2.5. The Lorentz Transformations.
If one synchronizes the clocks in a given inertial frame on the assumption that light propagates with the one speed c in all directions relative to that frame (as experiment indicates is the case), then it is a simple matter to transform the space-time coordinates of events in one inertial frame to the corresponding coordinates in another inertial frame. The resulting, "Lorentz" transformation is slightly more complicated than the pre-relativistic "Galilean" transformation. But it takes into account length contraction and time dilation, and in general is consistent with the results predicted by Maxwell-Lorentz-Newton.
Knowing the transformation formulas for x, y, z and t, it is a matter of algebra to derive transformations for the components of velocity, acceleration, and even da/dt. Transformations for the components of Finertial and Fradiative follow, and even a set of general transformations for the components of the electromagnetic field vectors E and B can be derived.
With regard to the field vectors, it is often desired to have a "snapshot" of the fields in a given frame at some particular instant. It should perhaps be borne in mind, in this regard, that the field transformations transform a snapshot in frame K to the fields at many different instants in K’. However, a computer routine can be employed to build a "snapshot" of the fields in K’, given a knowledge of E and B in frame K.