Solution to tensor differential equations

[jfy4]

hello all,

I need two solutions to two different tensor diffeqs. I think I may have the solution to the sourceless equation, however I am in the dark about the one with the source.


$$\left(\partial_{\gamma}\partial_{\alpha}+\imath k^{\beta}g_{\alpha\beta}\partial_{\gamma}\right) \phi=T_{\gamma\alpha}\phi$$

and

$$\left(\partial_{\gamma}\partial_{\alpha}+\imath k^{\beta}g_{\alpha\beta}\partial_{\gamma}\right) \phi=0$$.

Any help would be appreciated.

 

[jfy4]

here is my solution for the source less equation, feel free to check it please.

$$\phi^{\gamma\alpha}=Ae^{-\imath\left(\delta^{\gamma}_{\alpha}k_{\gamma}x^{\alpha}\right)}+Be^{-\imath\left(k_{\alpha}x^{\alpha}-k_{\gamma}x^{\gamma}\right)}$$

thanks.

 

[jfy4]

here is my solution for the source less equation, feel free to check it please.

$$\phi^{\gamma\alpha}=Ae^{-\imath\left(\delta^{\gamma}_{\alpha}k_{\gamma}x^{\alpha}\right)}+Be^{-\imath\left(k_{\alpha}x^{\alpha}-k_{\gamma}x^{\gamma}\right)}$$

thanks.

I also made the replacement $$k_{\beta}=k^{\alpha}g_{\alpha\beta}$$