The transformation of electric and magnetic fields under a Lorentz boost we established even before Einstein developed the theory of relativity. We know that E-fields can transform into B-fields and vice versa. For example, a point charge at rest gives an Electric field. If we boost to a frame in which the charge is moving, there is an Electric and a Magnetic field. This means that the E-field cannot be a Lorentz vector. We need to put the Electric and Magnetic fields together into one (tensor) object to properly handle Lorentz transformations and to write our equations in a covariant way.

The simplest way and the correct way to do this is to make
the Electric and Magnetic fields components of a
**rank 2 (antisymmetric) tensor**.

The fields can simply be written in terms of the
**vector potential**, (which is a Lorentz vector)
.

Note that this is automatically antisymmetric under the interchange of the indices.
As before, the
**first two (sourceless) Maxwell equations are automatically satisfied** for fields
derived from a vector potential.
We may write the
**other two Maxwell equations** in terms of the 4-vector
.

Which is why the T-shirt given to every MIT freshman when they take Electricity and Magnetism should say

``... and God said and there was light.''

Of course he or she hadn't yet quantized the theory in that statement.

For some peace of mind, lets
**verify a few terms in the equations**.
Clearly all the diagonal terms in the field tensor are zero by antisymmetry.
Lets take some example off-diagonal terms in the field tensor, checking the (old) definition of the fields in terms of the potential.

Lets also
**check what the Maxwell equation says** for the last row in the tensor.

Jim Branson 2012-10-21