text: # Electromagnetic waves in tensor notation
  
  I'm trying to derive the wave equations for the electric and magnetic fields in covariant (tensor) formulation. Starting with Gauss-Ampere law, $$\partial_\alpha F^{\alpha\beta}=\frac{4\pi}{c}J^\beta$$ and Gauss–Faraday law $$\partial_{\alpha}(\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}) = 0$$ I would like to derive the values of $$\square \vec{E}$$ and $$\square\vec{B}$$, or in general get to $$\partial^{\nu} \partial_{\nu} F^{\alpha\beta}$$. Is there an elegant way to do this? I did try using summation in each of the equations, but got quite confused with all the indices.
  
  • Do you want the wave equations in vacuum? So with $J^\beta=0$? Or do you want the wave equation for inside materials? Commented Jan 26, 2021 at 8:23
  • Preferably inside material, which is more general (so I can plug in $J^\beta=0$ afterwards to get them in vacuum). Commented Jan 26, 2021 at 8:26
  • Try simplifying $\partial^\alpha ( \partial_\alpha F_{\beta\gamma} + \partial_\gamma F_{\alpha\beta} + \partial_\beta F_{\gamma \alpha} ) = 0$. The thing in the bracket is the Bianchi identity (or Gauss-Faraday law). Commented Jan 26, 2021 at 10:11
  
  In the Lorentz gauge your first equation becomes the wave equation for the potential $$\partial_\alpha \partial^\alpha A^\beta = \frac{4\pi}{c}J^\beta ~.$$ By deriving left and right hand side you obtain $$\partial_\alpha \partial^\alpha F^{\gamma\beta} = \frac{4\pi}{c} \left(\partial^\gamma J^\beta - \partial^\beta J^\gamma \right) ~,$$ which is the required wave equation.