iSoul In the beginning is reality

# Modes and measures

What is the “distance” between two point events? That would include the length in both space and time. The measurement of the length of time between events depends on the mode of travel between them. For example, the time between leaving one’s residence and arriving at work depends on how one commutes. If the trip is by a slower mode such as walking, it will take longer than the same trip by a faster mode such as rapid transit. Another factor is the route that will vary by mode.

The situation is similar with space. What is the distance between two points? It depends on how the measurement is taken. The coastline paradox is the best-known example of this. Rigid measuring rods of different lengths will take different measurements. The length of a coastline depends on the unit of measure. The diagonal paradox shows another way the method of measurement makes a difference. In the commuting example, the distance of the commute depends on the route taken.

In order to have consistent measurement of space or time there must be a standard of measurement. For distance this is the shortest path using the fastest mode: light. Laser technology has made this method convenient and reliable. Can the light standard be applied to time, too? No. A laser would not help to measure a commute because a ray of light would simply go faster than any commute and arrive early.

The observation of an event is just-in-time so there’s no waiting time involved. So the motion from one event to another must meet it just as it happens. That means the shortest distance velocity is less than the speed of light in most cases.

Consider the 1+1 dimensions of time (t) and space (r) with the speed of light set to one, as above. The “distance” between two point events is the space-time distance between them. This depends on whether equal distances make circles or hyperbolas. For space and time, distance is hyperbolic since the speed of light is a barrier. So d² = Δt² – Δr².