satisfy Maxwell’s equations.

Similarly, an alternate heuristic manipulation of the continuity equation in the Minkowski spacetime leads to the existence of an electromagnetic tensor satisfying the covariant form of Maxwell’s equations. This is a consequence the following existence theorem [@7]: Given the localized four-vector ${\cal J}^{\mu}$ satisfying the continuity equation $\partial_\mu {\cal J}^\mu=0$ there exists the antisymmetric tensor field $${\cal F}^{\mu\nu}=
\int  {\cal G}(\partial'^\mu {\cal J}^\nu-\partial'^\nu {\cal J}^\mu)\,d^4x',$$ that satisfies the field equations: $\partial_{\mu} {\cal F}^{\mu\nu} = {\cal J}^\nu$ and $
\partial_{\mu}{\cal^*F}^{\mu\nu} = 0,$ where $^*{\cal F}^{\mu\nu}=(1/2)\varepsilon^{\mu\nu\alpha\beta} {\cal F}_{\alpha\beta}$ is the dual of ${\cal F}^{\mu\nu}$ and ${\cal G}=\delta\{t'\!-t+R/\mathscr{C}\}/(4\pi R)$ with $\mathscr{C}$ being a constant whose units are of velocity. If we make the identification $\mathscr{C}=c$ then ${\cal G}=G$ and if in addition we make ${\cal J}^\mu=\mu_0J^\nu$ with $J^\nu=(c\rho,\textbf{J})$ then ${\cal F}^{\mu\nu}={F}^{\mu\nu}$ is the electromagnetic field tensor in SI units.

The point to remark is that in the proof of an existence theorem of an object, one is generally free to use all heuristic devices that allows one to exhibit the explicit form of such an object. This is the more essential aspect in a constructive approach.

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Most authors agree that the continuity equation is a *consequence* of Maxwell’s 