Is the equivalence principle good for anything?

[yuiop]

1. For these reasons I conjecture that in the absence of other fields, a charged particle will follow the same geodesic as an uncharged particle.
2. a charged body freely falling in the absence of any other fields will not radiate.

Are we getting anywhere ?

In another thread "To radiate or not to radiate" https://www.physicsforums.com/showthread.php?t=369612&highlight=EP+charge I think the consensus was that:

1) In the absence of other fields, a charged particle (falling in a gravitational field) will not follow the same geodesic as an uncharged particle.
2) A charged body freely falling (in a gravitational field and) in the absence of any other fields will radiate.

A popularist statement of the EP (even used by Einstein himself in the falling elevator thought experiment) is that there is no experiment that can be carried out in within the confines of a lab that can distinguish between:

A) A lab that is stationary in a gravitational field and a lab artificially accelerating in flat space.
B) A lab free falling in a gravitational field and an inertial lab in flat space.

A small refinement of the popularist statement of the EP would be along the lines of:

There is no experiment (not involving charged particles) that can be carried out within the confines of a lab, that can distinguish between a lab in flat space and a lab in a gravitational field. If a difference is detected by a given experiment, then the experiment should be conducted over a shorter time period in a smaller lab, until the differences are negligible.

The clause specifying the experiment should be carried out over a smaller region of space and time means the explicit exclusion of charged particles is not required, but I have added it because it is important to this thread and to make it clear that charged and neutral particles do behave differently in a gravitational field over extended regions of space and time.

For example the other thread it was concluded that:
1) A charged mass inside an orbiting lab will fall relative to the lab while a neutral mass in the same lab will remain stationary, while both charged and neutral masses in a lab in flat space will remain stationary.
2) A charged mass in a rocket accelerating in flat space will radiate while a charged mass in lab on the surface of a gravitational body will not radiate.

By “concluded” I mean I made those statements in the other thread and no one challenged them.

 

[CuriousKid]

bcrowell and atyy,
Thank you very much for your on point answers. I feel those answered my questions to the best that is possible before I read up more (no doubt with some of the great sources you've pointed out). I started to worry after seeing how many people here did not understand. Thanks to people like you I feel like this place is a great source to discuss physics! Thank you.

I think so.
But I think you already formulated the resolution:

The point is that a free falling frame is supposed to be locally equivalent to an inertial frame.

Can a particle at rest in an inertial frame feel a proper force due to its field? No.

You omitted "locally" in your second statement. Include it, and everything is fine again.
The EP is valid locally, that means, if there is some backreacktion with the field from inside the region you are considering, that's ok. You just make the region smaller if you want less deviation, and it goes to zero for a point like region.

How can an electromagnetic force on the charged particle NOT be local to the charged particle's position? I would expect the electromagnetic force to depend only on the fields at the particle's position. If this were the case, then the EP would hold. It doesn't (or at least not exactly), so it must depend on second derivatives as atyy is mentioning. Neither maxwell's equations, nor the lorentz force law have second derivatives, so somehow the effect must be sensitive to the curvature of spacetime (and hence how it can violate the EP). This is fascinating to me. Your "make the region smaller" solution is missing the entire problem here. So I wanted to point that out so you too could hopefully enjoy the awesomely counter-intuitive results here.

In the presence of an electromagnetic field, the motion of a test particle is characterized by

$$d^2x^{\mu}/ds^2+\Gamma^{\mu}_{\alpha \beta}(dx^{\alpha}/ds)(dx^{\beta}/ds) + e/{m_0}F^{\mu}_{\alpha}(dx^{\alpha}/ds)=0,$$

where $$s$$ is the affine parameter of the curve along which the particle with charge $$e$$ is travelling, $$m_0$$ is the rest mass of particle and $$F_{\alpha}^{\mu} = g_{\alpha\beta}F^{\mu\beta}$$ is the electromagnetic field tensor. This of cource isn't a geodesic equation so charged particles won't freely fall unless we neglect their charge or take the electromagnetic field tensor to be zero.

AB

That is not entirely correct.
There is only one part that depends on the charge in what you wrote there, and that is just the Lorentz force law. Which cannot be the whole story, since what you wrote would actually predict charged and uncharged particle would follow the same path if there are no external fields. Which is not the case, and the whole point of this thread.

What do you mean by "completely mathematically/rigorously define"? Can you give an example of a mathematical definition that is not complete and/or rigorous (of the EP or anything else), so that we have an idea what you would like to avoid?

Sure, I would like to avoid defining it with terms that are not well defined. Because all that does is shift the problem.

For instance, what is wrong with this statement of the Einstein EP from An Introduction to General Relativity Spacetime and Geometry by Sean M. Carroll, Sec. 2.1, p. 50:
"It is impossible to detect the existence of a gravitational field by means of local experiments."

For example, while I can probably guess what is meant here, this requires me to define what it means to "detect the existence of a gravitational field" and what is and is not a "local experiment".

Charged particles may behave differently than neutral particles; the Einstein EP makes no statement regarding this difference. The issue of the EP is whether a given particle, charged or not, behaves differently in a kinematically accelerating frame compared to a gravitational frame. Can a scientist do an experiment on the charged particle to determine that he/she is in a gravitational field rather than an accelerating rocketship?

Emphasis added.
Please reread what you wrote here, as I'm not entirely sure what is preventing you from seeing the main point here. YES, because of the EP violation, we could (in principle, although as bcrowell mentions the effect is small) detect a difference in those two situations by dropping a charged particle and a neutral particle and noticing a difference in their paths.

That is the whole point that started this discussion.

---------------

Alright, so to try to combine what I'm learning, let me try restating the EP as rigorously as possible:
Pre-condition: SR and causality require state evolution to be local.
Then the EP requires:
1) Curvature depends on the total stress energy tensor (ie it can't couple to different contributions to the stress energy tensor differently ... so we are garaunteed we can define m_inertial = m_gravitational without internal contradictions in the theory)
2) Physics describing the local evolution to not depend on the curvature

If the second were rigorously true, then "local" experiments couldn't distinguish between a lab with proper-acceleration due to a rocket, or due to being "stationary in a gravitational field". And similarly for "local" experiments and an inertial frame and a freefalling frame.

