Electric Field in Accelerating Elevator

[jartsa]

According to an inertial observer the electric field of a charged ball sitting on the floor of an accelerating elevator is contracted more than the elevator is contracted. So the inertial observer concludes that an observer inside the elevator will measure that the Coulomb-force from the ball is weaker when the elevator is accelerating compared to when the elevator is not accelerating.

Is the above correct?

Oh yes, the reason that the field is contracted more than the elevator is contracted, according to the inertial observer, is that the floor where the charge is, is the fastest moving part of the contracting elevator, according to that observer.I mean, when the elevator moves at constant speed, everything is contracted the same amount, and inside the elevator all forces seem normal, like in a still standing elevator. So if things contract different amounts inside the elevator - then I would guess that everything does not seem normal inside the elevator.

 

[Dale]

So the inertial observer concludes that an observer inside the elevator will measure that the Coulomb-force from the ball is weaker when the elevator is accelerating compared to when the elevator is not accelerating.

How does the elevator observer make this measurement?

 

[Sorcerer]

I’m an undergrad so take this with a grain of salt, but the observer in the elevator should feel as if s/he is in a gravitational field, so define “normal.” If anyone would feel “abnormal,” it would be the person in the inertial reference frame, since we don’t ever actually experience that (except for the lucky few who get extended experiences in free fall, like astronauts).
Anyway, first of all I don’t think this is a Lorentz invariant situation, so you probably need general relativity to deal with the transformations. Second, one man’s E field is another man’s B field, so, combined with the above, the inertial (FREE FALL observer) and the accelerated observer need not necessarily agree on certain things.This wiki article may be illuminating, pun intended.https://en.m.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field

 

[jartsa]

How does the elevator observer make this measurement?

1: Buy some ordinary E-field measuring device.
2: Put it in an elevator that accelerates at 1 g.
3: Near the Rindler-horizon of that elevator put a ridiculously large charge. Accelerate the charge so that the proper distance between the elevator and the charge stays constant.
4: Now you can measure the E-field

But, I should have asked this question first:

How is the E-field of an accelerating charge contracted?

(I guessed that it's uniformly contracted, like a inertially moving stick. But maybe it's contracted like an accelerating stick: more at the rear, less at the front)

 

[Ibix]

How is the E-field of an accelerating charge contracted?

Google Lienard-Wiechert potentials. I started calculating the E field for your particular case, but it rapidly got beyond my ability to do algebra on the back of an envelope. It isn't simply length contraction, unless a lot of stuff cancels out that I don't see.

Edit: maybe approximately? Don't know.

 

[pervect]

How does the elevator observer make this measurement?

1: Buy some ordinary E-field measuring device.
2: Put it in an elevator that accelerates at 1 g.

If your measurement of the E-field is based on an ordinary E-field meter, you need to consider the effects of time dilation as well as length contractions. As far as I can tell, you have not.

I gather you don't appreciate 4-vector analysis at all, and I'm not willing to try to do a 3-vector analysis of the problem. Plus I'd be concerned about making an error if I did a 4-vector analysis of the problem that you didn't understand, and you'd most likely ignore it anwyay as you wouldn't know if you could trust the answer and wouldn't be able to check it. Therefore I'll just point out the general difficulty, you need to figure out how your ordinary "E-field meter" responds to time dilations, and you haven't considered the issue at all.

I don't think the relativity of simultaneity has any major effect here, but I could be wrong about that. Your general approach (in my past experience) totally ignores any issue related to the relativity of simultaneity, considering only time dilation and length contraction. In this instance, you've totally ignored time dilation, though.

3: Near the Rindler-horizon of that elevator put a ridiculously large charge. Accelerate the charge so that the proper distance between the elevator and the charge stays constant.
4: Now you can measure the E-field

You lost me here - all I can guess is you're interested in the E-field as you approach the Rindler horizon (?). Otherwise I don't see what it has to do with the problem. Once you've defined how you're going to measure the E-field, you can measure it anywhere, well, at least if you know what the E-field is in an inertial frame.

One of the challenging parts of the problem is figuring out the E-field in the inertial frame. I've glanced through a paper on the topic, I don't feel I fully understand the calculation. http://www.hep.princeton.edu/~mcdonald/examples/schott.pdf in particular. Perhaps there are better papers out there, I don't know.

In particular:

However, the potentials found above suffer from a defect apparently first noticed only in
1955 by Bondi and Gold [9], that the corresponding electromagnetic fields do not satisfy
Maxwell’s equations in the plane x+ct= 0. This can be attributed to the creation of the
charged particle at t=−∞
with speed vx=−c and with singular fields and potentials,while the Li enard-Wiechert forms (9)
tacitly assume there is no singular behavior at earlytimes. The defect can be remedied by expressing
the singular behavior in terms of delta functions,

Once one has the fieldsl in an inertial frame, tensor methods using frame fields and a transform to an orthonormal basis of the accelerating oberver make it fairly straightforwards to answer the question about what a standard E-field meter would measure. But I don't believe you use those methods as I previously mentioned.

 

[jartsa]

You lost me here - all I can guess is you're interested in the E-field as you approach the Rindler horizon (?). Otherwise I don't see what it has to do with the problem. Once you've defined how you're going to measure the E-field, you can measure it anywhere, well, at least if you know what the E-field is in an inertial frame.

Everything in my experiment was designed to produce large effects, which can be measured by the "ordinary device" - because I don't know how a very sensitive device should be calibrated to give correct readings when said device is accelerated.

... So , an inertial observer observing my experiment would see the elevator accelerating at approximately 1 g, and the charge accelerating at some large number of gees. After some time he would see that the charge - elevator distance has contracted quite a lot, and he would see that the E-field of the charge has contracted quite a lot.

