Galilean Transformation, Lorentz Transformation, and General Transformation

Abstract In the Galilean transformation system, a body contacts and moves on the moving frame and is completely carried by the moving frame and hence the body and the moving frame have the same time rule but velocity of the body moving in the stationary frame is larger than in the moving frame. In the Lorentz transformation system, however, a body is free in all frames and hence not carried by the moving frame and the body moving in the moving and stationary frames has the same velocity but different time rules. We mathematically proved that the classical Lorentz transformation works for movement of a body with light speed in all frames. Invariance of light speed in all frames is an instance of Lorentz velocity invariance. Galilean linear velocity addition can be derived from “complete carry” mechanism. We proved that Lorentz transformation cannot be reduced to Galilean transformation in any situation. A general transformation was given from “partial carry” mechanism. The general transformation can be reduced to Lorentz transformation when refraction coefficient n = 1 or reduced to Galilean transformation when n is large enough so that 1/n2 ≈ 0. Non-linear velocity addition can be obtained from the general transformation, and can be used to well explain Fizeau experiment.