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Sycobob

- 1

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## Homework Statement

From Sean Carroll's notes on general relativity (chapter 1, pg. 20):

Show that [itex]F_{[\alpha\beta,\gamma]} = 0[/itex] is equivalent to half of the Maxwell equations.

## Homework Equations

[itex]F_{\mu\nu}[/itex] is the electromagnetic tensor

[itex]\Phi_{,\nu} \equiv \partial_{\nu}\Phi[/itex]

[itex]F_{i0}= E_{i}[/itex]

[itex]F_{ij}= \epsilon^{ijk}B_{k}[/itex]

## The Attempt at a Solution

I'm specifically looking to turn Maxwell's (homogeneous) equations into tensor form, not just show that they fall out of [itex]F_{[\alpha\beta,\gamma]} = 0[/itex]. I sort of have a solution, but I feel like I'm missing a step.

[itex]

\begin{eqnarray*}

\nabla×\textbf{E} + \partial_{t}\textbf{B} = 0 \\

\nabla\cdot \textbf{B} = 0

\\

\\

\epsilon^{ijk}\partial_{j}E_{k} + \partial_{0}B^{i} = 0 \\

\partial_{i}B^{i} = 0

\\

\\

\epsilon^{ijk}\partial_{j}F_{k0} + \frac{1}{2}\epsilon^{ijk}\partial_{0}F_{jk} = 0 \\

\frac{1}{2}\epsilon^{ijk}\partial_{i}F_{jk} = 0

\end{eqnarray*}

[/itex]

which can be rewritten as:

[itex]\epsilon^{\mu\nu\rho\sigma}\partial_{\rho}F_{\mu \nu} = 0[/itex]

which, up to a normalization constant, is just:

[itex]F_{[\alpha\beta,\gamma]} = 0[/itex]

My question is about going from step 3 to step 4. I sort of pulled it out of my hat and checked that it was correct (term by term). I'm looking for some kind of justification for this step, or a nudge in the right direction if I'm approaching this all wrong.

Also, I'm still getting the hang of tensor notation, and I feel like equation 4 doesn't make sense. Only 3 indices are contracted, leaving the right side a vector, not a scalar. On the other hand, trying to use the Levi-Civita tensor with 3 indices here seems wrong too, as the indices run from 0 to 4 leaving you with stuff like [itex]\epsilon^{013}[/itex].