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I know how the contravariant and covariant components of a vector transform when the coordinate system is changed (⇒ the known relation between the old coordinate system and the new one, I multiply by partial derivatives properly)

Do the same transformation laws still holds when the relation between the two coordinate systems is time-dependent? Or in this case I should somehow involve partial-time-derivatives, too?

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Are you talking about space-time here? If you are, then time-dependent is not a co-variant statement. What one observer precieves as time-dependent, another one will preceive as space- and time-dependent. So your vector transformations will be simply position-dependent in four-dimensional space-time.

In principle, this is easily handeled by the machinery of the tensor calculus and differential geometry, but you will not be able to apply the simple techniques from introductory special relativity. There is a middle-ground. You may be after ways of working with accelerating objects in flat space-time. This can be done relatively easy. In a hand-wavey way, what you do is introduce instantaneous reference frames, where your object is at rest at some specific time in its history. You do have to be careful when using these. Misner-Thorne-Wheleer have a section on acceleration in Special Relativity. I think this may be helpful.

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