Understanding the STR Paradox of Electron's Rotating Magnetic Field

[Lasand]

A paradox?...In STR.

A thought experiment;

This could take place in a hallway or in space, whichever you prefer.

Consider an electron somewhere around the center of your monitor. Now have an observer A at the right of your screen closing the distance between the electron and himself in a uniform manner. The STR allows this observer to claim to be at rest and that the electron is moving toward him and emitting a rotating magnetic field in the correct manner if he could see it.

At the same time have observer B coming from the left of your monitor closing the distance between the electron and himself. Observer B is moving in a uniform manner. Observer B can also claim to be at rest and that the electron is moving toward him emitting a rotating magnetic field in the correct manner. The electron would have to emit two fields rotating in opposite directions to satisfy each observer.

Because the electron has a rotating magnetic field, does it have to be placed in a preferred reference frame?

How does the electron know which way it is moving in order to emit the proper field.

Does the electron have to be accelerating in order to produce a rotating magnetic field?

I sent this "paradox' to two physicists. They didn't reply so something must be wrong with the way I phrased it. It was unsolicited so was probably trashed. Any ideas?

 

[JesseM]

I'm confused, why should an electron emit a rotating magnetic field, and why should the two observers see it rotating in opposite directions?

 

[Dale]

Hi Lasand,

There is no paradox here. If you understand how SR unified the separate ideas of space and time into a unified concept of spacetime then you will find a similar thing here. The electric and magnetic fields are seen not as separate things, but rather as the components of the unified concept of the electromagnetic field. Different reference frames disagree on the electric and magnetic fields, but all agree on measurable things like proper accelerations of charged particles.

 

[Lasand]

Hello JesseM;

I'm still a bit confused also. It is understood that a moving charge has a magnetic field rotating around it. That accounts for the magnetic field around a current carrying conductor. There can be not doubt about this, that the field around a wire is an ensemle of moving charges that can deflect a compass needle. I still speculate about an alternative.

 

[Lasand]

Hello DaleSpam;

I know that is how the moving magnet and coil problem was resolved. The unified field could be detected differently depending on if the detector was co-moving with the source or in relative motion to it.

It's the rotation of the field, and it's direction of rotation to different observers that puzzles me. I seem to end up having to give the electron absolute motion.

Certainly, when it comes to either a flaw in the Special Theory or my thought experiment, my thought experiment will have to be the one that is flawed.

I must be switching reference frames when I'm not supposed to. Thanks for the input. I need to ponder this somemore.

 

[Lasand]

Maybe JesseM does have a point.

Just because electrons moving in a conductor do produce a rotating magnetic field, maybe we should'nt assume a free moving electron does the same thing. Lorentz thought so, but he thought that a moving charge induced something in the aether.

Consider a strong magnetic field between the poles of a magnet. Place a conductor in the field at a right angle to the field. Send an electric current thru the conductor and the Lorentz force will kick the conductor out of the field. Send a beam of electrons into the field from the same direction and the electrons won't be kicked out but will circle the field. Something different is happening.

Maybe a physicist will stop by and answer the part about if a free moving electron has a rotating magnetic field around it.

 

[Vanadium 50]

Let's back up a bit. Where exactly does SR come into this?

Let's pretend the universe is Newtonian for a moment. We have two frames where the electron direction is different. In these same two frames the magnetic field direction is also different. Why is the second statement more problematic than the first? Why is the first statement problematic? And how does SR allegedly chage things?

 

[Dale]

It's the rotation of the field, and it's direction of rotation to different observers that puzzles me. I seem to end up having to give the electron absolute motion.

Have you actually worked this out numerically? You should wind up with the same force on the test charge regardless of the field direction. If not, please post your work and we will be glad to help.

 

[Lasand]

Hello Vanadium 50;

Good idea. Let's back up a bit. If a neutron was under consideration could'nt both observers claim the neutron was coming toward them? I thought the rotation of the field of the electron would add an interesting twist to the thought experiment.

I tried to not give a particular motion to the electron. I must have set up three reference frames, one for each observer, and one for the electron. I'll have to put the electron in the frame of observer A and give it a direction of motion. Then observer B could tell by the field of the electron that the electron was moving in the same direction, but that he was overtaking it.

I'll have to clear my head and rethink this, it just seems to give one frame preference. Do all objects have to be stationary in frames when comparing them? It wasn't my intention to have the electron moving in two directions in two frames. How can I word the thought experiment so that people don't get the impression that the electron is moving in two directions in two frames?

I thought the paradox was, how could one and only one electron have two directions and two fields.

Evidently one can only consider one frame at a time. The theory is easier to understand when trains on parallel tracks are considered, or two rockets are coming toward each other. When three objects in line are considered, it almost requires one object to be given a preferred motion.

Thanks for the input, anything to help me focus.

