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Relativity: The Special and General Theory
value ct in the first and fourth equations of the Lorentz transformation, we obtain:
from which, by division, the expression
x1 = ct1
immediately follows. If referred to the system K1, the propagation of light takes place according to
this equation. We thus see that the velocity of transmission relative to the reference-body K1 is also
equal to c. The same result is obtained for rays of light advancing in any other direction
whatsoever. Of cause this is not surprising, since the equations of the Lorentz transformation were
derived conformably to this point of view.
Next: The Behaviour of Measuring-Rods and Clocks in Motion
Footnotes
1)
A simple derivation of the Lorentz transformation is given in Appendix I.
Relativity: The Special and General Theory
23
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
The Behaviour of Measuring-Rods and Clocks in Motion
Place a metre-rod in the x1-axis of K1 in such a manner that one end (the beginning) coincides
with the point x1=0 whilst the other end (the end of the rod) coincides with the point x1=I. What is
the length of the metre-rod relatively to the system K? In order to learn this, we need only ask
where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of
the system K. By means of the first equation of the Lorentz transformation the values of these two
points at the time t = 0 can be shown to be
the distance between the points being .
But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a
rigid metre-rod moving in the direction of its length with a velocity v is of a metre.
The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving,
the shorter is the rod. For the velocity v=c we should have ,
and for stiII greater velocities the square-root becomes imaginary. From this we conclude that in
the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be
reached nor exceeded by any real body.
Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations
of the Lorentz transformation, for these became meaningless if we choose values of v greater than
c.
If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to K, then we
should have found that the length of the rod as judged from K1 would have been ;
this is quite in accordance with the principle of relativity which forms the basis of our
considerations.
A Priori it is quite clear that we must be able to learn something about the physical behaviour of
measuring-rods and clocks from the equations of transformation, for the magnitudes z, y, x, t, are
nothing more nor less than the results of measurements obtainable by means of measuring-rods
and clocks. If we had based our considerations on the Galileian transformation we should not have
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Relativity: The Special and General Theory
obtained a contraction of the rod as a consequence of its motion.
Let us now consider a seconds-clock which is permanently situated at the origin (x1=0) of K1. t1=0
and t1=I are two successive ticks of this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks :
t = 0
and
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the
time which elapses between two strokes of the clock is not one second, but
seconds, i.e. a somewhat larger time. As a consequence of its motion the clock goes more slowly
than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.
Next: Theorem of the Addition of Velocities
Relativity: The Special and General Theory
25
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
equation
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
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