Is the equivalence principle good for anything?

[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