Time for spacetime

Consider a worldline in one dimension of space and one dimension of time that tracks the position of a point that moves from position 20 to 10 to 15. This could represent the movement of a point in the E-W dimension. Another worldline could track the movement of the same point in the N-S dimension. All would agree that the two diagrams together represent two dimensions of space. But the case with time is completely analogous; the two diagrams together represent two dimensions of time.

To see this consider someone traveling on city streets arrayed in a grid oriented N-S and E-W with two stop watches. To keep it simple say they are traveling only north and east. They use one stop watch when they travel north and the other stop watch when they travel east. So we would have two travel times: one going north and the other going east, which would correspond to two dimensions of travel distance. As we would all agree that the travel distances are associated with two dimensions, we should agree that the two travel times are associated with two dimensions.

One objection might be that the dimensions here are all “spatial” rather than “temporal”. But the travel times are measured in units of time independently of the travel distances (which might not even be known). It seems arbitrary to say that there are two dimensions in the case of travel distances but not in the case of travel times.

There is a tendency to associate dimensionality with space rather than time (although one strange dimension is granted to time). But dimensionality is a mathematical concept that can be applied to many things, as multivariate analysis shows. As we apply concepts of scalar and vector to spatial quantities, so we can apply these to temporal quantities. Both space and time are multi-dimensional.