It is number #2 that is violated by the charged particle example. I also don't like how vague parts of that are (which could lead it to imply things I don't mean). But we could hopefully at least agree on this:
If the energy density of some new physics was proportional to the scalar curvature, this would greatly violate the EP.

Can we agree on that?
In some sense, that example seems to "maximally" violate the EP.
As I finish more of the many things atyy and bcrowell pointed out, maybe I can come up with a better statement of the principle.

 

[yuiop]

For example the other thread it was concluded that:
1) A charged mass inside an orbiting lab will fall relative to the lab while a neutral mass in the same lab will remain stationary, while both charged and neutral masses in a lab in flat space will remain stationary.
2) A charged mass in a rocket accelerating in flat space will radiate while a charged mass in lab on the surface of a gravitational body will not radiate.

Now the big question is does either of the above thought experiment violate the EP principle. Perhaps not. In the first experiment the scientists in the orbiting lab notice the the charged particle is moving relative to the uncharged particle. (I am taking the liberty of conducting the experiment over a longer interval of time and space). The scientist might conclude that they are free falling in a gravitational field but equally they can not be certain that they are not in flat space and the charged particle is responding to a field emmanating from an electromagetic source outside the lab (that they can not see). They can not elliminate the possibility that there is an external electromagnetic field because that would be done by observing the behavior of a charged particle.. catch 22. I suppose they could enclose the experiment in a Faraday cage to elliminate the supposed external EM field but my guess is that if an orbiting lab was enclosed in a Faraday cage, then the charged and uncharged particles would follow the same geodesics.

In the second experiment, it would seem that the detection of radiation from a charged mass that is at rest with the lab would allow the scientists to conclude that their lab is in rocket accelerating in flat space and not in a real gravitational field. This would depend upon whether or not radiation from an artificially accelerating charged mass can be detected by observers that are co-accelerating and I suspect the answer is no.

If my above assumptions are correct, then neither experiment would determine whether the lab was in flat space or a gravitational field even when the experiments are carried out over a larger interval of time and space.

 

[CuriousKid]

They can not elliminate the possibility that there is an external electromagnetic field because that would be done by observing the behavior of a charged particle.. catch 22.

This catch 22 can be resolved.

We have to agree with Newtonian limits, so imagine a trajectory around a large planet. A charged particle (regardless of sign) will spiral in, not out. In otherwords, the effect is related to the magnitude (not sign) of the charge and therefore can be distinguished from an external electric field.

The other one you mention I don't want to get into here, as I don't want to get into trying to define "measuring radiation". That's why I used the trajectory example instead.

 

[Mentz114]

This statement and its variants

"It is not possible to detect if a body is at rest in a gravitational field or undergoing proper acceleration"

which has been elevated to a Principle, is worth nothing. First, we have add two conditions, "provided we don't look out of the window" and "provided we are infinitesimally small". In other words it's never true ! So why is it a principle when one can always tell the difference ?

Charges undergoing proper acceleration will radiate sometimes, but charges at rest in a gravitational field will not. Nothing is violated, there is actually nothing further to say.
One cannot discuss violations of the EP ( the real EP not the statement above) in the context of GR - because GR would no longer be valid.

To the OP I would say, the statement above is worthless and should never be mistaken for a principle.

Equivalence is whether inertial mass is gravitational mass. This cannot be discussed in the context of GR because it must be true in GR - any deviation renders GR a useless theory.

The proper framework to discuss inertial mass vs gravitational mass is a theory which treats gravity as a force-field, so $m_i$ and $m_g$ explicitly appear in the equations.

Such a theory exists but has been ignored in those of the cited papers I have read.

 

[CuriousKid]

First, we have add two conditions, "provided we don't look out of the window" and "provided we are infinitesimally small". In other words it's never true ! So why is it a principle when one can always tell the difference ?

Let me make some general statements which I hope we can agree on:
1) Something being true only locally, is not the same as it never being true.
and
2) If a principle is purely local, while that means any real experiment will in fact violate it to some extent, it is still immensely useful as it puts strong mathematical constraints on the theory.

A follow up on #2, not being as general, is local poincare invariance in GR. This is a huge restriction on theories in curved spacetime. So it is useful theoretically. We could consider terms which violate this (and experiments have, and put great limits on such terms), so experiment can also still test such local principles.

Equivalence is whether inertial mass is gravitational mass. This cannot be discussed in the context of GR because it must be true in GR - any deviation renders GR a useless theory.

As already explained to you by other posters, the equivalence principle encompasses more than just m_inertial is proportional to m_gravitational. If you are going to continue to ignore any material that does not fit this pre-conceived notion of yours, then it is really not that useful to have this discussion with you.

What you are referring to is a very truncated version of the "weak" equivalence principle. I say truncated, because if you actually tried to operationally define what you mean, you'd probably end up with something like:
All test particles at the alike spacetime point in a given gravitational field will undergo the same acceleration, independent of their properties, including their rest mass
Paul S. Wesson, 2006
Five-dimensional physics: classical and quantum consequences of Kaluza-Klein cosmology
pg 81

So even the "weak" equivalence principle you are try to reduce this to, is violated by the electric charge example.

As another poster mentioned, there are the weak, Einstein, and strong formulations of the equivalence principle. Let us call these the WEP, EEP, and SEP in further discussion.

So please, can you please stop saying the equivalence principle is "purely graviational"?
You can find discussion in many of the articles or textbooks mentioned previously, but a decent starting point is here:
http://en.wikipedia.org/wiki/Equivalence_principle

 

[yuiop]

This catch 22 can be resolved.

We have to agree with Newtonian limits, so imagine a trajectory around a large planet. A charged particle (regardless of sign) will spiral in, not out. In otherwords, the effect is related to the magnitude (not sign) of the charge and therefore can be distinguished from an external electric field.

