#### BELIEVE ME NOT! **-** **-** A SKEPTIC's GUIDE

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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.

** Next:** Lorentz Transformations
** Up:** The Special Theory of Relativity
** Previous:** The Special Theory of Relativity

Jess H. Brewer -
Last modified: Mon Nov 23 10:49:18 PST 2015