Direction and units of magnitude

I want to clarify the statement in the previous post that “the three dimensions of direction are the same for space and time”. I have made the point that vectors in physics have various units of magnitude but direction is the same for all of them. That is accurate in the sense that directionality is the same concept in all cases. But that does not mean that the particular directions are necessarily the same. They are not.

Here’s a simple example: if someone travels 10 miles east in 17 minutes, then 17 miles north in 10 minutes, the distance direction will be about 60 degrees northeast but the duration direction will be about 30 degrees. Their directions are different but east and north are the same in both cases. This is no different than vectors with other units (velocity, acceleration, etc.). We don’t notice these other differences because we almost always relate them to an underlying distance space.

Multiple dimensions of time have their own “space” as it were, even if directionality is the same concept. That’s why mapping the travel time between cities as map distances proportional to durations results in a “distorted” map. We’re so used to a distance map that anything else looks distorted. But if travel time is more important to us than travel distance (as it often is), a map of travel times is more useful.