Well the point is that when the observations in the orbiting lab are confined to measurements within the lab (not allowed to "look out the window") there is no concept of spiralling in or out just as there is no concept of up or down in an inertial lab in flat space. (All directions are equal). It would not practical to compare a positively charged particle alongside a negatively charged particle at the same time in the same lab, as their mutual electromagnetic attraction would be many orders of magnitude larger than the effects we want to study here. Now if we store one charge in a perfectly shielded cage while we tested the oppositely charged particle and then swap them over you might have a case.

We then get into the realm of a concept introduced by Einstein that I will call "coincidentiality" in which when you ignite the engine of a rocket in flat space you can not be certain that the acceleration you feel is not due to coincidental appearance of a real gravitational field and that the thrust of the engine is merely holding the rocket stationary in this new gravitational field. Presumably it could be argued when you when swap the charge of the test particle inside the lab, the fact that the direction that the test charge moves is unchanged could be due to a coincidental change in the electromagnetic field external to the lab and because we are excluded from taking measurements outside the lab we can not prove that has not happened.

 

[yuiop]

This statement and its variants

"It is not possible to detect if a body is at rest in a gravitational field or undergoing proper acceleration"

which has been elevated to a Principle, is worth nothing. First, we have add two conditions, "provided we don't look out of the window" and "provided we are infinitesimally small". In other words it's never true ! So why is it a principle when one can always tell the difference ?

I personally believe that when Einstein first had the concept that there is no experiment that can determine if a body is at rest in a gravitational field or undergoing proper acceleration, provided we don't look out of the window and provided we are infinitesimally small, he had in mind that there is literally no experiment and that includes experiments using charged or spinning particles or any other physical test you can devise.

This is some ways similar to the impossibility of detecting absolute motion in Special Relativity. If we could conduct experiments in an inertial lab using charged or spinning particles that could determine absolute motion, we would consider that a serious problem for the validity of SR.

To the OP I would say, the statement above is worthless and should never be mistaken for a principle.

I think its usefulness is that some effects can not be transformed away no matter how small the experimental volume of time and space. For example the proper acceleration measured by an accelerometer (whether due to gravity or artificial acceleration) can not be removed and made equivalent to purely inertial motion, by simply considering an infinitesimal region of spacetime.

 

[Mentz114]

My post #40 is way off-beam. I tried to change it but you were too quick for me.

I withdraw it - it is not worthy of this discussion and not what I want to say.

Apologies.

 

[Ich]

Your "make the region smaller" solution is missing the entire problem here.

I don't think so. The deviation is not from a weird behaviour of the fields at the location of the test particle. It's from the particle's field that goes out to the universe, encounters curvature there, and brings back some change to the particle's position (epically spoken, not physically).
If you'd contain the field in a small region, there'd be no such effect.

 

[Altabeh]

I don't understand what that first paper has to do with the discussion.

It discusses a version of EP that possibly fits within the framework of electromagnetism in the sense that the electromagnetic interactions of particles are supposed to be creating an effective geometry and a given geometry creates an effective medium so a "resemblance" exists, as the one B. Mashhoon et. al. give away in terms of an analogy between EM and GR, which can justify the presence of EP in EM! See also the related articles there!

The EM equations dictate how charge moves. The partial derivatives in the EM equations should be replaced by covariant derivatives in GR. So it has everything to do with "this stuff" (meaning this thread).

It has nothing to do with "this stuff" meaning "how EP can be made compatible with charged test particles" (I assume this is what OP asked about.) In the Einstein-Maxwell equations the discussion of geodesic equation leads to the motion equation

$$d^2x^{\mu}/ds^2+\Gamma^{\mu}_{\alpha \beta}(dx^{\alpha}/ds)(dx^{\beta}/ds) + e/{m_0}F^{\mu}_{\alpha}(dx^{\alpha}/ds)=0,$$

(see, for instance, Relativity, Thermodynamics and Cosmology by R. C. Tolman, p 259-261).

Until there is a charge, the particle cannot follow a geodesic which means if EP predicts all particles to fall at the same rate around a gravitating body locally, i.e. the proper acceleration from geodesic equations is equal to -g locally, then for charged particles this is not possible because there is some extra term involving the Lorentz force! How come a charged particle would be accepted within EP?

No, I would not, I would take it for granted.

I don't know if you have already hit GEM somewhere else, but I guess you haven't! (Period.)

I will admit that I do not know how spin (I presume you refer here to intrinsic angular momentum of fundamental particles) would modify the geodesic equation. However, the OP mentioned a concern due to charge, not spin. Even neutral particles have spin, so spin is a separate issue that is unrelated to the charge, and so unrelated to this thread.

The relation between The spin four-vector

$$S_{/alpha}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\theta} J^{\beta\gamma}U^{\theta}},$$

where $$\epsilon_{\alpha\beta\gamma\theta}$$ is the antisymmetric Levi-Civita tensor density, $$J^{\beta\gamma}$$ is the angular momentum or spin and finally $$U^{\theta}$$ is the four-velocity of particle,
the charge $$e$$ and the geodesic equations is that the perturbations due to both spin and charge enter the geodesic equation as independent terms (see Papapetrou's paper I quoted in an early post) which sounds problematic to the OP:

That is not entirely correct.
There is only one part that depends on the charge in what you wrote there, and that is just the Lorentz force law. Which cannot be the whole story, since what you wrote would actually predict charged and uncharged particle would follow the same path if there are no external fields. Which is not the case, and the whole point of this thread.

What else would there be other than the Lorentz force acting on charged particles that Einstein-Maxwell equations offer!?

All extra terms including charge, spin and other qualities discussed beyond GR enter as perturbations into the geodesic equation. For Einstein-Maxwell equations, where one deals with influences of the electromagnetic and gravitational fields on each other, EP which was defined above in my perspective doesn't work.

What? Why are EM and GR "like two brothers"? Where does that come from?

