Chapter 4: Differentation

### Maxwell's Equations III

A remarkable simplification occurs if we rewrite Maxwell's equations using differential forms in 4-dimensional Minkowski space, as we now show. We will assume that the orientation is given by \begin{equation} \omega = dx\wedge dy\wedge dz\wedge dt \end{equation}

We first need to introduce some temporary notation, so that we can consider both 3- and 4-dimensional quantities in the same computation. We will therefore add a “hat” to all 3-dimensional quantities. For 1-forms, such as $\hat{B}$ and $\hat{E}$, the hat indicates the absence of a $dt$-component, not normalization to $1$. We also have the hatted operators, $\hat{d}$ and $\hat{*}$, which ignore $t$.

It is straightforward to establish the following identities. \begin{align} df &= \hat{d}f + \dot{f}\,dt \\ d\hat{E} &= \hat{d}\hat{E} + dt\wedge\dot{\hat{E}} \\ d\hat{*}\hat{E} &= \hat{d}\hat{*}\hat{E} + dt\wedge\hat{*}\dot{\hat{E}} \\ {*}\hat{E} &= \hat{*}\hat{E}\wedge dt \\ {*}(f\,dt) &= \hat{*}f \end{align}

We can combine the 3 degrees of freedom of $\hat{E}$ with the 3 degrees of freedom of $\hat{B}$ into the 6 degrees of freedom in a (4-dimensional) 2-form, which we choose to do by defining \begin{equation} F = \hat{E} \wedge dt + \hat{*}\hat{B} \end{equation} and we note first of all that \begin{equation} {*}F = \hat{B} \wedge dt - \hat{*}\hat{E} \end{equation} so that the Hodge dual effectively interchanges the electric and magnetic fields.

Direct computation now shows that \begin{align} dF &= d\hat{E}\wedge dt + d\hat{*}\hat{B} \nonumber\\ &= \hat{d}\hat{E}\wedge dt + \hat{d}\hat{*}\hat{B} + dt\wedge\hat{*}\dot{\hat{B}} \nonumber\\ &= \hat{d}\hat{*}\hat{B} + (\hat{d}\hat{E} + \hat{*}\dot{\hat{B}}) \wedge dt \end{align} But the middle two of Maxwell's equations now imply that \begin{equation} dF = 0 \end{equation} and the implication goes both ways, since the two summands above are linearly independent. Similarly, \begin{align} d{*}F &= d\hat{B}\wedge dt - d\hat{*}\hat{E} \nonumber\\ &= \hat{d}\hat{B}\wedge dt - \hat{d}\hat{*}\hat{E} - dt\wedge\hat{*}\dot{\hat{E}} \nonumber\\ &= - \hat{d}\hat{*}\hat{E} + (\hat{d}\hat{B} - \hat{*}\dot{\hat{E}}) \wedge dt \end{align} and the remaining two of Maxwell's equations bring this to the form \begin{equation} d{*}F = - 4\pi\,\hat{*}\rho + 4\pi\,\hat{*}\hat{J}\wedge dt = - 4\pi\,{*}(\rho\,dt) + 4\pi\,{*}\hat{J} \end{equation} We are thus led to define the 4-current density \begin{equation} J = \hat{J} - \rho\,dt \end{equation} which combines the classical charge and current densities into a single object, and which brings the remaining two of Maxwell's equations to the form \begin{equation} d{*}F = 4\pi\,{*}J \end{equation}

Similarly, combining the scalar and vector potentials $\Phi$ and $\hat{A}$ into a single 4-potential via \begin{equation} A = \hat{A} - \Phi\,dt \end{equation} results, perhaps not surprisingly, in \begin{align} dA &= d\hat{A} - d(\Phi\,dt) \nonumber\\ &= \hat{d}\hat{A} + dt\wedge\dot{A} - \hat{d}\Phi\wedge dt \nonumber\\ &= F \end{align}

In summary, Maxwell's equations in Minkowski space can be reduced to the two equations \begin{align} F &= dA \\ d{*}F &= 4\pi\,{*}J \end{align}

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