TABLE OF CONTENTS In Special Relativity, we are only allowed to use inertial frames to assign coordinates to events. There are many different types of inertial frames. However, it is convenient to adhere to those with standard coordinates. That is, spatial coordinates which are right-handed rectilinear Cartesians based on a standard unit of length, and time-scales based on a standard unit of time. We shall continue to assume that we are employing standard coordinates. However, from now on, we shall make no assumptions about the relative configuration of the two sets of spatial axes, and the origins of time, when dealing with two inertial frames. Thus, the most general transformation between two inertial frames consists of a Lorentz transformation in the standard configuration plus a translation (this includes a translation in time) and a rotation of the coordinate axes. The resulting transformation is called a general Lorentz transformation, as opposed to a Lorentz transformation in the standard configuration, which will henceforth be termed a standard Lorentz transformation.
In Section 10.3, we proved quite generally that corresponding differentials in two inertial frames S and S' satisfy the relation
Thus, we expect this relation to remain invariant under a general Lorentz transformation. Since such a transformation is linear, it follows that
where (x 1,y 1,z 1, t 1) and (x 2,y 2,z 2,t 2) are the coordinates of any two events in S, and the primed symbols denote the corresponding coordinates in...
