# 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?

• The first tensor has two lower indices, the second has two upper indices. – jacob1729 Apr 25 '19 at 20:33
• Yes that is correct. That is how the paper I referenced has it (pg.13). This is why I am confused as to what electromagnetic stress energy tensor I can place into the energy-momentum tensor of Einstein's field equations. – Jay Apr 25 '19 at 20:42
• Both of the expressions you give are only correct for flat spacetime I think. I've never done EM in curved spacetime, so am not sure but Wikipedia gives $T_{\mu\nu} = - \frac{1}{\mu_0} \left ( F_{\mu \alpha} g^{\alpha \beta} F_{\beta \nu} - \frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma} \right )$ as the correct expression for curved spaces. – jacob1729 Apr 25 '19 at 21:01
• @Jay: is the Einstein field equation you're using one with a stress-energy tensor with raised or lowered indices in the Stress-energy tensor? – Jerry Schirmer Apr 25 '19 at 21:43
• Your answer is different if it is $R^{ab} - \frac{1}{2}Rg^{ab} = 8\pi T^{ab}$ versus $R_{ab} - \frac{1}{2} R g_{ab} = 8\pi T_{ab}$ – Jerry Schirmer Apr 25 '19 at 21:44

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.