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in general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let

*ξ*the coordinates in the local inertial frame, so we get

^{α}*ds²=η*. If we switch the frame of reference to coordinates x

_{αβ}dξ^{α}dξ^{β}^{μ}: ξ

^{α}=ξ

^{α}(

*x*) and with

^{0},x^{1},x^{2},x^{3}*g*we get:

_{μν}(x)=η_{αβ}∂ξ^{α}/∂x^{μ}∂ξ^{β}/∂x^{ν}*ds²=g*

_{μν}(x) dx_{μ}dx_{ν}I don't understand why it isn't possible to find a transformation to get

*ds²=η*on the whole or almost the whole mannifold? Because

_{αβ}dξ^{α}dξ^{β}*g*is still the same on the whole mannifold?

_{μν}(x)Thanks

Neutrino