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Altabeh said:This of course does not occur for flat spacetimes for which all Christoffel symbols vanish!
This is incorrect. For example, the Rindler coordinates, with [itex]ds^2=(1+ax)^2dt^2-dx^2[/itex], give [itex]\Gamma\indices{^x_{tt}}=-a(1+ax)[/itex] and [itex]\Gamma\indices{^t_{xt}}=a(1+ax)[/itex], but the spacetime is flat.