 

[Al68]

Hello JesseM;

I'm still a bit confused also. It is understood that a moving charge has a magnetic field rotating around it. That accounts for the magnetic field around a current carrying conductor. There can be not doubt about this, that the field around a wire is an ensemle of moving charges that can deflect a compass needle. I still speculate about an alternative.

Are you referring to the normal magnetic field produced by constant current flow? The magnetic field "lines" go around the conductor, but the field is static. In those little pictures with an arrow on the circular magnetic field lines, the arrows indicate the polarity of the magnetic field, not rotation. There is no mass there to "rotate".

If the current is constant, the magnetic field will not move around at all, it will remain "stationary". If the current is varying, the magnetic field may get bigger or smaller, or change polarity (direction). Nothing "rotates".

 

[Lasand]

Thanks for your reply Al68;

Doing this kind of thing helps me focus my concentration. I now think I see the flaw in the thought experiment. Since the Special Theory of Relativity is backgroud dependent the state of the electron is indeterminate while outside of your local inertial frame of reference. The electron would have to be brought into your local frame to determine it's motion against your background. The same should hold for the other observer. So maybe one can't say much about the electron between the two reference frames.

This brings up some interesting philosophical questions. How big can a 'local" inertial frame be? Can there really be two frames? Can't they both be placed in a larger frame?

 

[JesseM]

Doing this kind of thing helps me focus my concentration. I now think I see the flaw in the thought experiment. Since the Special Theory of Relativity is backgroud dependent the state of the electron is indeterminate while outside of your local inertial frame of reference. The electron would have to be brought into your local frame to determine it's motion against your background. The same should hold for the other observer. So maybe one can't say much about the electron between the two reference frames.

No, that's definitely wrong, inertial frames in SR are global coordinate systems, covering every point in the entire (infinite) spacetime. It's only in GR that inertial frames can only be defined locally.

Anyway, Al68's comment wasn't about locality, it was about the fact that the magnetic field of a constant current moving down a wire doesn't "rotate", the magnetic field vectors at every point are constant, although they are all tangent to a circle around the wire. http://www.britannica.com/EBchecked/topic-art/357048/1318/Magnetic-field-of-a-long-wire has a good diagram of the magnetic field vectors around a wire:

 

[Vanadium 50]

I thought the paradox was, how could one and only one electron have two directions and two fields.

What's the problem with it having two directions? This happens in Newtonian physics all the time.

Both observers see the electron moving towards them.
Both see an electric current consistent with the electron's motion as they observe it.
Both see a magnetic field consistent with the current as they observe it.

Where is the problem? (And again, the above statements are true even without relativity)

BTW, I also agree that you shouldn't use "rotating magnetic field" to describe the field - that implies a time-varying field, and this field is static. See JesseM's pictures.

 

[yuiop]

Einstein discusses this issue in the very first paragraph of his famous 1905 paper http://www.fourmilab.ch/etexts/einstein/specrel/www/" .
Also see http://www.phys.vt.edu/~demo/demos/q90Notes.html for how relativity can be demonstrated in a school lab.

 

[yuiop]

Thanks for your reply Al68;

Doing this kind of thing helps me focus my concentration. I now think I see the flaw in the thought experiment. Since the Special Theory of Relativity is backgroud dependent the state of the electron is indeterminate while outside of your local inertial frame of reference. The electron would have to be brought into your local frame to determine it's motion against your background. The same should hold for the other observer. So maybe one can't say much about the electron between the two reference frames.

This brings up some interesting philosophical questions. How big can a 'local" inertial frame be? Can there really be two frames? Can't they both be placed in a larger frame?

I think most experts would say SR is not background dependent. Pure SR is normally introduced in the context of an ideal universe that is flat all the way to infinity. In that context, if someone walks past you, their inertial RF consists of all objects in the universe that are moving at the same speed and direction as they are, while your your inertial RF consists of all objects (however far away they are) that are at rest with respect to you. Normally a RF is illustrated as an imaginary array of an infinite number of clocks, rulers and observers that are all at rest with respect to each other. The concept of a local inertial RF is only required in a practical universe that is curved by gravity and accelerating expansion, so that a small enough region has to be considered, that looks flat enough to aproximate the ideal SR conditions of perfectly flat space. In ideal flat space, two bodies are not in the same inertial RF, if they are moving relative to each other and it has nothing to do with how far apart they are. Hope that makes some sort of sense, despite the informal description.

 

[Hurkyl]

I think most experts would say SR is not background dependent.

SR specifies a priori both the topology and the geometry of space-time -- I don't know how you could get any more background dependent than that!¹ But... I suppose we can wait for an expert to chime in with what the phrase 'background dependence' means.