No, I didn't. GR accommodates EM perfectly well. What else do you want me to say about it?

No, the EP does not require charged test particles to follow geodesics. If it did, then it would certainly be violated.

It comes from Maxwell-Einstein! Ask them! You're acting like they have found a unified theory of GR and EM wherein every law of GR works as fine as it can! Does getting an analogy between Maxwell equations in SR and GR necessarily leads to EM admitting all kinds of laws of GR within it? Maybe you and I are looking at the problem through different angles. So I'd be interested in knowing what exactly your definition of EP is and please give us some reference to go find it in there to only make sure such definition can also be taken into account because there is, you know, a bunch of them and across this thread people seem to go in different directions about how EP is defined! And please don't use multi-quotes in tremendously grisly numbers!


I have no idea what this is supposed to mean.

I disagree. The amount of mass of a particle and the condition of freefall are completely independent.

Sorry, I would have written 'charge' in place of mass! This is exactly in accord with the equation of motion of a charged particle!

Keep in mind that we have different agenda's in this respect. I am trying to help the OP understand that the EP does work for charged particles.

I don't know how, but I'd be glad to know what your EP means!

What are you talking about? When did I say this? I believe that you must have misunderstood me. GR is a framework in which EM can be formulated. That does not mean that "a GR version of EM exists".

I'm starting to feel nauseous by seeing you quote just tiny parts of my sentences which by no means flesh out my purposes and I think you are wasting my time through this business of picking out what you seem to be in opposition to. The complete sentence is

If you are insisting a GR version of EM exists that is unified not analogous

But your answer proves my guess was correct about you not knowing what GEM is, a sort of incomplete GR version of EM or gravito-electromagnetism that brings together the analogous Maxwell's equations within the framework of GR using linearized field equations!

See for more information:

1- B. Mashhoon, Phys. Lett. A 173, 347 (1993).

2- B. Mashhoon, in Reference Frames and Gravitomagnetism, edited by J.-F. Pascual-Sanchez, L. Floria, A. San Miguel and F. Vicente (World Scientific,
Singapore, 2001), pp. 121-132.

3- D. Bini and R.J. Jantzen, in Reference Frames and Gravitomagnetism, edited by J.-F. Pascual-Sanchez, L. Floria, A. San Miguel and F. Vicente
(World Scientific, Singapore, 2001), pp. 199-224.

The issue raised in the OP is charged vs. neutral; not scalar vs. spinor. I think that you need to start a new thread to address the spin issue.

What are you talking about? Who did say something about spinor or scalar!? I'm saying that qualities like spin and charge will disturb the geodesic equations so they don't make EP work. Explain to us explicitly what theory you are making out of EM that accepts EP well within it! If you'd mind elaborating things, first define your special EP that is like a rampart against all qualities of particles added up to GR to only get EP to be still workin' fine!

AB

 

[Altabeh]

Emphasis added.
Please reread what you wrote here, as I'm not entirely sure what is preventing you from seeing the main point here. YES, because of the EP violation, we could (in principle, although as bcrowell mentions the effect is small) detect a difference in those two situations by dropping a charged particle and a neutral particle and noticing a difference in their paths.

Isn't this what I've been talking about for 2 days now? Charged particles would take different paths than neutral particles due to 1- Lorentz force 2- external forces which enter the geodesic equations as perturbative terms; so EP is violated but I am not sure about the size of difference and the size of regions in which EP is guaranteed! This is because when it comes to local discussion, we take, for example, the distance from moon to Earth a small distance with respect to the distance of Earth to sun within it EP is meant to work!

AB

 

[Altabeh]

I personally believe that when Einstein first had the concept that there is no experiment that can determine if a body is at rest in a gravitational field or undergoing proper acceleration, provided we don't look out of the window and provided we are infinitesimally small, he had in mind that there is literally no experiment and that includes experiments using charged or spinning particles or any other physical test you can devise.

This is some ways similar to the impossibility of detecting absolute motion in Special Relativity. If we could conduct experiments in an inertial lab using charged or spinning particles that could determine absolute motion, we would consider that a serious problem for the validity of SR.



I think its usefulness is that some effects can not be transformed away no matter how small the experimental volume of time and space. For example the proper acceleration measured by an accelerometer (whether due to gravity or artificial acceleration) can not be removed and made equivalent to purely inertial motion, by simply considering an infinitesimal region of spacetime.

I think you are realistic like me. But I hope we can agree on two things here:

1- Charged or spinning particles would make the path anything but geodesic. So in "small" distances (we don't know about the wor "small") that is even true to say EP doesn't work because such particles do not fall at the same rate!

2- If we mean by EP, every spacetime is locally flat, then in small regions we have still a non-vanishing curvature tensor except at the central point of the region where the metric is made flat! But if in such spacetime we take the motion of such particles unaffected by their qualities then, though questionable, but it is acceptable due to the smallness of influences of them on the motion! [I still have doubt it could be possible to accept this in Einstein's sense.]

AB

 

[yuiop]

2) A charged mass in a rocket accelerating in flat space will radiate while a charged mass in lab on the surface of a gravitational body will not radiate.

The above statement does not explicitly state the state of motion of the observer and since this is relativity, that is an important factor! If it was true that an observer inside an accelerating rocket could observe radiation from a charged mass at rest inside the rocket, while an observer in a stationary lab on the surface of a gravitational mass could not, then I would consider that quite a strong violation of the EP, because that discrepancy could not be removed by considering a smaller volume for the lab. The calculations to determine whether or not "an observer inside an accelerating rocket could observe radiation from a charged mass at rest inside the rocket" are probably quite complex and the experiment has not actually been carried out, but using the EP alone I can easily predict that the observer inside the accelerating lab would not see the charge radiating. That should be a demonstration of the usefulness of the EP.

 

[Altabeh]

If you'd contain the field in a small region, there'd be no such effect.

Who can clarify the region of EP is sufficiently small so as to let the effects of field be ruled out? In Enstein's sense, a very large distance from Earth to moon would also count as a region where EP works in it! How come in such a large distance the effects of fields can be small!?

