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## Main Question or Discussion Point

Hello! I am reading "A First Course in General Relativity" by Schutz and in chapter 8 (second edition) he introduces Nearly Lorentz coordinate system. He says that we can always find some coordinates such that the metric is: $$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$$ with $$|h_{\alpha\beta}|<<1$$ then he defines $$\bar{h}_{\alpha\beta}=h_{\alpha\beta}-\frac{1}{2}\eta^{\alpha\beta}h^\alpha_\alpha$$ Then using background Lorentz transformations and gauge transformations (which as far as I understand they are coordinate transformations), he obtains $$G^{\alpha\beta}=-\frac{1}{2}\Box \bar{h}^{\alpha\beta}$$ where ##G## is the Einstein tensor and from this he gets: $$\Box\bar{h}^{\alpha\beta}=-16\pi T^{\alpha\beta}$$ Now this seems like a tensorial equation, so it should hold true in any frame. But there are 2 things I am confused about: ##\textbf{1.}## Being a tensor equation, it should hold true in a locally flat frame, too. But there, ##h## is zero because ##g=\eta## which implies that ##T=0##. But we can apply this to any point on the manifold, which implies that ##T=0## everywhere, including inside the source (we should be allowed to do this inside the source because, as he does, we have a non-relativistic system so the approximations should hold there, too). But this is clearly not true (right?) so what is wrong with my reasoning? ##\textbf{2.}## In general when we want to use this equation, how do we know what is ##T##? For example, in cartesian coordinates, for a perfect fluid, ##T=diag(\rho,p,p,p)##. But in order to reach this equation, we did lots of coordinate transformations (which were not explicitly done, but it was only shown that they can be done). Doesn't this means that in this new coordinate system ##T## will have a more complicated form, which we need to know in order to find ##h##, which is the purpose of this formalism? How can we get ##T##, when we didn't really keep track of all the coordinate transformations we made? Sorry for the long post, but I hope someone can explain to me what I am missing here. Thank you!