Galilean Transformation

Figure 3.9: Inertial reference frames.

Consider two inertial frames of reference, $S$ and $S'$. Let frame $S'$ move at velocity ${\bf v}$ with respect to frame $S$. Let us set up right-handed Cartesian coordinate systems in both frames. Suppose that the coordinate systems are in the so-called standard configuration in which the corresponding coordinate axes are parallel, the $x$-axis in each system is parallel to ${\bf v}$, and the origins of the systems coincide at time $t=0$. See Figure 3.9. Consider an instantaneous `event' with a definite spatial location, such as the flashing of a light-bulb. Suppose that the event occurs at time $t$ and has displacement ($x$, $y$, $z$) in frame $S$. Suppose that the event occurs at time $t'$ and has displacement ($x'$, $y'$, $z'$) in frame $S'$. What is the relationship between ($x$, $y$, $z$, $t$) and ($x'$, $y'$, $z'$, $t'$). Well, according to standard Newtonian physics, the “common sense” relationship between the two sets of coordinates is

$$x' = x - v\,t,$$ (3.88)

$$y' =y,$$ (3.89)

$$z' = z,$$ (3.90)

$$t' = t.$$ (3.91)

(See Section 1.5.4.) As we have already mentioned, this transformation of coordinates is known as the Galilean transformation. Consider, now, a moving event whose coordinates in $S$ are $x=x(t)$, $y=y(t)$, and $z=z(t)$. The Cartesian components of the instantaneous velocity of our event in $S$ are $u_x=dx/dt$, $u_y=dy/dt$, $u_z=dz/dt$, whereas the corresponding components in $S'$ are $u_{x}'=dx'/dt'$, $u_{y}'=dy'/dt'$, $u_{z}'=dz'/dt'$. Hence, we can derive the following Galilean velocity transformation from Equations (3.88)–(3.91):

$$u_{x}' = u_x - v,$$ (3.92)

$$u_{y}' =u_y,$$ (3.93)

$$u_{z}' = u_z.$$ (3.94)

However, if the event in question is the path of a light ray that moves with velocity $c\,{\bf e}_x$ in $S$ then, according to the Galilean velocity transform, the light ray moves with velocity $(c-v)\,{\bf e}_{x}$ in $S'$. In other words, the light ray travels at difference speeds in the two frames of reference. However, this state of affairs is forbidden by Einstein's first postulate. Hence, we deduce that the Galilean transformation, (3.88)–(3.91), is actually inconsistent with the theory of relativity.