AB

 

[Ich]

In Enstein's sense, a very large distance from Earth to moon would also count as a region where EP works in it!

How so? You can't transform gravitation away when it even changes sign.
The allowed region depends on the allowed deviation from the EP. The smaller the region, the smaller the deviation.

 

[Altabeh]

How so? You can't transform gravitation away when it even changes sign.
The allowed region depends on the allowed deviation from the EP. The smaller the region, the smaller the deviation.

But my point wasn't this. I'm saying in large distances which EP allows, how can you neglect something like the Lorentz force on a charged particle in the equation of motion? If you do, then you're probably taking for granted that qualities like spin or charge has no effect in a small region (while this is a not small region) which is highly impossible! However, I agree that if we pick a region inside the region allowed by EP (e.g. neighbourhood of an arbitrary point), then the deviation can be taken so small but not vanishing. This of course makes EP violated slightly in the chosen region, but let's say we don't see it just because to justify EP for charged particles!

AB

 

[turin]

... because of the EP violation, we could (in principle, although as bcrowell mentions the effect is small) detect a difference in those two situations by dropping a charged particle and a neutral particle and noticing a difference in their paths.

That is the whole point that started this discussion.

I appologize if I have misunderstood your OP. I identified two basic questions that I considered to be separate:

1. Does a charged particle behave differently than a neutral particle, under only gravitational influence?

2. Does the behavior of a charged particle under only gravitational influence, presuming that it differs from the behavior of a neutral particle, violate the EP?

I will admit once again that I do not know how to adress Question 1. If that is your only real question, and the whole issue of the EP was just a side note, then I have nothing to offer to the discussion.

However, if you are interested in Question 2:

The deviation of a charged particle from a neutral particle is not sufficient to signal an EP violation. The EP does not regard what will happen to a particle; that is the job of the applicable nongravitational laws of physics in a given reference frame. The EP regards the interpretation of the (local) reference frame in which the behavior of a physical system is observed.

A particularly applicable law of physics for this thread is the law that says that accelerating charge induces EM radiation. If you observe an accelerating charge that is not radiating, then that would certainly violate the EP, regardless of the behavior of the neutral charge next door. However, according to your OP, I have understood that: a) The charge does not follow a geodesic (i.e. it experiences a proper acceleration), and b) The charge radiates. Therefore, so far I see no reason to claim EP violation.

 

[CuriousKid]

This catch 22 can be resolved.

We have to agree with Newtonian limits, so imagine a trajectory around a large planet. A charged particle (regardless of sign) will spiral in, not out. In otherwords, the effect is related to the magnitude (not sign) of the charge and therefore can be distinguished from an external electric field.

Well the point is that when the observations in the orbiting lab are confined to measurements within the lab (not allowed to "look out the window") there is no concept of spiralling in or out just as there is no concept of up or down in an inertial lab in flat space. (All directions are equal).

I gave that explanation to help show why the direction was independent of the charge sign, instead of just stating it as a fact. Therefore what you felt was a catch 22 is resolved: this effect can be distinguished in principle from an external electric field.

Your other comments fall into the bin of "experimental difficulties" due to the effect being so small.

Actually, MANY people's comments here fall into that bin.
So let me try to focus discussion here:

IF there is someway to precisely state the EP, THEN it is a mathematical consistency check on a theory. GR + EM seems to fail this check for how we are currently wording the EP. I want to know WHY. I don't care that the effect is small. The math is clear, the effect is there. So why?
is it because:
1) The EP is just an approximation? That is, its statement if made rigorous would have to include something like: to second order the deviations from blah blah
or
2) The EP does NOT fail, it is just that due to the EP currently being worded "non-rigorously", there are some subtleties in what one can use the EP for? That is, it has less "range of applicability" than the EP statement implies.In particular, I've heard claims that GR can be derived by assuming the EP along with some other requirements. Which does fit mathematically with this effect. So something interesting can be learned here.Since there still seems to be confusion on the details of this effect, we might as well discuss that too.

That is not entirely correct.
There is only one part that depends on the charge in what you wrote there, and that is just the Lorentz force law. Which cannot be the whole story, since what you wrote would actually predict charged and uncharged particle would follow the same path if there are no external fields. Which is not the case, and the whole point of this thread.

What else would there be other than the Lorentz force acting on charged particles that Einstein-Maxwell equations offer!?

The Lorentz force is a local law, and locally spacetime can always be described by the minkowski metric: spacetime is homogenous and isotropic locally.

Therefore, the Lorentz force alone should not be able to explain deviations from a geodesic due to a particle's own fields, since the spacetime is isotropic and the fields will be as well and thus the proper-force on the particle due to its own fields while at rest in a local inertial frame due to the Lorentz force must be zero.

Think of it this way. Consider an observer in the same local rest inertial frame as the charged particle. To the observer, the particle will just sit there, and then once they enter curved spacetime, the particle will start to move away... and you want me to believe this is the Lorentz force? How did it pick a direction for its own field to push it? This would look like a violation of Maxwell's equations and the Lorentz force law in the local rest inertial frame. That cannot be what is happenning.

Somehow the curvature must appear as a source even in the local inertial frame.
This is also the key according to the heuristics atyy posted for when the EP fails. The only way for the EP to fail, is to find physics that locally feels the local curvature.To make this more productive, let me ask:
Does anyone know how to calculate such deviations?
If so, HOW is the curvature coupling to this?

EDIT: Removed some incorrect statements.

 

[Mentz114]

Sorry to be a bore, but I want to come back to this

1. "It is not possible to determine if a body is at rest in a gravitational field or undergoing proper acceleration"

Obviously 'determine' here means some method that does not rely on external observation ( otherwise we could see if we were moving, say ) and we are also restricted to regions where the space-time is flat, i.e. tidal forces are too small to detect ( otherwise we could detect them, and rule out proper acceleration).

