iSoul In the beginning is reality.

Time and length clocks

An ordinary clock measures time. But what if a clock shows a length instead of a duration? For example, an old-fashioned clock with a face and hands points radially with a constant angular velocity. What if the output came from the circumference instead? A one o’clock reading would be equivalent to 1/12th of the circumference or πr/6, where r is the radius, in units of length.

That would be using length to measure time based on a particular clock. That means we can convert time and space based on this clock. Interesting, perhaps, but not very useful since it’s all relative to one particular clock. What if we get everyone to agree on a standard clock, that is, a standard radius for all clocks? Then everyone could specify time by giving a length: the length along the circumference of a circular clock.

The point is that clocks can be based on angular or linear movement, so length and duration can be interconverted based on a standard speed or angular velocity. The circumferential movement is like a wheel on an axle so we could say there are axial (length) and radial (duration) clocks, just as there are radial (polar) and axial (circumferential) vectors.

[An axial vector (or pseudovector) is a quantity that transforms like a (polar) vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. This is as opposed to a true or polar vector, which on reflection matches its mirror image.]

Angular movement can be used to measure length as well. A measuring wheel traveling at a constant angular velocity along the ground measures the length and duration between ends of an object or between objects.

What really distinguishes a clock is the fact that it ordinarily doesn’t stop, whether the measure is angular or linear. What distinguishes a ruler is the fact that it ordinarily does stop or has a non-standard movement that isn’t suitable for a standard measure of duration. So we have two kinds of rulers: angular and linear, and two kinds of clocks: angular and linear.

What then is time? Time is a measure associated with constant speed or angular velocity. What is space? Space is a measure that need not have a constant speed or angular velocity. But if there is a constant speed or angular velocity, then the same measure may be used for either time or space.

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