# The Behaviour Of Measuring–rods And Clocks In Motion

I place a metre-rod in the x’-axis of moving `K'`

in a way that its beginning coincides with the point `x' = 0`

while its end coincides with the point `x' = 1`

.

What is the length of the metre rod relative to the non-moving `K`

?

This is answered by asking where the beginning and end of the rod lie with respect to non-moving `K`

at a time `t`

of non-moving `K`

. The first equation of the Lorentz transformation shows that the values of these two points at the time `t = 0`

is:

```
x (beginning of rod) = 0 √ (1 − (v2/c2)
x (end of rod) = 1 √ (1 − (v2/c2)
```

the distance between the points being=

```
√ (1-(v2/c2))
```

But the metre-rod is moving with the velocity `v`

relative to non-moving `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
```

For faster speeds, the square-root becomes imaginary. This means 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**.

This `c`

follows from the equations of the Lorentz transformation. These become meaningless if `v`

becomes greater than `c`

.

If, on the contrary, we had considered a metre-rod at rest in the x-axis with respect to non-moving `K`

, then the length of the rod as viewed from `K'`

would have been `1 − v 2 c 2`

. This is in accordance with my principle of relativity.

The magnitudes `x, y, z, t`

, are merely the results of measurements of the measuring-rods and clocks. If we used Galilei transformation, the rod would not contract 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 1st and 4th equations of the Lorentz transformation give for these two ticks= `t = 0`

and

```
t = 1 / √ (1-(v2/c2))
```

As judged from non-moving `K`

, the clock is moving with the velocity `v`

as judged from this viewpoint, the time which elapses between two strokes of the clock is not 1 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.