So, we can rewrite 1
1. "It is not possible to determine if a body is at rest in a gravitational field or undergoing proper acceleration provided the gravitational field has the same configuration as the force-field caused by the proper acceleration".

or even

1. "It is not possible to determine if a body is at rest in a gravitational field or undergoing proper acceleration provided that anything that might distinguish them is too small to detect( including radiating charges)".

So we admit that they can be distinguished but in order to get a resounding 'principle' we pretend otherwise.

A proper 'principle' would say 'it is never possible to ...'.

If two things are the same, they are always the same, not just in the dark.

Proper acceleration is not the same as coordinate acceleration, but in some circumstances they may be mistaken for each other. Now that's a principle.

CuriousKid, you might find this helpful

How does the electromagnetic field couple to gravity, in particular to metric, 
nonmetricity, torsion, and curvature ?

Friedrich W. Hehl and Yuri N. Obukhov

arXiv:gr-qc/0001010v2 3 May 2000

 

[TcheQ]

So, we can rewrite 1
1. "It is not possible to determine if a body is at rest in a gravitational field or undergoing proper acceleration provided the gravitational field has the same configuration as the force-field caused by the proper acceleration".

I am mystified how people have confused this simple rule. If they weren't the same acceleration field, they wouldn't be equivalent(sic)

 

[CuriousKid]

Here's a paper showing the force is from coupling to the curvature:
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0012/0012057v2.pdf

For a charged particle freely falling with no external electromagnetic fields, once it gets to a section of spacetime with non-zero Ricci tensor, it will move off the geodesic according to:
$$m {u^\alpha}_{;\beta} u^\beta = \frac{1}{3}e^2 ({R^\alpha}_\beta u^\beta + u^\alpha R_{\beta\gamma} u^\beta u^\gamma)$$I can't find the paper I was remembering. Which looked at a charged particle orbiting a gravitational body. It too found deviation, but must have involved another term, since that would depend on the Weyl tensor, not the Ricci tensor.

Does anyone know the paper I am thinking of? If we can find where someone worked out the exact coupling to the curvature, it would be enlightening for this discussion.

EDIT: Hmmm.. there was another term involving Green's functions that I thought I could ignore in absence of external fields. It sounds like maybe no. So there could still be a Weyl coupling term in there. If only I could find that other paper.
EDIT(2): Yes it is somehow in there
Other derivations with the same result
http://relativity.livingreviews.org/open?pubNo=lrr-2004-6
"Radiation reaction of point particles in curved spacetime"
http://www.iop.org/EJ/article/0264-9381/21/16/R01/cqg4_16_r01.pdf
EDIT(3): Tries to discuss the Green's function term in more depth
http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.0464v1.pdf

 

[yuiop]

1- Charged or spinning particles would make the path anything but geodesic. So in "small" distances (we don't know about the wor "small") that is even true to say EP doesn't work because such particles do not fall at the same rate!

Charged particles would follow a geodesic, just not the same geodesic as an uncharged particle. The geodesic of a particle depends on its location in the gravitational field, its velocity, its charge and probably a few other factors.

Lets try this scenario. I have a bet with Joe. I say I will place him in a small lab and for a million dollars all he has to do is tell me whether or not he is in a gravitational field. He is not allowed to look out the window of the lab or use tidal effects, but he is allowed to use a neutral or negatively charged test particle in his experiments. Joe has been reading this thread and thinks he is on to a good thing and accepts the bet. He carries out his experiments and notices that the both the charged and neutral particles do not move relative to the lab even over extended periods of time and bets that he is not in a gravitational field.

The blind is removed from the lab window and to his horror he finds he is in orbit around a gravitational body, but it happens to be a negatively charged gravitational body which prevents the charged test particle spiralling inwards because of electromagnetic repulsion. He never had a chance, because he had no way to test if there is an external electromagnetic field other than use a charged test particle. Joe owes me a million bucks

Now I have a similar bet with Fred. Fred notices that the charged particle moves relative to the lab while the neutral particle does not and bets he is in orbit. The blind is removed and he finds he is flat space with an external electromagnetic field and he also owes me a million bucks.

So, the moral of the story is that Joe and Fred could not determine whether they are in a real gravitational field or not, even when they have a charged test particle available, so maybe the EP is safe.

 

[CuriousKid]

Charged particles would follow a geodesic, just not the same geodesic as an uncharged particle.

You can't call it a geodesic if it has a proper acceleration.

So, the moral of the story is that Joe and Fred could not determine whether they are in a real gravitational field or not, even when they have a charged test particle available, so maybe the EP is safe.

DAMN IT!
I already explained to you twice that in principle you can distinguish between this effect and an external electric field. This effect will not depend on the sign of the charge, while the interaction with an external electric field WILL.

I am making mistakes. So are plenty of others. But let's please try to learn from our mistakes and move on. The thread is already quite cluttered as is.

 

[yuiop]

DAMN IT!
I already explained to you twice that in principle you can distinguish between this effect and an external electric field. This effect will not depend on the sign of the charge, while the interaction with an external electric field WILL.

I understand your point and that is why I deliberately allowed Fred only one negatively charged particle. I think you on the other hand miss my point that if an observer can not make measurements outside the lab, he can not be sure that conditions outside the lab remain constant, so when he swaps the charge of the test particle he can not be certain that the external electromagnetic field coincidentally changed direction at the same time. I know this particular argument is stretching a point, but Einstein also used such an argument, talking about gravitational fields coincidentally "springing up" when a rocket burns its thrusters.

 

[Ich]

Here's a paper showing the force is from coupling to the curvature

I see there some agreement:

The physical effect responsible for the force, in DeWitt and DeWitt’s view [48], is a signal that originates from the particle at an earlier time t0, propagates toward the central mass M at the speed of light, bounces off the central mass, and comes back to the particle at the current time t.

