Velocity

An alternate title for this note is “the concept of momentum”, suggesting that it’s an unpartisan inquiry into the genealogy and prospects of this concept. In case you were thinking “who cares about momentum?” the answer is “all of physics from at least the late middle ages until today”. In short, the canonical presentation of a physical theory takes two variables to be basic: position and momentum.

We should, however, come clean that our motivation is somewhat partisan. Niels Bohr urged physicists not to surrender the powerful tool for reasoning that is encapsulated in the principle of the conservation of momentum. Insofar as he would have been willing to throw around the world “real”, Bohr would have insisted that momentum is real. What were his reasons for saying this, and should we listen to him? (And, ultimately, what difference does it make?)

In more simple, but technical, terms (see beables): Bohr’s interpretation of quantum mechanics would have different quantities being definite in different contexts. There are some contexts in which position is definite, and other contexts in which momentum is definite. In contrast, Bohmian mechanics demands that position is always definite — and that momentum, if we are to talk about it at all, is a derivative concept. Did Bohr give good reasons, then, to think that the Bohmian picture misses out on important features of reality?¹

What velocity could (not) be

Either velocity is fundamental, or it is not (see (Arntzenius 2000; Lange 2005)). And isn’t it pretty obvious that velocity is not fundamental, since it can be defined in terms of positions over time? In logicians’ lingo: velocity is explicitly definable in terms of position and time (hence reducible to them). In philosophers’ lingo: velocity supervenes on positions over an arbitrarily short period of time.

No, the view that velocity is defined to be change of place over time mistakenly assumes that it makes sense to talk about being in the same place at two different times. Setting aside deep metaphysical questions (which seem never to get settled), no theory in modern physics assumes a primitive notion of identity between different places at different times. To see this clearly, consider the case of “Galilean spacetime”, which is all the spacetime structure we need to do Newtonian particle mechanics: Galilean spacetime is assumed to be composed of an infinite number of copies of a three-dimensional Euclidean space \(E_t\), indexed by a time parameter \(t\in\mathbb{R}\).² Most importantly, there is no sense to be made of writing “\(a=b\)” or “\(a\neq b\)” when \(a\) is a point in \(E_t\) and \(b\) is a point in \(E_{t'}\) for \(t\neq t'\). Since Galilean spacetime “washes out” the identity of points at different times, there is no sense in which an object can be at the same (or at a different) place at different times.

To summarize: in post-Galilean physics, “change of position” is not well-defined, and so velocity cannot be defined as change of position with respect to time. Does it follow, then, that velocity is fundamental? No, not necessarily. What is definable in Galilean spacetime is the distance between two objects; and velocity of one object relative to another can be defined in terms of changes in the distance between them. In one important sense, the basic facts could be relative distances between objects, and then relative velocities could be defined in terms of those distances.