I am trying to derive geodesic motion for photons from the Lagrangian of electromagnetism coupled to General Relativity.
I tried to use the covariant conservation of the Stress energy tensor: $$\nabla_aT^{ab}=0$$ where the stress energy tensor is: $$T^{ab}=F^{ac}F^{b}_{c}-\frac{1}{4}F_{cd}F^{cd}g^{ab}$$ In this way, supposing $F_{cd}F^{cd}=\,\text{constant}\;$ and using the equation of motion $\nabla_aF^{ab}=0$ I get to the equation: $$g_{cd}F^{ac}\nabla_{a}F^{bd}=0$$ which looks fairly similar to the geodesic equation but is not, due to the contraction between the two $F$.
I would like to know if there is any way to interpret geometrically the equation obtained for $F^{ab}$ or the corresponding equation for $A_b$. Maybe one can write $A_b$ as plane waves with non constant wave vector and show that this vector follows a geodesic and that the polarization is always orthogonal to it?
