The electric and magnetic fields, E and B, do not transform independently under Lorentz transformations. Instead, they mix with one another, indicating that they are not components of 4-vectors but rather parts of a larger geometric object: the Faraday tensor.
To represent electromagnetism in a covariant way, we define the Faraday tensor, also called the electromagnetic field tensor, as an antisymmetric second-rank tensor:
This object contains all six independent components of the classical electromagnetic field: the three components of E and the three of B. The antisymmetry ensures that Fμν=−Fνμ, as required for it to represent a physical field under Lorentz transformations.
To gain insight into the rotational symmetries and to help write Maxwell’s equations in a compact form, it is useful to define the dual of the Faraday tensor, denoted ⋆Fμν or sometimes Gμν, as
⋆Fμν=21ϵμναβFαβ,
where ϵμναβ is the antisymmetric Levi-Civita symbol. This "Hodge dual" effectively rotates the field tensor within spacetime, exchanging electric and magnetic fields (up to factors of c).
The Hodge dual comes from differential geometry: in four dimensions, the space of 2-forms has dimension 6 (matching the 6 independent components of Fμν). The Hodge dual is a map from this space of 2-forms to itself.
In electromagnetism, Fμν encodes the sources (charges and currents), while ⋆Fμν encodes the field’s structure, e.g., Faraday’s Law and the absence of magnetic monopoles. Writing Maxwell’s equations with both tensors allows for a beautifully symmetric formalism.
Let's now imagine a cloud of charge Q moving at velocity v. The charge density is
ρ=VQ,
and the current density is
J=ρv.
It would be nice to express these quantities in terms of the proper charge density ρo. As such, let's start with the density in the rest frame of the moving charges:
ρ0=V0Q,
where V0 is the rest volume of the cloud. To determine how this is related to ρ, remember that one dimension is Lorentz contracted so that
V=γV0,
and hence
ρ=γρ0.
Therefore,
J=ρ0γv=ρ0u,
where in the last line we have identified the quantity u≡γv as the 3-component "proper velocity". Evidently charge density and current density go together to make a 4-vector, known as the current density 4-vector:
Jμ=ρ0uμ=<cρ,Jx,Jy,Jz>.
The continuity equation,
∇⋅J=−∂t∂ρ
can now be expanded as
xi∂Ji=−c1∂t∂J0=−x0∂J0.
Thus, bringing both terms to one side, it can easily be seen that the continuity equation can be compactly written as
∂xμ∂Jμ=0.
With this new found current density 4-vector, we can now express Maxwell's equations very compactly as
∂xν∂Fμν=μ0Jμ,∂xν∂Gμν=0
where the μ=0 case gives you Gauss's law (left equation) and the no-monopole law (right), and the μ=1,2,3 cases give you the x,y,z components of the Maxwell-Ampere law (left equation) and Faraday's law (right equation).
This can further be compactified by employing the potential formulism, whereby the electric and magnetic fields are expressed in terms of a scalar potential V and a vector potential A:
E=−∇V−∂t∂A,B=∇×A.
As you might have guessed, V and A together constitute a 4-vector:
Aμ=(cV,Ax,Ay,Az)
This 4-potential can be used to construct the Faraday tensor:
Fμν=∂xμ∂Aν−∂xν∂Aμ
where the derivatives are taken with respect to the covariant coordinates xμ.
Using the Lorentz gauge condition, which in relativistic notation becomes:
∂xμ∂Aμ=0,
all four of Maxwell's equations can be compactly written as: