*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.*

*The heading below is linked to
the original materials.*

.

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

I place a metre-rod in the ** x’-axis** of

**in such a manner that one end (the beginning) coincides with the point**

*K’***, whilst the other end (the end of the rod) coincides with the point**

*x’ = 0***. What is the length of the metre-rod relatively to the system**

*x’ = 1***? 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***K**at a particular time

**of the system**

*t***. By means of the first equation of the Lorentz transformation the values of these two points at the time**

*K***can be shown to be**

*t = 0*the
distance between the points being **√(1 –
v ^{2}/c^{2}).**

But
the metre-rod is moving with the velocity ** v** relative to

**. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity**

*K***is**

*v***√(**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

*1 – v*)^{2}/c^{2}**we should have**

*v = c***√(**= 0, and for still greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity

*1 – v*)^{2}/c^{2}**plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.**

*c**Only an
infinite velocity shall be the same in all inertial systems. A very large
velocity, such as, that of light, may be the same for material inertial systems
whose velocities are much smaller than that of light. *

*Based
on Michelson-Morley’s experiment, the speed of light of 3 x 10 ^{8}
meters/second was not affected by the velocity of the earth, which is 3 x 10^{4}
meters/second relative to the sun. This velocity of the earth is 1/10,000 of
the speed of light. The “v/c ratios” of most material bodies in the universe
are of this order or less. Therefore, the Lorentz transformation is good for v/c
ratios of 1/10,000 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

**greater than**

*v***.**

*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

**would have been**

*K’***√(**this is quite in accordance with the principle of relativity which forms the basis of our considerations.

*1 – v*);^{2}/c^{2}*As
shown earlier, the motion of a body in space decrease as its inertia increases.
If K’ is moving relative to K such that inertia of K’ is increasing and the
rod in K’ is shrinking with respect to K,
then the opposite shall occur in K with respect to K’. The inertia of K should
be decreasing and the rod in K should be expanding. Einstein’s theory predicts
the rod to be shrinking both ways. There is an inconsistency here.*

*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’***and**

*t’ = 0***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 =**

*t’ = 1*

*1/√*(*1 – v*).^{2}/c^{2}As
judged from ** K**, the clock is moving with the velocity

**; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but**

*v***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**

*1/√*(*1 – v*)^{2}/c^{2}**plays the part of an unattainable limiting velocity.**

*c**It must
be noted that K’ can gain velocity over K only when it is pushed. This applied
force is then balanced by increase in inertia. Therefore, K’ moving at a
velocity v with respect to K is equivalent to K’ increasing in inertia. This
then leads to length contraction and time delay.*

*Increase
in inertia is similar to quantization in the radiation spectrum. In the latter
case this length contraction and time delay Is more obvious. *

.