Publisher's Synopsis
Coordinate transformations Up until now we have assumed that we are given a specific coordinate system and its associated coordinates basis. However, we should also consider what will happen when we transform to a different set of coordinates. This is important as coordinates are in fact arbitrary labels for points in a manifold, and you might want to choose different coordinates under different circumstances. For instance, in the particular case of special relativity, you wish to use the coordinates associated with a given inertial frame, ot those associated with a different inertial frame and related to the first ones via Lorentz. You might also wish to consider arbitrary changes of coordinates. It is easy to see that under a change of coordinates the components of the displacement vector transform to the Jacobian matrix. In the particular case of special relativity, the Jacobian identity for the Lorentz transformations is given, but you can consider more general/non-linear, changes of coordinates. An important property of a change of coordinates is that, in the region of interest, the transformation should be one on one, as otherwise the new coordinates wouldn't be useful, which implies that the Jacobian is always invertible in that region. From the definition of the components of a vector it is easy to see that they transform just as the displacement vector.
