Why is magnetism a consequence of special theory of relativity

Question

Why is magnetism a consequence of special theory of relativity. Refrain from using mathematics as much as possible and try using physical arguments.

Answer 1

It isn't really true to say that magnetism is a consequence of special relativity, though the two are certainly related. The basis of the claim is that there is no such thing as a magnetic charge.

Anyone who has played with a Van der Graaff generator knows that electric charges cause electric fields. Once you charge the generator you can feel the field as a prickling on your skin. However as far as we know there are no magnetic charges. A magnetic charge would be called a magnetic monopole, and despite decades of searches for magnetic monopoles none have ever been found. But if electric charges cause electric fields, and no magnetic charges exist then how are magnetic fields created? well this is where special relativity comes in. I'll try and explain why this is, and as requested no mathematics will be involved.

Our starting point is the observation that the universe has a symmetry called Lorentz covariance. This is a bit involved, but we don't need the details for this explanation. We just have to accept that it exists. If you've ever heard the statement that the speed of light is constant for all observers this is a consequence of Lorentz covariance. Note that Lorentz covariance is an experimental observation. As far as we know there is no reason why the universe should have this symmetry, it's just that when we do the experiments we find it does.

Special relativity is the theory that results from Lorentz covariance i.e. starting from the fact the universe has this symmetry we end up with the theory of special relativity. So special relativity is the result of Lorentz covariance, and were going to discover that magnetic fields are also the result of this symmetry.

This happens because if we try to create a theory that describes just electric fields we find it is not Lorentz covariant. Likewise if we try to create a theory that describes just magnetic fields it is also not Lorentz covariant. The only way we can get a Lorentz covariant theory is to combine electric and magnetic fields into a single theory of electromagnetic fields. The equations that describe this theory are called Maxwell's equations, and indeed they were discovered before special relativity was.

The key result of the electromagnetic theory is that whether an electric field looks like an electric field, a magnetic field, or a mixture of the two, depends on the observer's velocity. In effect electric and magnetic fields are just different views of the single electromagnetic field. If you're looking at a static electric field generated by an electric charge, for example the charge on a Van der Graaff generator, and I am whizzing past you at some velocity, then to me the field looks like a mixture of an electric and a magnetic field. For me a magnetic field has appeared even though no magnetic charge is present, and even though to you it just looks like an electric field.

The way we calculate what your static electric field looks like to me is by using a Lorentz transformation, and this is what special relativity does. So this is where special relativity comes in. SR tells us how to calculate what the electromagnetic field looks like for different observers. But SR doesn't cause the magnetic field, it just tells how an electromgnetic field can look like electric and magnetic fields to the different observers.

So to summarise, that magnetic fields must exist is a result of a symmetry of the universe called Lorentz covariance, and special relativity allows to calculate what those magnetic fields look like. But magnetism is not a consequence of special relativity. Both magnetism and special relativity arise from the Lorentz covariance.

Answer 2

I might have a low math description.

Consider a particle of charge q stationary in the laboratory, called particle B. And in the lab, lets consider three different states for particle A also having charge q. Two of these states will be stationary, and one will be motion in a state of uniform velocity.

State 1: At some steady position I in the lab for particle A we can use Coulomb's law to calculate the Electric Field. The field depends only on the magnitude of the charges involved, the relative direction between the two particles, and the absolute distance between them.

State 2: Now consider particle A being moved to an arbitrary position II in the lab where it then remains stationary. Again, we have dependence only on charge, relative direction between A and B, and the absolute distance between them.

State 3: Instead of instantly moving, and then holding indefinitely, lets have a particle start at position I and travel continuously to position II at some constant speed in a straight line. The velocity can be broken up into components parallel and perpendicular to the direction of the velocity.

In State 3, the distance between the particles projected onto the component perpendicular to the velocity vector is a constant. We also find the Magnetic Field is 0 parallel to this direction. Why is that?

We know from Maxwell's Equations as well as measurements that the electro magnetic field travels at a finite speed. This happens to be the speed of light, but just understanding that this speed is finite might be sufficient to explain magnetism.

In State I we have no magnetic field, in State II we have no magnetic field. Yet in State III we have a magnetic field felt by particle B when particle A is at either position I or position II. The difference between particle A in the first two states and the particle in State III is that in State III it's moving.

Now what is the electric field at particle B when particle A is at position I vs position II having a non-zero velocity at either of those points? It cannot be the same strength as in the stationary cases. This would require instant communication of the location of particle A to particle B, but information about the electric field travels at a finite speed. Particle B "thinks" particle A is at some earlier point in its trajectory. If any one of these three components changes, we have a different Electric Field. We could maintain the same relative position and charge, but say, have an inverse cube dependence instead of inverse square.

Now, since the particle is moving, the distance between the particles is changing. The change in the distance between the two particles has one time dependence, the change in relative direction which dictates the direction of the electric field has its own functional time dependence. The rates of change are different and information about the changes propagate at a finite speed. Further despite the motion, the distance perpendicular to the velocity remains constant.

These three effects might be what gives rise to the characteristics of magnetism. The field is proportional to velocity, it has a specific symmetry with respect to the vector perpendicular to velocity of the moving charge.

In other words, if we assume an electric field is static, then modify the static field strictly as a parameterized position on a straight line, then factor in finite propagation of the speed of light, we essentially get correction factors that add up to the time-retarded electric and magnetic fields consistent with special relativity.

A thorough description would require integrating the charge density represented as a dirac delta potential factoring in the finite propagation of the field. Then out pops the geometric dependencies more explicitly, but hopefully the above is decent intuitively.