Is magnetic field due to an electric current a relativistic effect?

Question

I was reading a paper of the same name by Oleg D. Jefimenko; here is the concerned text:

[...] relativistic force transformation equations demand the presence of an electric field when the interactions between electric charges are assumed to be entirely due to a magnetic force. We could interpret this result as the evidence that the electric field is a relativistic effect. But the well known fact that similar calculations demand the presence of a magnetic field, if the interactions between the charges are assumed to be entirely due to an electric force, makes such an interpretation impossible (unless we are willing to classify both the magnetic and the electric field as relativistic effects, which is absurd).

Both fields—the electric field and the magnetic field are necessary to make interactions between electric charges relativistically correct.By inference then, any force field compatible with the relativity theory must have an electric-like ‘subfield’ and a magnetic-like ‘subfield’.

I, being a total novice, have no expertise in special relativity. But I read that magnetism is really due to Coulomb's Law.

This is taken from Chris White's answer here:

[...] magnetism is nothing more than electrostatics combined with special relativity.

And this is taken from the horse's mouth; from Purcell's Electricity and Magnetism:

[...] the magnetic interaction of electric currents can be recognized as an inevitable corollary to Coulomb's Law. If the postulates of relativity are valid, if electric charge is invariant, and if Coulomb's law holds, then the effects we commonly call "magnetic" are bound to occur.

These two statements speak volumes to the fact that magnetic field is due to electrostatics and relativity; which ought make it a relativistic effect, isn't it?

then why did Jefimenko tell otherwise?

He did deduce in that paper the electric field is due to the magnetic interaction combined with relativity!

Prior to the reading of that paper, I bore in mind that if magnetic interaction occurs due to Coulomb's Law, then electric field is more fundamental.

But, this is completely contrary to what Jefimenko said; if I consider the interaction between charges to be magnetic, then the relativistic force transformation equation makes sure the existence of electric fields; so magnetic field is more fundamental?

My questions are:

$\bullet$ Is Jefimenko contradicting the statement of Purcell? What did he actually want to say?

$\bullet$ What should be the actual interpretation if it is wrong to assert that 'magnetism is nothing more than electrostatics combined with special relativity'?

Answer 1

I don't think Jefimenko is unclear in the slightest. You have some choices:

1. given magnetic forces first, add SR and deduce electric forces, or
2. given electric forces first, add SR and deduce magnetic forces, or
3. learn SR first and expect to have electromagnetic forces from the get go then add the Lorentz Force.

I think it's wishful thinking to imagine you could describe anything at all as blah+SR and gain understanding while also not learning SR.

So first lets start by understanding relativistic kinematics and dynamics and then lets look at Maxwell from the viewpoint where we understand SR.

OK. So you have a 4d space-time, made up of events. You have curves which are functions from intervals of $\mathbb R$ into space-time. And there might be many parametrizations.

For instance the function from $\mathbb R$ into $\mathbb R\times \mathbb R^3$ that sends $t$ to $(tc,\vec 0)$ is one parametrization. And the function from $\mathbb R$ into $\mathbb R\times \mathbb R^3$ that sends $t$ to $(5tc,\vec 0)$ is another parametrization of the same path. It still goes along the $t$ axis but now a unit distance in $\mathbb R$ does not correspond to a unit proper time along the path.

Also not that the curve has a tangent everywhere along it and that the tangent is time-like (and future pointing). In general, when a curve corresponds to the motion of an object you can consider the instantaneously co-moving inertial frame to be the global inertial frame in SR that corresponds to the tangent of the curve at that event.

In that frame the object is instantaneously at rest. And you can consider parametrizing the curve so the unit tangent points in the direction of the energy momentum four vector, and thus we can scale the unit tangent by the (rest) mass of the object and get the energy momentum four vector.

So now if we have the curve and the mass, we effectively have the energy-momentum too.

