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in general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Letξthe 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}gwe 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 getds²=ηon the whole or almost the whole mannifold? Because_{αβ}dξ^{α}dξ^{β}gis still the same on the whole mannifold?_{μν}(x)

Thanks

Neutrino

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# Local inertial frame

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