Equality of space and time

“How far is it to X from Y?” That everyday question can be answered either by a distance or a duration with a mode of travel (e.g., walking, driving, flying). The interchangeability of a length of space and a length of time leads to two simple conclusions: (1) time has as many dimensions as space does, and (2) space and time are symmetric. In short, space and time are equals in an almost political way: we should not discriminate against one or in favor of the other if possible.

“What is the length of a coastline?” This question is used to point out that the coastline of a landmass does not have a well-defined length. If someone is walking the coastline or examining high-resolution aerial photographs, they will find a longer coastline than someone flying along the coastline or looking at low-resolution photographs. This reinforces that mode of travel or resolution are needed to specify a distance properly. It is similar for travel on land: flying between two cities is likely more direct than driving on a highway network. Which is the correct distance? It depends on the mode of travel.

In physics the “mode of travel” is a light wave — but there may be exceptions. For example, a sound studio or study of whale sounds would be more concerned with the distance that sound travels. Or a phase space might have its own kind of distance. In any case, the mode is there whether it is specified or implicitly understood.

Another aspect of distance and duration is the path that is taken. That may be clear from the mode or there may be alternate routes within a mode. For example, there are many ways to travel from X to Y on a highway network. One may be the shortest distance of travel and another the shortest travel time. So the question of “how far is it?” leads to the response, “with the shortest distance or travel time?”

There is a desire in physics to describe physical laws independently of any observer. If distance and duration aren’t convertible, e.g., by a conversion factor, then how can this be done? The answer is via knowledge of a field, which in transportation terms corresponds to a transportation network. Then physical laws can specify which path will be taken. The key way to do this is via the principle of least action (or stationary action).