Lorentz transformation, I
This chapter provides a systematic derivation of the Lorentz transformation (LT) and examines its kinematical consequences. It derives the LT first for frames in standard configuration and then for frames not in standard configuration. The chapter considers two assumptions about space and time, that space is isotropic isotropy and that spacetime is homogeneous homogeneity. The LT is a linear mapping between the spacetime coordinates assigned to events by different inertial observers. All inertial observers see straight worldlines for free particles, and straight lines are preserved under homogeneous, linear mappings. The most general linear homogeneous mapping between four-dimensional spaces has 16 parameters. For frames in standard configuration, that number can be reduced considerably by invoking homogeneity and isotropy. The LT is a linear mapping between the coordinates of inertial reference frames in relative motion. Velocity and acceleration involve ratios of differences between space and time coordinates.