1. Okay, I fibbed. One could further assert that things like location in space-time are actually meaningful. (SR does not assert that -- it is content with absolute location being meaningless, but relative position being meaningful)

 

[Steely Dan]

It seems to me like the very nature of relativity indicates that there is no such thing as an "absolute" reference frame, so it is meaningless to ask which frame is correct; they both make sense from their reference frames. The electron's magnetic field could only be pointing in different directions if there were some 'absolute' frame from which an external observer were watching, but by SR that doesn't exist. At least, that's how I see this.

 

[yuiop]

SR specifies a priori both the topology and the geometry of space-time -- I don't know how you could get any more background dependent than that!¹ But... I suppose we can wait for an expert to chime in with what the phrase 'background dependence' means.

See http://arxiv.org/abs/hep-th/0507235"

"This leads to a careful statement of what physicists mean when we speak of background independence. Given this we can characterize the precise sense in which general relativity is a background independent theory."

Also http://en.wikipedia.org/wiki/Background-independent"

"What Einstein discovered was that physical entities are located with respect to one another only and not with respect to the spacetime manifold. This is what background independence is!"

I can find many more quotes like it. Just about any discussion of quantum gravity mentions that GR is background independent. I assume if GR is background independent then SR is too. But then again... maybe not... Might be worth a thread of its own.

 

[Hurkyl]

I assume if GR is background independent then SR is too. But then again... maybe not... Might be worth a thread of its own.

That's a bad assumption. It's a very big deal that SR specifies geometry a priori, while GR does not, and that quality is what I have most often noticed being described by background (in)dependence.

Incidentally, your Smolin link even points out that GR is not fully background independent:

It is true that in general relativity the dimension, topology, differential structure and
signature are fixed. They can be varied from model to model, but they are arbitrary and
not subject to law. These do constitute a background.​

(I was reading that paper too before my previous post; I was disappointed he didn't give an explicit definition of a 'background' or of 'background (in)dependence')

 

[yuiop]

That's a bad assumption. It's a very big deal that SR specifies geometry a priori, while GR does not, and that quality is what I have most often noticed being described by background (in)dependence...

(I was reading that paper too before my previous post; I was disappointed he didn't give an explicit definition of a 'background' or of 'background (in)dependence')

I read the paper some more myself and tend to agree. I assumed in SR that particles only have motion relative to other particles and sinse there is no absolute reference frame I assumed that was the same thing as having no background, but that would seem to not be the formal meaning of the word in this context. In GR the geometry (of the background? spacetime?) alters as massive bodies move through it, so it certainly not a fixed background. I too would like to find out more what exactly is meant by background independent, but that is perhaps something for another thread, as the OP only mentioned it in passing.

 

[Lasand]

JesseM :

Thanks for the info. It doesn't look like this thing is going to rise to the status of the Twin Paradox or the Pole-Barn Paradox.

It is going to be difficult for me to get "rotation" out of my head and replace it with vectors, but it seems to be required.

kev;

If this leads to a topic on backgrounds, I will follow along with interest.

Thanks to Hurkyl, Steely Dan, and Vanadium 50 for opinions and information.

Next for me will be to ponder" privately, not think out loud on a forum" how the field of the electron changes as the electron goes from uniform motion to being accelerated. Acceleration should give an "absolute" character to the motion and direction of motion.

 

[Chrisc]

That's a bad assumption. It's a very big deal that SR specifies geometry a priori, while GR does not, and that quality is what I have most often noticed being described by background (in)dependence.

Incidentally, your Smolin link even points out that GR is not fully background independent:

It is true that in general relativity the dimension, topology, differential structure and
signature are fixed. They can be varied from model to model, but they are arbitrary and
not subject to law. These do constitute a background.​

(I was reading that paper too before my previous post; I was disappointed he didn't give an explicit definition of a 'background' or of 'background (in)dependence')

I'm no expert, but as I understand it, a common confusion over the background independence of relational theories is the idea that if a theory "has" background independence it "is" a background independent theory.
This is not quite right. (Unless of course I am as confused as I think others are)
It is impossible for a theory to be fully background independent and still have physical relevance to observation.
A relational theory can be independent of any absolute, specific, a-priori background, but dependent on the background defined by the solution to its relations on any choice of subject.
So GR "has" background independence, but its solutions define backgrounds upon which it depends. (for validity or falsifiability)
Which is why the metric is the field...or the background is the solution.

 

[eep]

JesseM :
Next for me will be to ponder" privately, not think out loud on a forum" how the field of the electron changes as the electron goes from uniform motion to being accelerated. Acceleration should give an "absolute" character to the motion and direction of motion.

I recommend pondering with pen and paper if you really want to understand it

 

[Lasand]

Hello all;

Perhaps it is time to let the " paradox" thing drop. If anyone wants to continue background discussions here, it is OK with me.

I read that a major obstacle to unification was that QM and STR are background dependent, while GTR with it's covariance is background independent.

I did bring up STR so backgrounds can be discussed relating to that.