So really, now we can look at what forces do. The unit tangent parametrized by proper time now determines the kinematics. Because we effectively know which direction in space-time it is going by knowing the tangent. The unit tangent gives the direction in space-time.

And so if the object always has an instantaneously co-moving inertial frame, then that tangent is always pointing towards the future and is a time-like vector. And if it is the unit tangent (or the rest mass times that) then really it is a future pointing unit tangent vector turning into another future pointing unit tangent vector.

So really all forces could possibly do is rotate a future pointing unit tangent vector into another future pointing unit tangent vector, at a certain rate as given either by an inertial frame, or at a certain rate per proper time along the curve.

So now describing changing kinematics, with rate of change means giving a rotation as per time rate that tangent vector rotates. And there are different independent rotations in 4d, three boosts and three rotations. Everyone agrees on the need for six generators, though what one frame considers a pure boost would not be a pure boost for a different frame.

So we now expect all relativistic forces to be described by a rank two tensor. And that is just how the force describes a rotation of the unit tangent.

The unit tangent is the kinematics, and the rotations reflect that a unit tangent turning into a slightly different unit tangent is a rotation of the first tangent.

Now we can get to Jefimenko. You could start with saying how your magnetic forces act in a frame, so you would describe some rotations for tangents that don't start in one fixed direction and then find out how those work.

Or you could start with tangents that start in that direction (electric forces) and figure out the others.

Why you'd want to start with just one and get the other is unknown. Historical reasons could be justified but otherwise it would be better to sit down and learn SR and note that forces should rotate unit tangents.

Answer 2

This is indeed an interesting question. Let me share a couple of considerations that have not been mentioned. The consensus in the community seems to be that electric and magnetic fields are on equal footing, and that one is as much a product of the other as the other way around. This is exactly the position Jefimenko takes in the article you cite.

I do, however, believe that there is a blind spot in Jefimenko’s argumentation. He uses a special case to come to a general conclusion. Situations where we have a purely magnetic field is the product of a many body system where positive and negative charges cancel each other’s electric field. Such a frame does not always exist. Take electromagnetic radiation (EMR) as an example. In the rest frame of a test charge the force of the radiation is purely electric, and in all other frames of reference the force is a combination of magnetic and electric. The same can be said about the field produced by single charge. No frame exists where the force of the charge is purely magnetic. So, it is not generally true that the electric field can be seen as a relativistic effect (a product of the magnetic force) as Jefimenko states. I believe this erodes the foundation of his argument.

Jefimenko later states “Both fields—the electric field and the magnetic field are necessary to make interactions between electric charges relativistically correct.” The following argument speaks against this.

In the rest frame of a charge only electric forces are present. If the electric force is evaluated in the rest frame, the combined electric and magnetic force in any other frame can be obtained by relativistic force transformation. If we decide to always follow this approach, it is possible to correctly predict all acting forces, using only the electric laws and special relativity. In the case of EMR and single charges, no frame exists where the force is solely magnetic. It is, therefore, not possible to predict all acting forces using only the magnetic laws. If an element can be removed, without affecting the prediction power of the theory, this element can be viewed as non-essential to the theory. Neither the electric field or relativistic transformations can be left out and are thus essential, while the magnetic laws are not.

This method is of course a nightmare from a computational standpoint, but there is no law in the universe telling us that simple is correct and complex is not.

In other posts on this forum people argue that the magnetic field does not disappear just because the test charge is at rest and doesn’t interact with the field. This is indeed true from a mathematical standpoint. But what is the meaning of a field that you cannot interact with? We cannot directly measure fields, we can only measure forces and then infer the existence of a field. In that sense the field is a theoretical construction. So yes, the magnetic field is still present when the test charge is at rest, but is could simply be viewed as a handy mathematical tool ready to tell us what the force on a second test charge in motion will be, not necessarily a representation of something real and physical.

Disclamer: Some of the arguments above has not yet been tested in peer review.