The Relativity of Lengths and Times

The following reflexions are based on:

• the principle of relativity
• the principle of the constant speed of light

These 2 principles I define as follows:

1. Assume there are 2 systems of co-ordinates in uniform translatory motion. These systems change in their state. These changes do not affect the laws that facilitate those changes.

2. Any ray of light moves in the “stationary” system of co-ordinates with the speed c, whether the ray is from a stationary or a moving body. Hence:

…

where time interval is to be taken in the sense of the definition in § 1.

Let there be a stationary rigid rod with length l as measured by a measuring-rod which is also stationary.

The following are then imparted to the rod:

• the axis of the rod lying along the axis of x of the stationary system of co-ordinates
• a uniform motion of parallel translation with speed v along the axis x in the direction of increasing x

What is the length of the moving rod?

I imagine its length to be ascertained by 2 operations:

1. The observer moves together with the measuring-rod and the rod to be measured.

He measures the length of the rod by superposing the measuring-rod, in the same way as if all three were at rest.

2. There are stationary clocks in the stationary system.

These synchronize in accordance with § 1.

The observer ascertains at what points of the stationary system the 2 ends of the rod to be measured are located at a definite time.

“The length of the rod” is the distance between these 2 points measured by the measuring-rod already employed, which in this case is at rest.

According to Relativity, the length measured by this operation:

• in the moving system must be equal to the length l of the stationary rod
• in the stationary system as the length of the moving rod
  • Based on my 2 principles, this should differ from l.

Current kinematics* tacitly assumes that:

• the lengths measured by these 2 operations are precisely equal
• a moving rigid body at the epoch t in geometrical respects can be perfectly represented by the same body at rest in a definite position

At the 2 ends A and B of the rod, place clocks which synchronize with the clocks of the stationary system.

• Their indications correspond at any instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”

Each clock has a moving observer.

These observers apply to both clocks the criterion established in § 1 for the synchronization of 2 clocks.

Let a ray of light depart from A at the time4 tA ,

It is reflected at B at the time tB, and reaches A again at the time t0A.

Because of the constant speed of light we find that:

…

4 “Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated at the place under discussion.”

where rAB denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus find that the 2 clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.

So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.