One surprising result is that an object’s velocity and its inverse, which I’m calling legerity, may have different directions. Here’s what I mean:

Suppose someone drives 30 miles East in 50 minutes, then turns North and drives 40 miles in 50 minutes. Overall, they have driven 70 miles in 100 minutes but as the crow flies they ended up 50 miles from where they started (30^{2} + 40^{2} = 50^{2}). And a crow flying at the average of the speeds would have taken only 71 minutes to get there: 30/50 = 36 mph, 40/50 = 48 mph, 50/((36+48)/2) = 71.

It may seem strange at first, but the direction of this hypothetical crow is different for the space, time, and velocity vectors. The reason is that their units are different, and so their vector spaces are different. The meaning of direction is different with different units for the magnitude.

But what about the inverse velocity, the legerity – does that have the same direction as the velocity? Yes, in the sense that the same turns are there but no, in the sense that the units are different. As curves they have different parameterizations.

There’s another reason they’re different: because the direction of the velocity is that of the distance interval (relative to a unit of time), but the direction of the legerity is that of the duration interval (relative to a unit of space).

There’s really not much new here. We usually take the spatial directions to be the only ones that matter, which is fine, but we could as well take the temporal directions as the ones that matter.