Is the equivalence principle good for anything?

[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?

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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.

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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