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 1:
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 vA
relative to observer O will be moving
at a different velocity 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'.
First of all, it is self-evident that t'=t, otherwise
nothing would make any sense at all.
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 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 u.
In this case we picked u along x,
so that the vector subtraction
v'A = vA - u
becomes just
while 
and 
.
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.