A new geometry for space and time

This blog has described how as the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. One way of looking at this is as a map with the distance and duration given between places, such as this from the Interstate Drive Times and Distances:

There are two numbers for each leg or link in the map; one number in red for the distance in miles and another in blue for the time in hours and minutes. The durations are not proportional to the distance, which reflects the differing local conditions and topography.

It is common to adopt a speed that represents the typical speed for the mode of travel, which I’m calling the modal speed (or rate). This serves various purposes, one of which is for pre-trip planning to estimate the travel time to a destination. It can also serve as the conversion of distance and duration for the mode, much as the speed of light serves for relativity.

Divergence from the modal rate may be represented through a third dimension (z), similar to a topographic map. The modal rate then would be represented by a flat topography. A link with greater distance than the modal one would be like a positive z coordinate, and a link with less than the modal one would be like a negative z coordinate. The resultant link would be determined by the Pythagorean theorem for positive values of z, and by a hyperbolic version (as if z were imaginary) for negative values of z.

That is, |A|² = x² + y² + z² if z > 0 (Pythagorean), and |A|² = x² + y² – z² if z < 0 (hyperbolic). For positive values of z, the magnitude of the resultant link is greater than unity, that is, greater than the modal rate. But for negative values of z, the magnitude of the resultant link is less than unity, that is, less than the modal rate.

The result may be mapped like a contour or isoline map, except that negative values have a hyperbolic geometry. The simplest way to see this is to take Δt = 1 and Δr = > 1, which uses the Pythagorean theorem. If either the time interval changes to Δt < 1 or the space interval is reduced to Δr < 1, the hyperbolic version is used for the link.