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Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
Theorem of the Addition of Velocities.
The Experiment of Fizeau
Now in practice we can move clocks and measuring−rods only with velocities that are small
compared with the velocity of light; hence we shall hardly be able to compare the results of the
previous section directly with the reality. But, on the other hand, these results must strike you as
being very singular, and for that reason I shall now draw another conclusion from the theory, one
which can easily be derived from the foregoing considerations, and which has been most elegantly
confirmed by experiment.
In Section 6 we derived the theorem of the addition of velocities in one direction in the form which
also results from the hypotheses of classical mechanics− This theorem can also be deduced
readily horn the Galilei transformation (Section 11). In place of the man walking inside the carriage,
we introduce a point moving relatively to the co−ordinate system K1 in accordance with the
x1 = wt1
By means of the first and fourth equations of the Galilei transformation we can express x1 and t1 in
terms of x and t, and we then obtain
x = (v + w)t
This equation expresses nothing else than the law of motion of the point with reference to the
system K (of the man with reference to the embankment). We denote this velocity by the symbol W,
and we then obtain, as in Section 6,
W=v+w A)
But we can carry out this consideration just as well on the basis of the theory of relativity. In the
x1 = wt1 B)
we must then express x1and t1 in terms of x and t, making use of the first and fourth equations of
the Lorentz transformation. Instead of the equation (A) we then obtain the equation
which corresponds to the theorem of addition for velocities in one direction according to the theory
of relativity. The question now arises as to which of these two theorems is the better in accord with
experience. On this point we axe enlightened by a most important experiment which the brilliant
physicist Fizeau performed more than half a century ago, and which has been repeated since then