It's from the particle's field that goes out to the universe, encounters curvature there, and brings back some change to the particle's position

Ok, as to the coupling with local curvature (inside matter only, obviously): I'm not sure what to make of that. Unless proven otherwise, I'd rely on something like that (the very next sentence in the paper):

Although the self-force is nonlocal, Eq. (1.10) involves the conditions at the current time only. This is because the time delay in not noticeable at the level of approximation maintained in the calculation. To leading order in a weak-field, slow-motion approximation, the electromagnetic self-force appears to be entirely local.

Maybe the local part of the equation is only an effective term - or I have to correct myself as to the validity of the EP inside of matter. I believe that in vacuum, my position is still quite firm.
So, for the time being, I maintain my claim that deviations from a geodesic are of nonlocal origin and thus no violation of the EP.

 

[CuriousKid]

I think you on the other hand miss my point that if an observer can not make measurements outside the lab, he can not be sure that conditions outside the lab remain constant, so when he swaps the charge of the test particle he can not be certain that the external electromagnetic field coincidentally changed direction at the same time.

Yes, I noticed you tried to make that point. I did not miss it, I just felt that because they are not equivalent, and therefore in principle an experiment can measure it. Yes you can make the scenarios more and more complicated, but since they are not equivalent, again in principle an experiment can measure it.

So here you want me to consider the possibility that there is an external electric field which can be time varying. In which case a magnetic field will be generated and I could detect that. Actually, this gives me an even simpler idea for ensuring it is not an electromagnetic field: just measure the potential drop across some resistors in various positions and orientations.

However, your point DID end up being very useful in just a second (albeit in a much more complicated way). Basically, imagine there was an external field that was correlated to the particle's movement through spacetime only when there was gravity. Could we ever distinguish these even in principle? While it would be weird for scientists to dismiss continual "coincidences", they can't really know there are coincidences in the first place (since that would require knowing the "full" spacetime situation). Second, even if they knew there were correlations, they can't absolutely distinguish a "correlated" interaction from just a coincidental external field. But how could an interaction ever be correlated like this?

--------------

As a follow up on the previous posts/paper links:
I continued to think there was some way to couple to the Weyl curvature like the Ricci curvature coupling term. I'm finally starting to see and accept that there is just no such (at least local) coupling.

It appears the Green's function term cannot be reduced to a local interaction as explained with simple counter-examples in one paper (this interaction does not violate relativity, it is non-local in the sense that it depends on infinite things in the past light cone). So while it doesn't contribute if spacetime is flat, it doesn't "couple" to the Weyl curvature in a simple local way.

Furthermore, since it depends on the past, one could consider it as "fields from the past" collecting here. So, while curvature was needed to make this non-zero, it is a real electric field causing it: the radiation.

If the integral over all history is generically dominated by near times (say an exponential dependence in time or something), then this still seems like a violation to me. But if it is dominated by far times (since it needs to be as non-flat as possible? sample more spacetime?), then it would just appear as an external field that is correlated oddly with the particle positions. And HERE, I can see a corollary to your point. In that case I wouldn't consider that a violation.

And there seems to be no easy cutoff.
So damn. Made it full circle. After all we learned: What is the definition of the EP? And indeed as some argued earlier, it seems too vague to let me decide soundly whether an "integrate over all history" interaction violates it.Oh, and as for the Ricci coupling term. That can only come from stuff purely locally, and thus IN the experiment itself. So the "Stong Equivalence Principle" seems to handle it just fine.

Maybe not everyone will agree with me, but this seems to boil down to:

1) Does coupling with the Ricci curvature violate EP? I say no.
2) Would direct coupling with the Weyl curvature violate EP? I say yes, but we don't have evidence of that here.
3) Does indirect coupling through an "integrate over all history" interaction violate EP?
I can't answer this cleanly right now.

-------------
EDIT:
I change my mind.
Here is a clear answer: If the interaction from "history" ONLY depended on the history later than L/c ago, where L is the size of the "lab" we're riding in, then I feel this would be a clear "no violation" since no experiment could distinguish it absolutely from external fields.

But the EP is a LOCAL principle. And while the force is local, the interaction propagated from previous in time... a different location in spacetime. Essentially, just take the L->0 limit of my above lab argument.

The only problem would be if there is a non-vanishing contribution from t=now in the "integrate over history" term. I cannot prove it doesn't (Can someone please?), but since it is beyond me in math, and I can see everything falling in place, I'll accept that for now.
The EP has emerged a lot different in my mind. I feel I have learned a lot. Thanks to all of you!

Weirdly enough, it looks like the EP is NOT violated here, but for very subtle reasons (Ricci curvature is determined purely locally, and coupling based on correlations with history (even if they involve the Weyl curvature) don't violate local principles). I don't think I could have fully appreciated this without having thought through it all myself. It probably would have just sounded like a brush off "well, its subtle" answer. It really is amazing.

Thank you everyone!
Unless there are some big gaps in my logic, I think I'm done.

The most rigorous statement I can make of the EP currently is:

The EP requires:
1) Curvature depends on the total stress energy tensor (ie it can't couple to different contributions to the stress energy tensor differently ... so while it is sometimes difficult to define mass, when we can, we are guaranteed we can define m_inertial = m_gravitational without internal contradictions in the theory)
2) Physics describing the local evolution at spacetime point X can only depend on field values (or their derivatives?) at X, but cannot depend on the the Weyl curvature at XWooHoo!

EDIT(again):
Just saw your post Ich.
Yep, now that I understand more, I understand what you meant by that in retrospect. At the time, because I knew there was direct coupling to curvature, I incorrectly thought that was counter-example enough. But in the end, it looks like we came to the same conclusion. Thanks for the discussion!

 

[Altabeh]

IF there is someway to precisely state the EP, THEN it is a mathematical consistency check on a theory. GR + EM seems to fail this check for how we are currently wording the EP. I want to know WHY. I don't care that the effect is small. The math is clear, the effect is there. So why?[/tex]

I don't think you mean EP can be deduced mathematically! In every Lorentzian spacetime, we can define EP as

If a region S in spacetime is found wherein the curvature tensor $$R^{\alpha }_{\mu\nu\beta}$$ vanishes (*) (note I don't say "vanishes approximately" to give a precise definition not a poor one) then all uncharged test particles would follow geodesics in S.

