Given the energy momentum tensor in E&M: $T_{\mu\nu} = -F_{\mu\alpha} g^{\alpha \beta} F_{\beta \nu} +\frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma}$ I want to show it is homogenous and isotropic in Robertson-Walker spacetime. How would you go about performing a rotation and a translation to show that $T_{\mu \nu}$ is left invariant?

  • $\begingroup$ You're going to need to pick a form for $F_{\mu\nu}$. The maxwell tensor is not generally homogenous and isotropic. For instance, pick a static monopole charge. $\endgroup$ – Jerry Schirmer Jul 10 '14 at 23:52

I don't think performing a Lorentz rotation/translation will get you anywhere. $T_{\mu\nu}$ and $F_{\mu\nu}$ being tensors automatically makes them frame independent so performing a Lorentz transformation gives you back the same tensor-equation by defininion (though with different components).

I will assume that you mean having a large scale EM field described by $T_{\mu\nu}$ that is driving the expansion of the Universe (possibly together with other fields descibed by $T_{\mu\nu}^{\rm other}$ that are homogenous and isotropic).

To show your claim we use the Einstein equation. Since

$$G_{\mu\nu} - 8\pi GT_{\mu\nu}^{\rm other} = 8\pi GT_{\mu\nu}$$

and if the LHS is homogenous and isotropic (which it is by assumption) then so is the right hand side.

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