Galilean Invariance

Abstract

The freedom of choice for reference frames includes more than rotations: one can displace the origin, translate it by a constant vector; or one can let that translation grow proportionally with time; the two frames are in relative motion at constant velocity. We’ll consider only relative speeds that are small on the scale of the speed of light; see Problems 4-3 and 4-4 for other circumstances. Then time has an absolute significance (Galilean*-Newtonian relativity) apart from the freedom of displacing its origin. The infinitesimal transformations of these types are displayed by the space-time changes

$$\begin{array}{*{20}{c}} {\overline t = t - \delta t,} \\ {\overline r = r - \delta r,} \\ {with\quad \delta r = \delta \varepsilon + \delta \omega \times r + \delta vt,} \\ \end{array}$$

(4.1.1)

where δt is a constant, as are the vectors δε, δw, δv. The accompanying unitary operator is

$$U = 1 + iG$$

(4.1.2)

where, now

$$G = \delta \varepsilon \cdot P + \delta w\cdot J + \delta v\cdot N - \delta tH + \delta \varphi $$

(4.1.3)

and we want to recognize that we always have the freedom of a phase transformation.