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I'm struggling to grasp the physical reason behind the fact that, in a curved spacetime, a change of metric implies, in general, a change of connection, i.e. if I have two metrics [tex]g_{ab}[/tex] and [tex]\hat{g}_{ab}[/tex], in general [tex]\nabla_a \neq \hat{\nabla}_a[/tex].

Besides this, is there any relationship between the two connections? In other words, if I know [tex]\nabla_aT[/tex] for a given tensor T, is there a general formula which converts it into [tex]\hat{\nabla}_aT[/tex]?

Thanks