This is enough to claim EP works in S accurately. For example, assume that you have a long rope hung between your hands in the air as the rope isn't completely pulled. Gently descend the rope over a flat plane and keep doing so until the lowest fairly small segment of the rope is spread on the plane and call the segment on plane ROP. Thus $$(R^{\alpha }_{\mu\nu\beta})_{ROP}=0.$$ This has a simple result: All uncharged particles along ROP (assuming that particles can slide over the threads of rope) are at rest but moving from one side to the other one from an observer's perspective being at rest in the frame K moving in the direction of the ROP at some constant velocity. All the particles that make ROP move in a straight line from observer's viewpoint just because the curvature tensor vanishes for ROP.

The above definition of EP must be cast in our wish list since it is highly idealistic and almost any curved spacetime cannot be locally as such. So people have to get a jump on the inaccurate and poor definitions such as local flatness or visualization of curved spacetimes in small regions. The latter has a very available example: Look at the Earth locally and you'll find out it's flat so the theorem of "local flatness" was born:

Every curved spacetime is locally flat.

This generally means in mathematical terms that you can find a coordinates system $$\bar{x}^{\alpha}$$ in which the metric $$g_{\mu\nu}$$ around some given point P, which is basically the origin of $$\bar{x}^{\alpha}$$, can be written as

$$\bar{g}_{\mu\nu}=\eta_{\mu\nu}+... ,$$

where the dots refer to terms involving $$g_{\mu\nu}$$ and all vanish at $$P.$$

The trouble is we cannot get a precise definition of EP according to this theorem. If we could, then it'd take us show the dots not only vanish at $$P$$ but also vanish, at the very least, in the neighborhood of $$P$$ to guarantee EP in a small region, but no dice! The only way out of this trouble is to consider this happens approximately. Now I think you can understand why EP is poorly defined though this doesn't mean it has no mathematical backbone to rely on it. We only make this mathematics very approximate, or in your sense non-rigorous, for EP by putting the first derivatives of dots equal to pure zero i.e. making the curvature tensor vanish, to meet something believed to be true experimentally.

Now how can one consider the situation for charged particles? The reason that I didn't use the rope example for charged particles is just their equation of motion and here you must be so conscious: According to the equation of motion of a charged particle,

$$\ddot{x}^{\alpha}+\Gamma^{\alpha}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\frac{e}{m_0}F^{\alpha}_{\mu}\dot{x}^{\mu}=0,$$ (**)

you can see that in ROP the Christoffel symbols vanish and thus the second term gets zero. leaving a proper acceleration and a Lorentz force term. This means the proper acceleration of the charged particle is position dependent through the presence of a four-vector potential in the electromagnetic field tensor. So even along ROP the observer in K measures different proper accelerations for charged particles (with the same charge and rest mass) spread in ROP. This is in agreement with kev's statement about "no absolute motion exists even in flat spacetimes" which I want to modify it as:

"no absolute motion exists for charged particles even in flat spacetimes."

The Lorentz force is a local law, and locally spacetime can always be described by the minkowski metric: spacetime is homogenous and isotropic locally.

Therefore, the Lorentz force alone should not be able to explain deviations from a geodesic due to a particle's own fields , since the spacetime is isotropic and the fields will be as well and thus the proper-force on the particle due to its own fields while at rest in a local inertial frame due to the Lorentz force must be zero.

No! You are making some big mistake as most people do. Yes, the Lorentz force is a local law but until the electromagnetic field does not vanish, as you behold above, the deviation is guaranteed by having a non-constant proper acceleration where the spacetime is flat. (Consider the uncharged particles case. They would follow geodesic in ROP with vanishing proper accelerations but now for charged particles -again with the same mass and charge-the situation is so squalid through dependency of their acceleration on position. Now which one would follow geodesic along ROP?) No spacetime can be made locally flat if by "locally" we mean anything but a point and we discussed this above. Due to this reason, the definition of a "geodesic coordinates system" by which any Christoffel symbol and consequently Riemann tensor would be made zero is always valid at only one given point $$P.$$ If you are speaking non-rigorously, then around the point $$P$$ we can make the spacetime approximately flat and thus retrieving the poor definition of EP. I don't know how you can through this poor definition make the electromagnetic field vanish, but I hope I've explained everything clearly.

Think of it this way. Consider an observer in the same local rest inertial frame as the charged particle. To the observer, the particle will just sit there, and then once they enter curved spacetime, the particle will start to move away

If you ignore the electromagnetic field, then the charge wouldn't affect the situation so let alone the charged particle and stick to an uncharged one. But if you take into account the electromagnetic field, then the charged particle cannot be at rest and before entering a curved spacetime it is already moving. So the reasoning isn't logical nor is true.

Also, the derivation of equations of motion is completely independent of curvature of spacetime and can be obtained only through the Lagrange-Euler equations (if not use some similar but boring way of Christoffel symbols.) If the perturbation of, for example, spin is taken into account, then the Riemann tensor plays a role. But as you can see, the motion of a charged particle under only an electromagnetic field is independent of the Riemann tensor. However, any extra term, I think, would drag the Riemann tensor in the perturbed geodesic equation. I've not read any article about how extra terms due to particle's own fields can enter in the equation of motion.

The only way for the EP to fail, is to find physics that locally feels the local curvature.

The physics always feels locally the local curvature in a very rigorous way if you can understand it from my above discussion.

To make this more productive, let me ask:
Does anyone know how to calculate such deviations?
If so, HOW is the curvature coupling to this?

My own questions.

(*) The stronger condition says that the Christoffel symbols must vanish in S that is because we have spacetimes for which the curvature tensor vanishes but the geodesic equations don't include constant proper accelerations. As an example, look at the http://arxiv.org/abs/physics/0601179" .

(**) For simplicity, take the electromagnetic field be only position-dependent so the scalar field is supposed to vanish.

AB