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latentcorpse

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i.e. [itex]g_{ab}=\eta_{ab}+h_{ab}[/itex] with [itex]|h_{ab}| \ll 1[/itex]

The system is also non relativistic meaning that time derivatives can be taken to be much smaller that spatial derivatives. This implies that the components of the stress energy tensor can be ordered [itex]|T_{00}| \gg |T_{0i}| \gg |T_{ij}|[/itex]

Under such circumstances, I want to show that stress energy conservation reduces to [itex]T^{\mu k}{}_{,k}=0[/itex]

Well we have to start with our standard GR defn [itex]T^{\mu \nu}{}_{; \nu}=0[/itex]

[itex]T^{\mu \nu}{}_{, \nu} + \Gamma^\mu{}_{\sigma \nu} T^{\sigma \nu} + \Gamma^\nu{}_{\sigma \nu} T^{\sigma \mu}=0[/itex]

[itex]T^{\mu \nu}{}_{, \nu} + \frac{1}{2} \eta^{\mu \rho} \left( h_{\rho \sigma , \nu} + h_{\rho \nu , \sigma} - h_{\sigma \nu , \rho} \right) T^{\sigma \nu} + \frac{1}{2} \eta^{\nu \rho} \left( h_{\sigma \rho , \nu} + h_{ \rho \nu , \sigma} - h_{\sigma \nu, \rho} \right) T^{\sigma \mu}=0[/itex]

Now I don't know where to go with this???