Four-Vectors and the Road to Tensors

Abstract

When crossing a rainy street, should we walk or run? Walking takes time, during which we get very wet, but while running reduces our time spent in the rain, it does force us to sweep through the raindrops more quickly, since the rain will be falling at a different angle and thus pounding into the front of our body in a way that it wasn't when we walked slowly. Probably most people would choose to run as fast as possible every time, and that's what we'll find, too, for the case of vertical rain and a simple "boxy" model of a human in the coming pages. (For other angles of rainfall, such as when it ploughs into our back, and for other bodily angles of running, we might adopt a different strategy. But the case of vertical rain will suffice for our needs here.) The problem of just how fast we should walk in the rain is interesting because it introduces two quantities that become fused into one when we take a relativistic viewpoint. These quantities are number density, which is how many raindrops there are in a unit volume (the more there are, the wetter we are going to get), and flux density, which is how many of those drops are crossing a unit area in a unit time (pouring rain delivers more water than light rain). That these two types of density become fused into one might only be of passing interest if it were not for the fact that this new quantity, a four-vector-as well as the bigger family of which it is a part, called tensors-also appears in a great many other diverse areas of physics. Understanding how four-vectors arise makes it possible to see how useful they are in those other areas, too. So we'll look more closely at these number and flux densities. This question of how fast to run in the rain is a great example of the utility of taking a whimsical problem to an extreme-in this case, the relativistic limit. Despite the apparent irrelevance of taking such a limit, we find that when we do, this fusion of two quantities into one happens, and we learn something interesting about Nature. Let's do just that.