While deriving vanishing of covariant derivative of metric tensor,Schutz says in pg 132-"But now this equation is a tensor equation, so its validity in one coordinate system implies its validity in all",how far is this true?
I mean $g_{\mu\nu}=\eta_{\mu\nu}$ for Minkowskian flat spacetime,but it isn't possibly true for all spacetime(globally)?
I have seen a lot of usage of this fact.Where can I find a formal proof of the same?
And how does it not contradict with what I said above?
I mean $g_{\mu\nu}=\eta_{\mu\nu}$ for Minkowskian flat spacetime,but it isn't possibly true for all spacetime(globally)?
I have seen a lot of usage of this fact.Where can I find a formal proof of the same?
And how does it not contradict with what I said above?
