Is Special Relativity Consistent with Classical Electromagnetism and Mechanics?

[snoopies622]

I thought I understood this matter, but the more I think about it the more confused I am becoming.

Suppose I have a very long wire which has a uniform negative charge per unit length ρ. I hold near it a particle with charge -q. Due to the electrical field around the wire there is a force that pushes the particle directly away from it, and if I release the particle it accelerates radially from the wire and contines on a particular radial trajectory.

Now suppose instead of simply letting go of the particle, I throw it in a direction parallel to the wire with initial velocity v. From the wire's frame of reference there is no magnetic field in the vicinity of the particle, so the component of the particle's acceleration that is perpendicular to the wire should be the same as in the case when it was released with initial velocity zero.

But is that what is seen?

I know that from the point of view of the particle, in addition to the electrical field around the wire there is a current running through it, and this creates a magnetic field which may also influence its radial acceleration (although I'm not even certain of that since in its own reference frame the particle's velocity parallel to the moving wire is always zero). But since the laws of electromagnetism are supposed to be the same and equally valid in every reference frame, can this magnetic field simply be ignored?

With special relativity there are the additional complications of an increased charge density in the wire from the moving particle's point of view, as well as a more massive moving particle (and smaller acceleration per unit force) from the wire's point of view. Maybe time dilation is a factor, too.

I simply don't know what influences are mutually consistent and which ones are mutually redundant. I was under the impression that special relativity was consistent with classical electromagnetism and conflicted with only classical mechanics, but then I have also read that magnetic fields can be ignored if one considers special relativity and the electric fields alone, which seems to imply the opposite.

 

[atyy]

In the frame of the charge moving parallel to the wire, there is B field, but it has no effect on the charge because the charge isn't moving in this frame, as you said.

The Lorentz transformations change E fields into B fields and vice versa. So it's more like if you have an E field and you assume special relativity, then you are forced to invent the concept of a B field. In simple situations with lots of symmetry, it's somewhat possible to see what is going on (like getting on and off trains and reasoning using light signals), but after a while it gets complicated, and it's usually safer to just do brute force calculate with the Lorentz transformations on the E and B fields.

Try Eq 154 - 156 of Woodhouse's http://people.maths.ox.ac.uk/~nwoodh/sr/sr06.pdf .

 

[snoopies622]

Thanks for the reference, atyy.

Actually, applying a Lorentz transformation to the Faraday tensor is where this question started for me. I began with an E-field pointing in the z direction and wanted to see what would happen if I applied a boost in the x-direction. Then I decided to check my results (which are not with me at the moment) to see if they matched what I "should have" gotten doing it the old fashioned way. That's when I began to realize that I'm not even clear on what the old fashioned way is.

It now seems to me that classical EM by itself cannot be completely right since it doesn't take into account the velocity-addition-formula/velocity-dependent-mass issue. That is, a charged particle in a uniform electric field won't accelerate at a constant rate (at least from the perspective of an inertial observer) and classical EM doesn't have a magnetic field appear in such a case in order to account for this. Of course, this example involves dynamics as well as the electromagnetic field, but then is it even possible to measure an EM field without mechanics somehow coming into play?

I don't know. I'll have to think about this some more..

 

[atyy]

My understanding is that Maxwell's equations are perfectly fine, but the Lorentz force law needs to be modified.

So the old fashioned way would be
(i) find an inertial frame
(ii) apply Maxwell's equations and the relativistic Lorentz force law
(iii) all experimental results will be the same as if you applied step (ii) in another inertial frame.

The tensor notation does exactly the same thing, just organized into matrices (or something like that).

 

[Naty1]

Coulomb's law must be modified to reflect effects of relativity: it is precisely accurate for stationary charges only.

"I simply don't know what influences are mutually consistent and which ones are mutually redundant."

Relativity suggests mutually inconsistent, but equally correct, "influences"...what is observed is dependent on one's frame of reference...

 

[snoopies622]

Coulomb's law must be modified to reflect effects of relativity

Can you elaborate? Is there a relativistic form of Coulomb's law? Are you referring simply to classical magnetism caused by moving charges?

 

[Born2bwire]

Can you elaborate? Is there a relativistic form of Coulomb's law? Are you referring simply to classical magnetism caused by moving charges?

Well the first step is to use retarded potentials.

 

[snoopies622]

Wikipedia gives the relativistic version of the Lorentz force as

$$\frac {dp_a}{d \tau} = q F _{ab} \frac {dx ^b}{dt}$$

I guess using this, the Faraday tensor and the Lorentz transformation matrix gives me all I really need.

Thanks, everyone.

Edit: the LaTeX isn't working, so here's the reference:

http://en.wikipedia.org/wiki/Formulation_of_Maxwell's_equations_in_special_relativity

It's about half way down the page under the heading, "Lorentz force".