Electromagnetic field tensor: cyclic derivative relation

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 7; Problem 7.6.

We’ve used the following relation between the derivatives of the electromagnetic field tensor {F^{ij}} to get several of Maxwell’s equations.

\displaystyle  \partial_{i}F_{jk}+\partial_{k}F_{ij}+\partial_{j}F_{ki}=0 \ \ \ \ \ (1)

Here we verify that this relation is true when {F^{ij}} is written in terms of the four-potential, that is

\displaystyle  F^{ij}=\partial^{i}A^{j}-\partial^{j}A^{i} \ \ \ \ \ (2)

We can lower both indices in this equation and the plug it into the first equation:

\displaystyle   \partial_{i}F_{jk}+\partial_{k}F_{ij}+\partial_{j}F_{ki} \displaystyle  = \displaystyle  \partial_{i}\left(\partial_{j}A_{k}-\partial_{k}A_{j}\right)+\partial_{k}\left(\partial_{i}A_{j}-\partial_{j}A_{i}\right)+\partial_{j}\left(\partial_{k}A_{i}-\partial_{i}A_{k}\right)\ \ \ \ \ (3)
\displaystyle  \displaystyle  = \displaystyle  \partial_{i}\partial_{j}A_{k}-\partial_{j}\partial_{i}A_{k}+\partial_{k}\partial_{i}A_{j}-\partial_{i}\partial_{k}A_{j}+\partial_{j}\partial_{k}A_{i}-\partial_{k}\partial_{j}A_{i}\ \ \ \ \ (4)
\displaystyle  \displaystyle  = \displaystyle  0 \ \ \ \ \ (5)

The terms in the second line cancel in pairs since the order of the partials doesn’t matter.

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