Magnetic field = electric field in some reference frame?

[rumborak]

This Wikipedia article

https://en.m.wikipedia.org/wiki/Relativistic_electromagnetism

seems (to me) imply that there is always a frame of reference in which a magnetic field can be rather viewed as an electric field modified by relativistic considerations. Is that always true? That is, disregarding the inconvenience of doing so, could one entirely remove the concept of the magnetic field and only describe it as electric field + relativity?

 

[Orodruin]

Is that always true?

No. You can construct two Lorentz invariants from the electromagnetic field. One of them is $\vec E^2 - c^2 \vec B^2$ and if this is negative the magnetic field will be non zero in all frames.

 

[jartsa]

This Wikipedia article

https://en.m.wikipedia.org/wiki/Relativistic_electromagnetism

seems (to me) imply that there is always a frame of reference in which a magnetic field can be rather viewed as an electric field modified by relativistic considerations. Is that always true? That is, disregarding the inconvenience of doing so, could one entirely remove the concept of the magnetic field and only describe it as electric field + relativity?

1: Some magnetic fields can be viewed as electric fields modified by relativistic considerations, in any frame except one, the one where the modifications by relativistic considerations are zero.

2: Some other magnetic fields can be viewed as magnetic fields in all frames. In other words these magnetic fields can be viewed as an electric fields modified by relativistic considerations, and the relativistic considerations do not become zero in any frame.

 

[jartsa]

No. You can construct two Lorentz invariants from the electromagnetic field. One of them is $\vec E^2 - c^2 \vec B^2$ and if this is negative the magnetic field will be non zero in all frames.

In all frames a moving charge detects a magnetic field = In all frames a moving charge feels electric fields that are modified by relativistic considerations

 

[Orodruin]

In all frames a moving charge detects a magnetic field = In all frames a moving charge feels electric fields that are modified by relativistic considerations

No, this is incorrect. It is not even clear what you are trying to say. There are perfectly valid examples of EM fields that cannot be transformed to pure electric or pure magnetic fields in any frame. Any field configuration for which $\vec E\cdot \vec B \neq 0$ will have this property.

 

[rumborak]

Awesome answers, thanks everybody. I somewhat suspected that if one was able to reduce EM to either just E or B, this would have been pointed out more in literature, but the wording in the Wikipedia page seemed suggestive.

 

[jartsa]

No, this is incorrect. It is not even clear what you are trying to say. There are perfectly valid examples of EM fields that cannot be transformed to pure electric or pure magnetic fields in any frame. Any field configuration for which $\vec E\cdot \vec B \neq 0$ will have this property.

Any occurrence of "magnetic field" can be replaced with "electric fields that are modified by relativistic considerations". That is the idea. If all sentences make sense after that translation, then we could forget "magnetic fields" if we wanted.

Let us try, here is a paragraph:

A current loop produces a magnetic field. A device that measures magnetic fields shows that there is a magnetic field. No matter how the device is moved, the magnetic field stays a magnetic field. The magnetic field can not be transformed away.And here is a "translation":

A current loop produces electric fields that are modified by relativistic considerations. A device that measures electric fields that are modified by relativistic considerations shows that there are electric fields that are modified by relativistic considerations. No matter how the device is moved, the electric fields that are modified by relativistic considerations stay electric fields that are modified by relativistic considerations. The electric fields that are modified by relativistic considerations can not be transformed away.Well that sounds terrible, but not wrong. Except if a magnetometer is absolutely not a device that measures electric fields that are modified by relativity.

 

[Orodruin]

Any occurrence of "magnetic field" can be replaced with "electric fields that are modified by relativistic considerations". That is the idea

That idea is wrong. See above.

 

[rumborak]

I mean, mathematically I guess it also makes sense that you can't reduce it. If you can describe it as a four-dimensional tensor, that sort of implies that it can not be reduced to lower dimensions, correct? Or could you actually have "redundancy" in a tensor?

 

[jartsa]

That idea is wrong. See above.

So, if I have a stream of electrons, the stream has a magnetic field, that can be transformed away, so the field can be said to be some electric fields with some relativistic effects.

And if I have another stream of electrons, that stream has a magnetic field, that can be transformed away, so the field can be said to be some electric fields with some relativistic effects.

But if I cross the two streams, there is a magnetic field that can not be transformed away.

And now I am supposed to be convinced that the field of crossed beams can not be said to be some electric fields with relativistic effects?

(Note: I'm not convinced at all)

 

[Orodruin]

And now I am supposed to be convinced that the field of crossed beams can not be said to be some electric fields with relativistic effects?

Yes. Because there is no frame in which the EM field is purely electric. This is a mathematical fact.

I could just as well consider a fundamental magnetic dipole (the electron is a magnetic dipole so classically the magnetic field actually dominates the electron's EM field at short distances).