Section 23

# Behaviour Of Clocks And Measuring-rods On A Rotating Body Of Reference

March 18, 2022

I have purposely refrained from speaking about the physical interpretation of space- and time-data in General Relativity (GR) because it would be very complicated to explain.

Assuming a space-time domain with no gravitational field relative to a reference-body K whose state of motion has been suitably chosen.

• `K` is then a Galileian reference-body as regards the domain considered
• The results of Special Relativity (SR) hold relative to `K`

The same domain referred to a 2nd body of reference `K'`, which is rotating uniformly with respect to `K`.

`K'` is a plane circular disc rotating uniformly in its own plane around its centre.

An observer who is sitting eccentrically on the disc `K'` feels a force acting outwards in a radial direction.

An observer at rest in the original reference-body `K` will see this as the effect of inertia (centrifugal force).

But the observer on the disc may regard his disc as a reference-body which is “at rest” according to GR.

The force acting on himself, and in fact on all other bodies which are at rest relative to the disc, he regards as the effect of a gravitational field.

The space-distribution of this gravitational field is of a kind that would not be possible on Newton’s theory of gravitation*.

*Einstein Note= The field disappears at the centre of the disc and increases proportionally to the distance from the centre as we proceed outwards.

But since the observer believes in GR, he believes that a general law of gravitation can be formulated — a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself.

The observer performs experiments on his circular disc with clocks and measuring-rods. He intends to arrive at exact definitions for time- and space-data with reference to the circular disc K’ based on his observations.

He places one of two identically constructed clocks=

• one at the centre of the circular disc
• the other on the edge of the disc, so that they are at rest relative to it

Do both clocks go at the same rate from the standpoint of the non-rotating Galileian reference-body `K`?

From this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to `K` because of the rotation.

According to Section 12 [time dilation and length contraction], the latter clock ticks permanently slower than the clock at the centre of the circular disc (from `K`).

The same effect would be noted by an observer sitting alongside his clock at the centre of the circular disc.

Thus on our circular disc and in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest).

This is why it is impossible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference.

A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case. But I do not wish to go any farther into this question.

Moreover, at this stage, the definition of the space co-ordinates also presents unsurmountable difficulties. If the observer applies his standard measuring-rod (a rod which is short as compared with the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than 1, since, according to Section 12, moving bodies suffer a shortening in the direction of the motion.

On the other hand, the measuring-rod will not experience a shortening in length, as judged from `K`, if it is applied to the disc in the direction of the radius.

If, then, the observer first measures the circumference of the disc with his measuring-rod and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the familiar number `π = 3.14..`, but a larger number*, whereas of course, for a disc which is at rest with respect to `K`, this operation would yield π exactly.

*Einstein Note: We have to use the Galileian (non-rotating) system `K` as reference-body, since we may only assume the validity of the results of Special Relativity relative to `K` (relative to `K'` a gravitational field prevails).

This proves that Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length 1 to the rod in all positions and in every orientation. Hence the idea of a straight line also loses its meaning.

We are therefore not in a position to define exactly the co-ordinates `x, y, z` relative to the disc by means of the method used in discussing the special theory, and as long as the co-ordinates and times of events have not been defined we cannot assign an exact meaning to the natural laws in which these occur.

Thus, all our previous conclusions based on general relativity would appear to be called in question.

In reality, we must make a subtle detour in order to be able to apply the postulate of general relativity exactly.