More on velocity

From time to time I’ve been playing with the question whether velocity just is rate of change of position over time in a philosophical elaboration of classical mechanics.

Here’s a thought. It seems that how much kinetic energy an object x has at time t (relative to a frame F, if we like) is a feature of the object at time t. But if velocity is rate of change of position over time, and velocity (together with mass) grounds kinetic energy as per E = m|v|²/2, then kinetic energy at t is a feature of how the object is at time and at nearby times.

This argument suggests that we should take velocity as a primitive property of an object, and then take it that by a law of nature velocity causes a rate of change of position: dx/dt = v.

Alternately, though, we might say that momentum and mass ground kinetic energy as per E = |p|²/2m, and momentum is not grounded in velocity. Instead, on classical mechanics, perhaps we have an additional law of nature according to which momentum causes a rate of change of position over time, which rate of change is velocity: v = dx/dt = p/m.

But in any case, it seems we probably shouldn’t both say that momentum is grounded in velocity and that velocity is nothing but rate of change of position over time.

A philosophical advantage of quantum mechanics over Newtonian mechanics

We often talk as if quantum mechanics were philosophically much more puzzling than classical mechanics. But there is also a deep philosophical puzzle about Newtonian mechanics as originally formulated—the puzzle of velocities—which disappears on quantum mechanics.

The puzzle of velocities is this. To give a causal explanation of a Newtonian system’s behavior, we have to give the initial conditions for that system. These initial conditions have to include the positions and velocities (or momenta) of all the bodies in the system.

To see why this is puzzling, let’s imagine that t₀ is the first moment of the universe’s existence. Then the conditions at t₀ explain how things are at all times t > t₀. But how can there be velocities at t₀? A velocity is a rate of change of position over time. But if t₀ is the first moment of the universe’s existence, there were no earlier positions. Granted, there are later positions. But these later positions, given Newtonian dynamics, depend on the velocities at t₀ and hence cannot help determine what these velocities are.

One might try to solve this by saying that Newtonian dynamics implies that there cannot be a first moment of physical reality, that physical reality has to have always existed or that it exists on an interval of times open at the lower end. On either option, then, Newtonian dynamics would have to be committed to an infinite temporal regress, and that seems implausible.

Another solution would be to make velocities (or, more elegantly, momenta) equally primitive with positions (indeed, some mathematical formulations will do that). On this view, that the velocity is the rate of change of position would no longer be a definition but a law of nature. This increases the number of laws of nature and the fundamental properties of things. And if it is a mere law of nature that velocity is the rate of change of position, then it would be metaphysically possible, by a miracle, that an object standing perfectly still for days would nonetheless have a high velocity. If that seems wrong, we could just introduce a technical term, say “movement propensity” (that’s kind of what “momentum” is), in place of “velocity”, and it would sound better. However, anyway, while the resulting theory would be mathematically equivalent to Newton’s, and it would solve the velocity problem, it would be a metaphysically different theory, since it would have different fundamental properties.

On the other hand, the whole problem is absent in quantum mechanics. The Schroedinger equation determines the values of the wavefunction at times later than t₀ simply on the basis of the values of the wavefunction at t₀. Granted, the cost is that we have a wavefunction instead of just positions. And in a way it is really a variant of the making-momenta-primitive solution to the Newtonian problem, because the wavefunction encodes all the information on positions and momenta.

Velocity and teleportation

Suppose a rock is flying through the air northward, and God miraculously and instantaneously teleports the rock, without changing any of its intrinsic properties other than perhaps position, one meter to the west. Will the rock continue flying northward due to inertia?

If velocity is defined as the rate of change of position, then no. For the rate of change of position is now westward and the magnitude is one meter divided by zero seconds, i.e., infinite. So we cannot expect inertia to propel the rock northward any more. In fact, at this point physics would break down, since the motion of an object with infinite velocity cannot be predicted.

But if velocity (or perhaps momentum) is an intrinsic feature that is logically independent of position, and it is merely a law of physics that the rate of change of position equals the velocity, then even after the miraculous teleportation, the rock will have a northward velocity, and hence by inertia will continue moving northward.

I find the second option to be the more intuitive one. Here is an argument for it. In the ordinary course of physics, the causal impact of physical events at times prior to t₁ on physical events after t₁ is fully mediated by the physical state of things at t₁. Hence whether an object moves after time t₁ must depend on its state at t₁, and only indirectly on its state prior to t₁. But if velocity is the rate of change of position, then whether an object moves via inertia after t₁ would depend on the position of the object prior to t₁ as well as at t₁. So velocity is not the rate of change of position, but rather a quality that it makes sense to attribute to an object just in virtue of how it is at one time.

This would have the very interesting consequence that it is logically possible for an object to have non-zero velocity while not moving: God could just constantly prevent it from moving without changing its velocity.

Zeno's arrow, Newtonian mechanics and velocity

Start with Zeno's paradox of the arrow. Zeno notes that over every instant of time t₀, an arrow occupies one and the same spatial location. But an object that occupies one and the same spatial location over a time is not moving at that time. (One might want to refine this to handle a spinning sphere, but that's an exercise to the reader.) So the arrow is not moving at t₀. But the same argument applies to every time, so the arrow is not moving, indeed cannot move.

Here's a way to, ahem, sharpen The Arrow. Suppose in our world we have an arrow moving at t₀. Imagine a world w* where the arrow comes into existence at time t₀, in exactly the same state as it actually has at t₀, and ceases to exist right after t₀. At w* the arrow only ever occupies one position—the one it has at t₀. Something that only ever occupies one position never moves (subject to refinements about spinning spheres and the like). So at w* the arrow never moves, and in particular doesn't move at t₀. But in the actual world, the arrow is in the same state at t₀ as it is at w* at that time. So in the actual world, the arrow doesn't move at t₀.

