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Relativity: The Special and General Theory by Albert Einstein

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Author Topic: Relativity: The Special and General Theory by Albert Einstein  (Read 529 times)
Jean
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« Reply #30 on: March 10, 2009, 12:20:23 am »

Section 12 - The Behaviour of Measuring-Rods and Clocks in Motion
Place a metre-rod in the x'-axis of K' in such a manner that one end (the beginning) coincides with the point x' = 0 whilst the other end (the end of the rod) coincides with the point x' = 1. 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

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Jean
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« Reply #31 on: March 10, 2009, 12:20:47 am »

« Last Edit: March 10, 2009, 12:21:11 am by Jean » Report Spam   Logged
Jean
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« Reply #32 on: March 10, 2009, 12:21:46 am »

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« Reply #33 on: March 10, 2009, 12:22:17 am »

the distance between the points being . But the metre-rod is moving with the velocity v relative to K. It therefore follows
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Jean
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« Reply #34 on: March 10, 2009, 12:23:51 am »

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 ,
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Jean
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« Reply #35 on: March 10, 2009, 12:24:38 am »

and for still 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 K' would have been
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Jean
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« Reply #36 on: March 10, 2009, 12:25:34 am »



; 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 x, y, z, 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 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 (x' = 0) of K'. t' = 0 and t' = 1 are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two ticks :

t = 0
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Jean
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« Reply #37 on: March 10, 2009, 01:01:38 am »

and

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Jean
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« Reply #38 on: March 10, 2009, 01:32:32 am »

http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I#Figure_1
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Jean
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« Reply #39 on: May 03, 2009, 05:30:05 am »

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
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Jean
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« Reply #40 on: May 03, 2009, 05:30:20 am »

motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity.
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Jean
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« Reply #41 on: May 03, 2009, 05:30:42 am »

Section 13 - 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 K ' in accordance with the equation
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« Reply #42 on: May 03, 2009, 05:32:06 am »

x ' = wt '
By means of the first and fourth equations of the Galilei transformation we can express x ' and t ' in terms of x and t, and we then obtain
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Jean
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« Reply #43 on: May 03, 2009, 05:32:25 am »

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,
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« Reply #44 on: May 03, 2009, 05:32:41 am »

W = v + w . . . . . . . . (A)
But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation
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