... And now it seems to me that the field and the distance must contract the same amount - otherwise the E-field measuring device would measure that the field decreases as time passes.

 

[Ibix]

... And now it seems to me that the field and the distance must contract the same amount - otherwise the E-field measuring device would measure that the field decreases as time passes.

But a device that works as an E-field detector in its rest frame does not work as an E-field detector when analysed in any other frame. Rather, it measures some combination of E- and B-field. You can see this by putting an E-field detector and a B-field detector at rest with respect to a charge. The E-field detector will react to the Coulomb field and the B-field detector will read zero. However, an observer sauntering past your lab regards the charge as in motion, so says that there is a non-zero B-field and a different E-field - yet your detectors' outputs must be invariant. Therefore, to this observer, your detectors cannot be regarded as distinct E- and B-field detectors - just because they are in motion.

So your conclusion that the E-field, viewed from an inertial frame, must length contract with time in a naive way doesn't follow from your (correct) observation that the readout of the detector must remain constant.

 

[Dale]

1: Buy some ordinary E-field measuring device.
2: Put it in an elevator that accelerates at 1 g.
3: Near the Rindler-horizon of that elevator put a ridiculously large charge. Accelerate the charge so that the proper distance between the elevator and the charge stays constant.
4: Now you can measure the E-field

Excellent. So to calculate the result you will need to use two reference frames: the non-inertial Rindler frame and the inertial Minkowski frame. Start in the Rindler frame, write the EM field tensor for a pure E field at the detector, and transform to the inertial frame. This will give you the EM field in the inertial frame which will be detected as an E field in the Rindler frame. Then use the Lienard Wiechert fields to calculate the EM tensor at the detector due to the charge. Use the results from these two steps to determine what part of the EM field in the Minkowski frame is detected as an E field in the Rindler frame.

 

[Tomas Vencl]

...
... And now it seems to me that the field and the distance must contract the same amount - otherwise the E-field measuring device would measure that the field decreases as time passes.

When you draw the situation to Rindler chart it seems, that the E-field must contract the same way as the elevator.
To the points "projection of some electric equipotentials" you can put your E-field detectors. Accelerated observer (blue simultaneity line)e.g. at charge position do not see any length changes so at his framework the detectors remain constant.

 

[Tomas Vencl]

When you draw the situation to Rindler chart it seems, that the E-field must contract the same way as the elevator.

... But then...how looks the E-field of accelerated charge for static observer behind Rindler horizon ? Solid body you can not extend behind, because of infinite acceleration at that point. There is something wrong there...

 

[Tomas Vencl]

On the other side, when we describe the situation from static observer, changes of E-field (caused by different position of charge) are expanding at c, and at high accelerations (and speeds) near rindler horizon we need to take it into account.

 

[pervect]

Everything in my experiment was designed to produce large effects, which can be measured by the "ordinary device" - because I don't know how a very sensitive device should be calibrated to give correct readings when said device is accelerated.

... So , an inertial observer observing my experiment would see the elevator accelerating at approximately 1 g, and the charge accelerating at some large number of gees. After some time he would see that the charge - elevator distance has contracted quite a lot, and he would see that the E-field of the charge has contracted quite a lot.

... And now it seems to me that the field and the distance must contract the same amount - otherwise the E-field measuring device would measure that the field decreases as time passes.

Excellent. So to calculate the result you will need to use two reference frames: the non-inertial Rindler frame and the inertial Minkowski frame. Start in the Rindler frame, write the EM field tensor for a pure E field at the detector, and transform to the inertial frame. This will give you the EM field in the inertial frame which will be detected as an E field in the Rindler frame. Then use the Lienard Wiechert fields to calculate the EM tensor at the detector due to the charge. Use the results from these two steps to determine what part of the EM field in the Minkowski frame is detected as an E field in the Rindler frame.

That's the way I'd suggest doing it. I"ve never seen Jartsa use a 4-vector, so I've been assuming that he doesn't understand tensors :(.

It's well known how the E-fields transform, these transformations can even be expressed in terms of 3-vectors, see for instance <<wiki link>>. The short version of how the E-fields transform is that if you have two E-field meters at the same point, the parallel component is unchanged betwwen the two, while the prependicular component gets boosted by a factor of gamma. And of course there are electric fields created by a moving magnetic fields in the complete transform.

I'm suspecting that Jartsa is not transorming the E-fields correctly, but he hasn't described what he's doing clearly enough for me to be sure. I also don't know what he's using for the E-fields in the inertial frame. I tried looking that up - it's quite a mess, apparently, and I'd feel happier if I saw a treatment that considered a charge that didn't accelerate forever, but started accelerating at some finite time, then took the limit as that starting time approached minus infinity.

 

[Dale]

And now it seems to me that the field and the distance must contract the same amount

The problem is that the field does not simply contract. The EM field is a tensor, so it transforms like a tensor, not a length. The E field is the time-space components of the tensor, so even just the E field does not transform like a length.

 

[jartsa]

So, if we inertial observers ask a guy inside an accelerating elevator to draw a field-line picture of the field of a charge somewhere in the elevator frame, then when we from our frame look at field and the picture of the field, we will say that the field is not like the picture?

(I'm talking about a field whose field lines are on a plane aligned with a wall of the elevator and a picture drawn on surface also aligned with the wall )

 

[Dale]

So, if we inertial observers ask a guy inside an accelerating elevator to draw a field-line picture of the field of a charge somewhere in the elevator frame, then when we from our frame look at field and the picture of the field, we will say that the field is not like the picture?

Correct. That is even true without the acceleration. If they are just moving inertially we will disagree.