A pretty standard response to The Arrow is that movement is not a function of how an object is at any particular time, it is a function of how, and more precisely where, an object is at multiple times. The velocity of an object at t₀ is the limit of (x(t₀+h)−x(t))/h as h goes to zero, where x(t) is the position at t, and hence the velocity at t₀ depends on both x(t₀) and on x(t₀+h) for small h.

Now consider a problem involving Newtonian mechanics. Suppose, contrary to fact, that Newtonian physics is correct.

Then how an object will behave at times t>t₀ depends on both the object's position at t₀ and on the object's velocity at t₀. This is basically because of inertia. The forces give rise to a change in velocity, i.e., the acceleration, rather than directly to a change in position: F(t)=dv(t)/dt.

Now here is the puzzle. Start with this plausible thought about how the past affects the future: it does so by means of the present as an intermediary. The Cold War continues to affect geopolitics tomorrow. How? Not by reaching out from the past across a temporal gap, but simply by means of our present memories of the Cold War and the present effects of it. This is a version of the Markov property: how a process will behave in the future depends solely on how it is now. Thus, it seems:

1. What happens at times after t₀ depends on what happens at time t₀, and only depends on what happens at times prior to t₀ by the mediation of what happens at time t₀.

But on Newtonian mechanics, how an object will move after time t₀ depends on its velocity at t₀. This velocity is defined in terms of where the object is at t₀ and where it is at times close to t₀. An initial problem is that it also depends on where the object is at times later than t₀. This problem can be removed. We can define the velocity here solely in terms of times less than t₀, as lim_h→0−(x(t+h)−x(t))/h, i.e., where we take the limit only over negative values of h.[note 1] But it still remains the case that the velocity at t₀ is defined in terms of where the object is at times prior to t₀, and so how the obejct wil behave at times after t₀ depends on what happens at times prior t₀ and not just on what happens at t₀, contrary to (1).

Here's another way to put the puzzle. Imagine that God creates a Newtonian world that starts at t₀. Then in order that the mechanics of the world get off the ground, the objects in the world must have a velocity at t₀. But any velocity they have at t₀ could only depend on how the world is after t₀, and that just won't do.

Here is a potential move. Take both position and velocity to be fundamental quantities. Then how an object behaves after time t₀ depends on the object's fundamental properties at t₀, including its velocity then. The fact that v(t₀)=lim_h→0(x(t₀+h)−x(t₀))/h, at least at times t₀ not on the boundary of the time sequence, now becomes a law of nature rather than definitional.

But this reneges on our solution to The Arrow. The point of that solution was that velocity is not just a matter of how an object is at one time. Here's one way to make the problematic nature of the present suggestion vivid, along the lines of my Sharpened Arrow. Suppose that the arrow is moving at t₀ with non-zero velocity. Imagine a world w* just like ours at t₀ but does not have any times other than t₀.[note 2] Then the arrow has a non-zero velocity at t₀ at w*, even though it is always at exactly the same position. And that sure seems absurd.

The more physically informed reader may have been tempted to scoff a bit as I talked of velocity as fundamental. Of course, there is a standard move in the close vicinity of the one I made, and that is not to take velocity as fundamental, but to take momentum as fundamental. If we make that move, then we can take it to be a matter of physical law that mlim_h→0(x(t₀+h)−x(t₀))/h=p(t₀), where p(t) is the momentum at t.

We still need to embrace the conclusion that an object could fail to ever move and yet at have a momentum (the conclusion comes from arguments like the Sharpened Arrow). But perhaps this conclusion only seems absurd to us non-physicists because we were early on in our education told that momentum is mass times velocity as if that were a definition. But that is definitely not a definition in quantum mechanics. On the suggestion that in Newtonian mechanics we take momentum as fundamental, a suggestion that some formalisms accept, we really should take the fact that momentum is the product of mass and velocity (where velocity is defined in terms of position) to be a law of nature, or a consequence of a law of nature, rather than a definitional truth.

Still, the down-side of this way of proceeding is that we had to multiply fundamental quantities—instead of just position being fundamental, now position and momentum are—and add a new law of nature, namely that momentum is the product of mass and velocity (i.e., of mass and the rate of change of position).

I think something is to be said for a different solution, and that is to reject (1). Then momentum can be a defined quantity—the product of mass and velocity. Granted, the dynamics now has non-Markovian cross-time dependencies. But that's fine. (I have a feeling that this move is a little more friendly to eternalism than to presentism.) If we take this route, then we have another reason to embrace Norton's conclusion that Newtonian mechanics is not always deterministic. For if a Newtonian world had a beginning time t₀, as in the example involving God creating a Newtonian world, then how the world is at and prior to t₀ will not determine how the world will behave at later times. God would have to bring about the initial movements of the objects, and not just the initial state as such.

Of course, this may all kind of seem to be a silly exercise, since Newtonian physics is false. But it is interesting to think what it would be like if Newtonian physics were true. Moreover, if there are possible worlds where Newtonian physics is true, the above line of thought might be thought to give one some reason to think that (1) is not a necessary truth, and hence give one some reason to think that there could be causation across temporal gaps, which is an interesting and substantive conclusion. Furthermore, the above line of thought also shows how even without thinking about formalisms like Hamiltonian mechanics one might be motivated to take momentum to be a fundamental quantity.

And so Zeno's Arrow continues to be interesting.