Frames of Reference and Galilean Transformations

For concreteness, you can think of the frames of reference as trains, each with meter stick and clock attached, traveling at constant speeds along parallel tracks (Strictly speaking for the analysis to be one dimensional they should be on the same track, but this would give rise to practical problems as they pass each other.) In Fig.11.1, the coordinates

Prior to Einstein it was assumed that identical clocks of any two observers could be synchronized so that they would always agree. In terms of equations it was expected that:

In other words, time was expected to have an absolute meaning, independent of the motion of the observer.
With regard to spatial coordinates, it
is clear that if two frames of reference are moving
relative to each other, they will measure different coordinates on
their respective meter sticks for the
same event, and also different velocities for the same object.
For example, if the frames in Fig.11.1 were exactly lined up
at *t* = *t'* = 0, then the coordinate of the explosion in frame *O* would be
equal to the coordinate *x'* as measured by *O'* plus the distance that the frames had moved relative to each other in that time interval:

Eqs[11.1] and [11.2] constitute the so-called * Galilean transformations* relating coordinates as measured in two
different frames. Common sense tells us that they must be
correct. It was therefore a complete shock to the scientific
community when Einstein realized that they contradicted the speed of
light
postulate, and suggested that they were, in fact, incorrect. To see
where they fail, we need to look at what they imply for the addition
of velocities. Suppose that just as the frames *O* and *O'* coincide, their
clocks are synchronized to read *t* = *t'* = 0. At precisely this instant
the observer in *O'* throws the ball to
the right with a velocity of *u'* = 3 cm/s relative to his frame.
(See Fig. 11.2 below).

Fig. 11.3 shows the situation at a later time

The ball is now at the position labelled by

We now have to extend our discussion to a situation which is easy to
imagine, but rather difficult to realize in nature. Such situations
are called *gedanken* or thought experiments. Suppose that
*O'* is moving relative to *O*, at three quarters of the speed of light,
instead of 2 cm/s. Moreover, instead of throwing a ball forward *O'* points a
flashlight in the forward direction and turns it off and on
quickly. This sends out a pulse of light moving at
300 million meters/second as measured relative to *O'*. How
fast would *O'* see the pulse of light moving relative to his frame? In order
to save writing, we will henceforth use the symbol *c* to denote the speed of
light. Everytime you see this letter, you should think ``300 million meters per second''.
According to the
above discussion the speed of the pulse relative to *O'* would
be
*c* + 3*c*/4 = 7*c*/4, or about 500 million meters/second. This
contradicts the speed of light postulate, which says that the speed of
light should be the same in every frame of reference.

So what has gone wrong? The easiest conclusion to draw is that the speed of light postulate is incorrect. However, Einstein realized that one cannot give up this postulate without giving up either Maxwell's wave theory, or the postulate of uniform motion, and he was willing to give up neither. He therefore made a brilliant intuitive leap and concluded that our common sense is wrong. Einstein's faith in the speed of light postulate turned out to be well founded. Its strange consequences have since been verified experimentally. We will now discuss them in turn.

1999-09-29