In perusing the linearized Einstein equation, it appears that even a classical electromagnetic plane wave would always have to be associated with a tensor perturbation to the background spacetime. For a wave in the "z" direction say:
$$\partial_{\alpha}\partial^{\alpha}h_{zz}=kT_{zz} $$
where the plane em wave has the $T_{zz}$ as the only non-zero component of the stress energy tensor and $h_{\mu\nu}$ is the perturbation to the background metric.
From a qualitative view, they could never separate as the wave would always generate such a perturbation about it as it traverses the vacuum. To be consistent effects of an expanding universe would have to "redshift" both waves precisely the same.
Is this the case? And how come I never see it mentioned? It seems strange or rather fundamental that electromagnetic propagation would always have to be associated with this.
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