Could magnetic fields really be completely substituted by relativity and electric fields?

Question

In many textbooks (especially those for undergraduate level), magnetic fields are described merely as a relativistic side product of electric fields when considering frames in motion relative to moving charges. Everybody knows the argument, so I will not repeat it here.

Is this view correct in terms of "deeper theory"? I learned that on university in the relativistic lectures, but nevertheless we are all talking about magnetic fields as "own entities", obeying Maxwell's equations.

Could we really remove magnetic fields from physics and still get the same laws? For my personal feeling, all we know is, that E and B are related by certain relativistic transformation rules which are intrinsically consistent, but I cannot imagine physics without B at all.

In particular, how could Faraday's law be deduced, which at first assumes no particular relative movement but makes a statement about $\partial \vec B/\partial t$ and $\nabla \times \vec E$ in fixed frames?

Answer 1

The argument that you are describing was originally put forth by Purcell and is described in a simplified fashion here: http://physics.weber.edu/schroeder/mrr/MRRtalk.html

In a nutshell, the argument is that in the rest frame of a charge there is no magnetic force on the charge and so as we transform to another frame what was a purely electric force becomes a combination of an electric force and a magnetic force. Through Coulomb’s law and knowledge of the relativistic transformations for forces we can deduce the magnetic force.

However, note that the above argument is an argument about forces and not fields. The argument does imply that you can consider magnetic forces to be relativistic side effects of electric forces. It does not imply that you can consider magnetic fields to be relativistic side effects of electric fields.

The reason is that at a given point we may have two identical charges with different velocities and therefore different electromagnetic forces. In order to attribute those two different forces at the same point to a local field we need a magnetic field. We can analyze the relativistic forces on a bunch of charges moving in different directions at each point to deduce the transformation laws for the total electromagnetic field, but we cannot remove the magnetic part.

In fact, as we do so we find that the most natural representation of the electromagnetic field is as a unified antisymmetric tensor with 6 independent components. No subset of 3 of these components has any right to claim supremacy. This is also borne out by the fact that the tensor has two invariants, one of which is $E^2-B^2$. If this invariant is negative then there is a frame where $E=0$ but there is no frame where $B=0$

Answer 2

No, this is a common misunderstanding of the relativistic argument.

Consider the three spatial dimensions of height, width, and depth. You describe paintings in terms of their height and width, but one day you discover that if you rotate the painting, some of the height and width can turn into depth. That tells us that rotational symmetry relates the three spatial dimensions.

It does not say that depth isn't real -- the whole point is that the three dimensions are on an equal footing. It does not say that we can always make every object have zero depth -- good luck building a house using that principle. It especially does not mean that invoking depth is a "mistake" due to not correctly considering rotational effects. The only thing it tells us is that if we have something with no depth, we can still rotate it into something with depth.

The exact same holds for electromagnetism; substitute "rotations" for "Lorentz boosts", "height/width" for "electric fields", and "depth" for "magnetic fields".