I am attempting to follow a derivation from a physics paper relating to covariant electromagnetism. It is given that,
$$ F^{\mu \nu} = u^{\mu}E^{\nu} - E^{\mu} u^{\nu} + \epsilon^{\mu \nu \alpha \beta} u_{\alpha} B_{\beta}$$
Also,
$$ B^{\mu} = - \frac{1}{\omega} \epsilon^{\mu \nu \alpha \beta} k_{\nu} u_{\alpha} E_{\beta}$$
Ultimately, I want to show that
$$ F^{\mu \nu} = \frac{1}{\omega} (k^{\mu} E^{\nu} - E^{\mu}k^{\nu})$$
Inserting the second equation into the first gives,
$$ F^{\mu \nu} = u^{\mu}E^{\nu} - E^{\mu} u^{\nu} + \frac{1}{\omega}\epsilon^{\mu \nu \alpha \beta} \epsilon_{\mu \nu \alpha \beta} k^{\nu} E^{\mu}$$
since $u_{\alpha}u^{\alpha} = -1$. But I am confused as to how to progress form here, especially in relation to the 4 dimensional levi-cevita tensors.
Thanks in advance
You must be careful to distinguish free indices from dummy indices. In your second equation you have
$$ B^{\mu} = - \frac{1}{\omega} \epsilon^{\mu \nu \alpha \beta} k_{\nu} u_{\alpha} E_{\beta}$$
and here the indices $\nu, \alpha $ and $\beta $ are already "in use", and they should not be mixed with $\nu, \alpha $ and $\beta $ from the first equation. This is avoided by using different indices:
$$ B^{\mu} = - \frac{1}{\omega} \epsilon^{\mu \sigma \rho \phi} k_{\sigma} u_{\rho} E_{\phi}$$
Now you can substitute to get
$$ F^{\mu \nu} = u^{\mu}E^{\nu} - E^{\mu} u^{\nu} + \frac{1}{\omega}\epsilon^{\mu \nu \alpha \beta} \epsilon_{\mu \sigma \rho \phi} k^{\sigma} u^{\rho} E^{\phi}$$
The tensors share one index and their product can be simplified to
$$\epsilon^{\mu \nu \alpha \beta} \epsilon_{\mu \sigma \rho \phi} = -\begin{vmatrix} \delta^{\nu}_{\sigma} & \delta^{\nu}_{\rho} & \delta^{\nu}_{\phi} \\ \delta^{\alpha}_{\sigma} & \delta^{\alpha}_{\rho} & \delta^{\alpha}_{\phi} \\ \delta^{\beta}_{\sigma} & \delta^{\beta}_{\rho} & \delta^{\beta}_{\phi} \end{vmatrix} $$
Expand this determinant and see what you get.