iSoul In the beginning is reality.

Space as time and time as space

Galileo parabola

Galileo used the length of uniform motion as a measure of duration, i.e., time (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914):

Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time-interval bc finds itself at the point i. p.199

Without getting into the details of the Figure 108, notice the shift of language: “the body moves from b to c” [i.e., a length-interval], then “the time-interval bc“. Galileo uses a length interval to measure a time-interval, which is justified since the motion is “with uniform speed”.

Let there be a ball dropped out the window by a passenger on a train in uniform motion. Consider the following four scenarios, in which the length or duration of a uniform motion is measured: (1) looking down above the moving ball, measuring the length of fall; (2) looking down above the moving ball, measuring the (uniform) duration of fall; (3) looking from the side, measuring the length of motion in two dimensions; and (4) looking from the side, measuring the (uniform) duration of motion in two dimensions.

Vertical axisVertical axis with timeTwo spatial axesTwo temporal axes

The point to make here is that a length-interval and a time-interval can be interchanged if the motion is uniform. That is exactly the function of a clock: to provide a standard time-interval for a length-interval. It seems the change from length to time is only a change of units.

4 dimensions

One could even have 2D space and 2D time, as in the figure above. Imagine a planar Earth with the Sun moving over in semi-circular fashion. In this context, space has two horizontal dimensions, and time has vertical and parametric dimensions. One could diagram this as a horizontal circle intersected by a vertical circle. Equations for spatial and temporal circular motion would be used. This 2 + 2 form of space and time is like an isochron map (see here) or a flat (2D) earth with celestial (2D) time.

But don’t space and time have different characters? Certainly, relativity recognizes this by using opposite signs for spatial and temporal coordinates in the invariant equation. The difference between space and time is also between the independent and dependent quantities.

Then switching from spatial to temporal coordinates inverts the independent and dependent coordinates. A temporal coordinate as the independent variable is changed into a spatial coordinate as the independent variable, or vice versa.

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