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):
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 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 . In this case we picked along , so that the vector subtraction becomes just v'_{Ax} = v_{Ax} - u while v'_{Ay} = v_{Ay} and v'_{Az} = v_{Az}. Let's summarize all these "coordinate transformations:"
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.