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Galilean Transformations

So what? Well, this innocuous looking claim has some very perplexing logical consequences with regard to relative velocities, where we have expectations that follow, seemingly, from self-evident common sense. For instance, suppose the propagation velocity of ripples (water waves) in a calm lake is 0.5 m/s. If I am walking along a dock at 1 m/s and I toss a pebble in the lake, the guy sitting at anchor in a boat will see the ripples move by at 0.5 m/s but I will see them dropping back relative to me! That is, I can "outrun" the waves. In mathematical terms, if all the velocities are in the same direction (say, along x), we just add relative velocities: if v is the velocity of the wave relative to the water and u is my velocity relative to the water, then v', the velocity of the wave relative to me, is given by v' = v - u. This common sense equation is known as the GALILEAN VELOCITY TRANSFORMATION - a big name for a little idea, it would seem.

With a simple diagram, we can summarize the common-sense GALILEAN TRANSFORMATIONS (named after Galileo, no Biblical reference):


  
Figure: Reference frames of a "stationary" observer O and an observer O' moving in the x direction at a velocity u relative to O. The coordinates and time of an event at A measured by observer O are $\{x,y,z,t\}$ whereas the coordinates and time of the same event measured by O' are $\{x',y',z',t'\}$. An object at A moving at velocity $\Vec{v}_A$ relative to observer O will be moving at a different velocity $\Vec{v}_A'$ in the reference frame of O'. For convenience, we always assume that O and O' coincide initially, so that everyone agrees about the "origin:" when t=0 and t'=0, x=x', y=y' and z=z'.
\begin{figure}
\begin{center}\epsfysize 2.0in
\epsfbox{PS/ref_frames.ps}\end{center} %
\end{figure}

First of all, it is self-evident that t'=t, otherwise nothing would make any sense at all.23.1 Nevertheless, we include this explicitly. Similarly, if the relative motion of O' with respect to O is only in the x direction, then y'=y and z'=z, which were true at t=t'=0, must remain true at all later times. In fact, the only coordinates that differ between the two observers are x and x'. After a time t, the distance (x') from O' to some obect A is less than the distance (x) from O to A by an amount ut, because that is how much closer O' has moved to A in the interim. Mathematically, x' = x - ut.

The velocity $\Vec{v}_A$ of A in the reference frame of O also looks different when viewed from O' - namely, we have to subtract the relative velocity of O' with respect to O, which we have labelled $\Vec{u}$. In this case we picked $\Vec{u}$ along $\hat{x}$, so that the vector subtraction $\Vec{v}'_A = \Vec{v}_A - \Vec{u}$ becomes just v'Ax = vAx - u while v'Ay = vAy and v'Az = vAz. Let's summarize all these "coordinate transformations:"

    % latex2html id marker 946
\fbox{\parbox{3.0in}{
\null \hfil {\large The {\sc G . . . 
 . . . v_{A_y} \qquad \qquad \\
v'_{A_z} &=& v_{A_z} \qquad \qquad
\end{eqnarray} } }

This is all so simple and obvious that it is hard to focus one's attention on it. We take all these properties for granted - and therein lies the danger.


next up previous
Next: Lorentz Transformations Up: The Special Theory of Relativity Previous: The Special Theory of Relativity
Jess H. Brewer
1999-03-19