[Physics] What does this quote about the four dimensional divergence of an antisymmetric tensor mean

Question

In the beginning, God said that the four dimensional divergence of an antisymmetric second rank tensor equals zero and there was light.

Can someone explain what is the meaning of this quote by Michio Kaku?

Answer 1

The antisymmetric second-rank tensor being referenced is the electromagnetic field tensor. It is defined as follows. Let $\varphi$ be the electrostatic potential (a scalar field), and let $\underline{A}$ be the magnetic potential (a 3-vector) from classical E&M. Concatenate them into a 4-vector $\vec{A}$. Now define the tensor of interest as the exterior derivative of $\vec{A}$: $$ \mathbf{F} = \mathrm{d}\vec{A}. $$ We can write this component-wise with partial derivatives: $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$ You can see that if thought of as a matrix, the components $F_{\mu\nu}$ of $\mathbf{F}$ are antisymmetic.

Now the use of this is that the four equations that govern classical electromagnetism (and hence light) are equivalent to: $$ \partial_\nu F^{\mu\nu} = J^\mu $$ ($\vec{J}$ is the 4-current composed of electric charge concatenated with 3-current) and $$ \partial_{[\alpha} F_{\mu\nu]} = 0 $$ (the brackets denote the summing all permutations of indices with a sign given by the parity of the permutation).

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Note that depending on your unit system there may be constants like $c$ or $\mu_0$ floating around in these equations.