Covariant electrodynamics

Using the machinery that we have defined in the previous sections, we will now start by defining the charge-current density \[\begin{equation} \require{physics} (j^\mu(x)) = % \begin{pmatrix} c \rho(x) \\ \vb{j}(x) \end{pmatrix} \thinspace , \end{equation}\] which is a Lorentz \(4\)-vector. The charge-current density \(j^\mu(x)\) can be used to conveniently state the continuity equation (cfr. equation \(\eqref{eq:maxwell_continuity}\)) in \(4\)-vector form as: \[\begin{equation} \partial_\mu j^\mu = 0 % \thinspace . \end{equation}\]

In addition to the \(4\)-charge current density, we will also introduce the electromagnetic gauge field \(4\)-vector as \[\begin{equation} (A^\mu(x)) = % \begin{pmatrix} \phi(x)/c \\ \vb{A}(x) \end{pmatrix} \thinspace , \end{equation}\] such that the inhomogeneous Maxwell equations (cfr. \(\eqref{eq:inhomogeneous_maxwell_charge_density}\) and \(\eqref{eq:inhomogeneous_maxwell_current_density}\)) can be elegantly rewritten as \[\begin{equation} \square A^\mu = \mu_0 j^\mu \end{equation}\] in Lorenz gauge.

General gauge transformations can now be written as \[\begin{equation} {A'}^\mu = A^\mu - \partial^\mu \chi % \thinspace , \end{equation}\] in which \(\chi\) is a sufficiently smooth gauge function and the Lorenz gauge condition is conveniently written as \[\begin{equation} \partial_\mu A^\mu = 0 % \thinspace . \end{equation}\]

Let us also introduce the electromagnetic field strength tensor: \[\begin{equation} (F^{\mu \nu}) = (\partial^\mu A^\nu - \partial^\nu A^\mu) % = % \begin{pmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \\ \end{pmatrix} \thinspace , \end{equation}\] which is an antisymmetric Lorentz contravariant tensor of rank 2. It can also be shown that it is a gauge invariant tensor: \[\begin{equation} \label{eq:field_strength_tensor_gauge_invariant} F^{'\mu \nu} = F^{\mu \nu} \end{equation}\] and the covariant components are obtained by changing the sign of the components of the electric field: \[\begin{equation} (F_{\mu \nu}) = % \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y & B_z & 0 & -B_x \\ -E_z & -B_y & B_x & 0 \\ \end{pmatrix} \thinspace . \end{equation}\] The components of the electric field can be derived from the electromagnetic field tensor as: \[\begin{equation} E_i = -F^{0i} \end{equation}\] and the magnetic field components as: \[\begin{equation} B_i = \frac{1}{2} \epsilon_{ijk} F^{jk} % \thinspace . \end{equation}\]

Using the electromagnetic field strength tensor, we can write the inhomogeneous Maxwell equations in covariant form (in Lorenz gauge) as \[\begin{equation} \label{eq:inhomogeneous_maxwell_field_tensor} \partial_\mu F^{\mu \nu} = \mu_0 j^{\nu} % \thinspace , \end{equation}\] whose \(\mu = 0\) component leads to the inhomogeneous Maxwell equation for the scalar potential \(\eqref{eq:inhomogeneous_maxwell_charge_density}\) and the other components lead to the inhomogeneous Maxwell equation for the vector potential \(\eqref{eq:inhomogeneous_maxwell_current_density}\).

The dual field strength tensor is now defined by \[\begin{equation} \tilde{F}^{\mu \nu} % = \frac{1}{2} \epsilon^{\mu \nu \sigma \tau} F_{\sigma \tau} % = \epsilon^{\mu \nu \sigma \tau} \partial_{\sigma} A_{\tau} = \begin{pmatrix} 0 & -B_x & -B_y & -B_z \\ B_x & 0 & E_z & -E_y \\ B_y & -E_z & 0 & E_x \\ B_z & E_y & -E_x & 0 \\ \end{pmatrix} \thinspace , \end{equation}\] in which we clearly see that the duality transformation \[\begin{equation} \vb{E} \rightarrow \vb{B} \qq{and} \vb{B} = -\vb{E} \end{equation}\] has taken place. Let us also list some frequently-used expressions concerning these field strength tensors: \[\begin{align} F_{\mu \nu} F^{\mu \nu} % &= 2 \qty( % \partial_\mu A_\nu \partial^\mu A^\nu % - \partial_\mu A_\nu \partial^\nu A^\mu % ) \\ % &= 2 \qty( \norm{\vb{B}}^2 - \norm{\vb{E}}^2 ) \end{align}\] and \[\begin{equation} F_{\mu \nu} \tilde{F}^{\mu \nu} = - 4 \vb{E} \cdot \vb{B} % \thinspace . \end{equation}\]