Analogue clock analyzed

This post is related to a previous post here.

Consider the dial and one hand of an analogue clock: there are two circular “axes” of reference. One is the circle of the dial, and the other is the circle of the hand (other hands point to the same circle but at different rates):

The dial and hand are in uniform angular motion relative to one another. The dial moves counter-clockwise relative to the hand. The hand moves clockwise relative to the dial. The observer is at rest relative to the dial.

These circles are like coordinate axes, and they could be replaced with lines in uniform linear motion. The clock is a mechanism, a system, with the output the time. Two frames of reference, represented by two circles, are required for a clock.

The hand points to the time of the observer’s frame of reference. This is strange because a characteristic of the observer’s frame of reference is in motion relative to it. There is a kind of inversion here: the observer’s frame is in a sense not the frame at rest relative to the observer but the frame in uniform motion relative to it.

The observer can say, “the moving hand points to my time.” Compare saying, “the moving train is my train,” even though the observer is not on it, which would be like missing the train. But the moving hand indicates the time for the frame at rest, the entire frame at rest.

This analysis leads to the development of a frame of reference system for space and time.