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Messenger
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Studying and looking through fluid tensors used in GR and have a question to make sure I understand correctly:
If I had an isotropic and homogeneous perfect fluid [itex]\Omega g_{\mu\nu}[/itex] and within this fluid I had a generic stress energy tensor [itex]\kappa T_{\mu\nu}^{generic}[/itex] but defined it so that [itex]\Omega g_{\mu\nu}=\kappa T_{\mu\nu}^{generic}+\kappa T_{\mu\nu}^{matter}[/itex] where the generic stress energy tensor was the inverse of the stress energy tensor of matter, would I still be able to equate this to successive contractions of the Riemann? Meaning is it still mathematically permissible to write
[itex]R_{\mu\nu}+\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}^{matter}=\Omega g_{\mu\nu}-\kappa T_{\mu\nu}^{generic}[/itex]?
If I had an isotropic and homogeneous perfect fluid [itex]\Omega g_{\mu\nu}[/itex] and within this fluid I had a generic stress energy tensor [itex]\kappa T_{\mu\nu}^{generic}[/itex] but defined it so that [itex]\Omega g_{\mu\nu}=\kappa T_{\mu\nu}^{generic}+\kappa T_{\mu\nu}^{matter}[/itex] where the generic stress energy tensor was the inverse of the stress energy tensor of matter, would I still be able to equate this to successive contractions of the Riemann? Meaning is it still mathematically permissible to write
[itex]R_{\mu\nu}+\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}^{matter}=\Omega g_{\mu\nu}-\kappa T_{\mu\nu}^{generic}[/itex]?