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##T_{\mu_\nu} = 0##

##\Rightarrow~R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 0##

Now multiply both sides by ##g_{\mu\nu}##

with definition of ##R = g^{\mu\nu} R_{\mu\nu}##

##R - \frac{1}{2} R = 0##

##R = 0##

Is that my imagination wrong? I thought 'empty space' might not be flat in general, like the Schwarzschild metric. And, I thought ##R = 0## meant 'the spacetime is flat.

What is the point I missed? Thanks.

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# Energy-stress tenor = 0 => flat spacetime?

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