Electromagnetic stress-energy tensor to be used in Einstein's Field Equations I am trying to put in the electromagnetic energy-stress tensor in for the energy-momentum tensor of Einstein's field equations. 
I am, however, unsure as to which tensor matrix to use. I found the following tensor matrix from "Introduction to Einstein-Maxwell equations and the Rainich conditions" by Wytler Cordeiro Dos Santos:
$$ T_{mv} =
    \begin{bmatrix}
    \frac{1}{2}(\epsilon |E|^{2} + \frac{1}{\mu}|B|^{2}) & -\frac{S_{x}}{c} & -\frac{S_{y}}{c} & -\frac{S_{z}}{c} \\
    -\frac{S_{x}}{c} & -\sigma_{xx} & -\sigma_{xy}  -\sigma_{xz} \\
    -\frac{S_{y}}{c} & -\sigma_{yx} & -\sigma_{yy}  -\sigma_{yz} \\
-\frac{S_{z}}{c} & -\sigma_{zx} & -\sigma_{zy} -\sigma_{zz} \\
    \end{bmatrix}
$$
However the textbook definition of the electromagnetic energy-stress tensor is:
$$ T^{mv} =
    \begin{bmatrix}
    \frac{1}{2}(\epsilon |E|^{2} + \frac{1}{\mu}|B|^{2}) & \frac{S_{x}}{c} & \frac{S_{y}}{c} & \frac{S_{z}}{c} \\
    \frac{S_{x}}{c} & -\sigma_{xx} & -\sigma_{xy}  -\sigma_{xz} \\
    \frac{S_{y}}{c} & -\sigma_{yx} & -\sigma_{yy}  -\sigma_{yz} \\
\frac{S_{z}}{c} & -\sigma_{zx} & -\sigma_{zy} -\sigma_{zz} \\
    \end{bmatrix}
$$
with $\sigma_{ij} = \epsilon E_{i}E_{j} + \frac{1}{\mu}B_{i}B_{j} - \frac{1}{2}(\epsilon E^{2} + \frac{1}{\mu}B^{2})\delta_{ij} $
So which matrix equation would I use in Einstein's field equation:
$G_{\alpha\beta} = R_{\alpha \beta} - \frac{1}{2}g_{\alpha \beta}R = -\frac{8 \pi G}{c^{4}} T_{\alpha\beta}$?
Thanks. If any further information is needed please let me know.
Jay
edit:Fixed Einstein's Field Equation. If I was to want free space energy tensor which would I use? 
 A: I am not sure if this answers your question, but I think it might help. 
First of all your field equation is messed up. What you want is 
$$R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R = \kappa T_{\mu\nu}$$ 
or this
$$R^{\mu\nu} -\frac{1}{2}g^{\mu\nu}R = \kappa T^{\mu\nu}$$ 
The stress energy tensor in electromagnetism can be derived from the Lagrangian
$$\mathcal{L} = -\frac{\sqrt{-g}}{4} F^{\alpha\beta}F_{\alpha\beta}$$
It comes out to be 
$$T_{\mu\nu} = F^{\beta}_{\mu}F_{\beta\nu} - \frac{1}{4}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}$$
The expression will be similar in covariant form. Plugging this into the Einstein field equations
$$R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R=\kappa(F^{\beta}_{\mu}F_{\beta\nu} - \frac{1}{4}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta})$$
Also a bit of advice: you should use 
$$R_{\mu\nu}=\kappa(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T)$$
Instead of $G_{\mu\nu}=\kappa T_{\mu\nu}$. It is easier to solve.
FYI: 1. All formulae are in natural units.
2. $F_{\mu\nu}$ is Electromagnetic field tensor. 
