## Einstein 1920: Rods and Clocks

##### Reference: Einstein’s 1920 Book

This paper presents Part 1, Chapter 12 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

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## The Behaviour of Measuring-Rods and Clocks in Motion

I 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

the distance between the points being √(1 – v2/c2).

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 √(1 – v2/c2) 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 √(1 – v2/c2) = 0, 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.

For Lorentz transformation to be applicable c must be infinitely large compared to v. Therefore, a finite v can never approach an infinite c. Einstein ignores this assumption in his interpretation of Lorentz transformation equations.

Based on Michelson-Morley’s experiment; the speed of light of 3 x 108 meters/second was not affected by the velocity of the earth, which is 3 x 104 meters/second relative to the sun. This velocity of the earth is 1/10,000 of the speed of light. Therefore, the Lorentz transformation is good for v/c ratios of 1/10,000 or less. The “v/c ratios” of most material bodies in the universe are of this order or less. Lorentz transformation may not be valid for v/c ratios that are significantly greater than 1/10,000 and closer to 1.

Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these become meaningless if we choose values of v greater than c.

Equations of Lorenz transformation may become meaningless much before v even gets closer to c, based on the reasoning given above.

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 √(1 – v2/c2); this is quite in accordance with the principle of relativity which forms the basis of our considerations.

When two inertial systems are moving at velocity v relative to each other Einstein’s interpretation predicts that the rod shall shrink in either system for the observer in the other system. That is inconsistent.

As shown earlier, the uniform velocity of an inertial system is inversely proportional to its inertia. The rod shall actually be shrinking in the system of greater inertia, and this would not depend on any observer.

Let K’ be moving faster than K by velocity v. Then K is moving slower than K’, by the same velocity. The inertia of K’ is less than K. Therefore, the rod in K will shrink relative to the rod in K’. This shrinking shall depend on inertia of the systems and not on “observers”.

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 Galilei transformation we should not have obtained a contraction of the rod as a consequence of its motion.

I believe that we should have obtained a contraction of the rod with increasing inertia, even under Galilei transformation. It is possible to show that mathematically.

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 and t = 1/√(1 – v2/c2).

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 1/√(1 – v2/c2) 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.

It must be noted that K’ can gain velocity over K only when it is pushed. This increase in velocity is then balanced by decrease in inertia. Therefore, K’ moving at a greater velocity v with respect to K is equivalent to K’ decreasing in inertia. This then leads to increase in length and time duration in K’ with respect to K (objectively).

Decrease in inertia is similar to decrease in quantization in the energy spectrum. As we go down the spectrum, velocity wavelength and period increase.

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