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?