field has no influence on the motion of matter that produces the field...
...It is therefore possible in principle, as pointed out by Stephani,1 that an exact solution, provided it could be found, could differ appreciably from the linearised solution. So we must beware, especially since the linear approximation may be used in cases where an exact solution is not known; and therefore the conclusions drawn may not be reliable."



From the first moment you have been claiming that the WEP, and test partcles must be understood in the weak field limit and as an approximative approach (low mass code) and now you say none of the discussion has any connection with linearized GR, I truly find hard to understand you too.
All the examples used thru the thread refer to Newtonian limit, weak field, linearized approximation.

I don't understand how you are reading what I write to say any of this. I have never mentioned linearized theory except perhaps in direct response to something you brought up about. Nothing I have said in this thread, as I wrote it, and read as written, has anything to do with linearized theory.

The statement that the WEP be considered for test particles is not, in my mind, connected in any way to the linearized theory. It is, in fact, connected (in the case of GR) with being able to use a geodesic of the background geometry - that is, a calculation in the full exact theory (in, e.g. the strongest field section of Kerr geometry), but limited to test particles that follow background geodesics. It is truly mind boggling how much your interpretation differs from the words I wrote.

In a few places, I have referred to the obvious fact that there is no known exact, non-static two body solution. However, the alternatives I've proposed for dealing with this have nothing to do with linearized GR - the papers I've linked (for the two body problem) involve using numeric solution of the full field equations, with sophisticated convergence control, to compute corrections to 3.5 order post Newtonian approximation, which is already way beyond linearized theory.

It really sometime feels like you have separate dictionary for English when reading what I write.

 

[TrickyDicky]

I don't understand how you are reading what I write to say any of this. I have never mentioned linearized theory except perhaps in direct response to something you brought up about. Nothing I have said in this thread, as I wrote it, and read as written, has anything to do with linearized theory.

The statement that the WEP be considered for test particles is not, in my mind, connected in any way to the linearized theory...

In my mind there is some connection, in the sense that if test particles are considered as only those bodies with not enough mass to perturb the background source(s), that is exactly linearized GR, a perturbative approach valid as long as the test body doesn't perturb the Minkowski background too much (weak field limit), in that sense in my opinion you have referred to the linearized approximative approach to GR most of the time when you have mentioned test particles and more explicitly:
In post #15 when referring to GW which are derived from the linearized equations.
In posts #37, #41, #44 and#89 when describing the low mass limit for test particles in terms of not perturbing the background metric.
In posts #94 and #106 when talking about gravitational radiation situations.

I think the english dictionary might not be the problem here after all.

 

[PAllen]

In my mind there is some connection, in the sense that if test particles are considered as only those bodies with not enough mass to perturb the background source(s), that is exactly linearized GR, a perturbative approach valid as long as the test body doesn't perturb the Minkowski background too much (weak field limit),

I agree that the linearized approach means ignoring non-linear corrections to Minkowski background. What has that got to do with computing geodesics in exact Kerr geometry (for example)?

A related usage is to perturb a background metric other than Minkowski, e.g. a simple approach to the two body problem. However, in discussing WEP my view is you simply limit it to cases where the geodesic equation of motion is valid. For cases where it isn't, nowadays, you needn't rely on linearized theory if you are concerned about its validity - you can use numeric solution of the full field equations.

Many of the calcualtions (even very old ones) showing that the goedesic equation of motion is true only in the limit use the full field equiations, not linearized equations. And to try to limit confusion, by geodesic equation of motion, I mean a geodesic of the background geometry.

As an aside, there are now numerous derivaitons of GW that make no use of perturbative approaches at all. I linked two of them (equal mass neutron stars, equal mass black hole papers; these use numeric solution of the full field equations). There are also arguments based not even on any numeric simulations, e.g. based on the 'relaxed' form of field equations which are exact as long as you are dealing with non-singular spacetimes (roughly; more precisly, you have to be able to cover the spacetime in harmonic coordinates (introduced by Vladimir Fock)).

 

[TrickyDicky]

It is obvious that given the ambiguous nature of all the versions of the EP there is little chance to reach an agreement.

But to me is enough with what is said in posts #106 to #109 by atty and PAllen in which a certain consensus is reached, at least I can agree with most of what is said in those.
So I guess it's been worthy. There's been some mutual misundertandings on both sides of the question, but at least I can get something out of it all. Thanks everyone.