I'm having trouble understanding how the lorentz transformations are posed intially. Why can different observers agree on their relative velocity? I don't understand why this velocity is exempt from all of the other relativistic phenomena that occur. Also, in the initial derivation, I'm told that one could make the following equations ##x'=G(x-ut)## ##x=G(x'+ut')## I understand the reasoning behind the signs, but I think I dont see any clear reason for wanting to use different times here. Of course, the point of this is to conclude that time intervals aren't absolute, however, I don't get why, for example: the term (x'+ut') is constructed that way because this coordinate is measured to be going away by O (unprimed frame). But why is the translation a function of the time measured by the O' frame? namely, why is it ##ut'## that is considered by the O frame. Why isn't it ##ut##? That makes more sense to me. Thanks.