How to get the Ricci tensor of an EM field? We have the Einstein equations $$G_{\alpha\beta}=\frac{8\pi G}{c^4}T_{\alpha\beta}\\R_{\alpha\beta}-\frac12 g_{\alpha\beta}R=\frac{8\pi G}{c^4}T_{\alpha\beta}$$

I have been asked to show that for an electromagnetic field these can be written as $$R_{\alpha\beta}=\kappa (F_{\alpha\gamma}F_\beta^\gamma-\frac14 g_{\alpha\beta}F^{\gamma\delta}F_{\gamma\delta})$$

I am not fully sure what $\kappa$ is, but I suppose it is just some constant. In this case, I'd believe this problem reduces to showing $R=0$, but I don't see how that is true in general. So I am not sure how to show this. Could I get a hint? Preferably not a full solution.

An idea I had was to multiply the above equation by $g^{\alpha\beta}$, which yields $$\frac12R=\frac{6\pi G}{c^4}F_{\beta\gamma}F^{\beta\gamma}$$ where $$T_{\alpha\beta}=F_\alpha^\gamma F_{\beta\gamma}-\frac14 F_{\gamma\delta}F^{\gamma\delta}g_{\alpha\beta}$$ is the energy-momentum tensor. But this does not do anything to help me (I think). I have included it since it may turn out to be helpful.
 A: Let Einstein Field Equations be given
$$R_{\alpha\beta}-\dfrac{1}{2}g_{\alpha\beta}R=k T_{\alpha\beta}.$$
Contract with $g^{\alpha\beta}$ and define $T = g^{\alpha\beta}T_{\alpha\beta}$, so that we have upon using $g^{\alpha\beta}R_{\alpha\beta}=R$
$$R-\dfrac{1}{2}g^{\alpha\beta}g_{\alpha\beta}R=kT,$$
but $g^{\alpha\beta}g_{\alpha\beta}=n$ the dimension of spacetime. Thus
$$R-\dfrac{n}{2}R=kT$$
In the case of $n = 4$ we get $kT = -R$. Thus we can substitute $R = -kT$ on Einstein's field equations
$$R_{\alpha\beta}+\dfrac{k}{2}g_{\alpha\beta}T=kT_{\alpha\beta}$$
Or also
$$R_{\alpha\beta}=kT_{\alpha\beta}-\dfrac{k}{2}g_{\alpha\beta}T$$
So you have Ricci's tensor written fully in terms of the energy-momentum tensor and its trace. This is just Einstein's equations rewritten. This is the form which you should work with here.
A: EDIT: The orijinal question was completely different when I first answered, so I changed the answer for the new version.

First of all, $\kappa \equiv\frac{8 \pi G}{c^4}$ and $G_{\mu\nu} \equiv R_{\mu\nu} - \frac12 g_{\mu\nu} R$ are the definitions.
You can first express $R_{\mu\nu}$ in terms of $T_{\mu\nu}$ by taking the trace of the equation and substitute it back in it. Then, you need to define your stress-energy energy tensor for the EM field.
Since the source-free Maxwell Lagrangian in vacuum is
$$
\mathcal{L} = -\frac14 F_{\mu\nu} F^{\mu\nu}
$$
you can calculate the stress-energy tensor by its variation with respect to the metric.
