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.

More on Newtonian velocity

Here is a big picture story about Newtonian mechanics: The state of the system at all times t > t₀ is explained by the initial conditions of the system at t₀ and the prevalent forces.

But what are the initial conditions? They include position and velocity. But now here is a problem. The standard definition of velocity is that it is the time-derivative of position. But the time-derivative of position at t₀ logically depends not just on the position at t₀ but also on the position at nearby times earlier and later than t₀. That means that the evolution of the system at times t > t₀ is explained by data that includes information on the state of the system at times later than t₀. This seems explanatorily circular and unacceptable.

There is an easy mathematical fix for this. Instead of defining the velocity as the time-derivative position, we define the velocity as the left time-derivative of position: v(t)=lim_{h → 0−}(x(t + h)−x(t))/h. Now the initial conditions at t₀ logically depend only on what happens at t₀ and at earlier times.

This fixed Newtonian story still has a serious problem. Suppose that the system is created at time t₀ so there are no earlier times. The time-derivative at t₀ is then undefined, there is no velocity at t₀, and Newtonian evolution cannot be explained any more.

Here’s another, more abstruse, problem with the fixed Newtonian story. Suppose I am in a region of space with no forces, and I have been sitting for an hour preceding noon in the same place. Then at noon God teleports me two meters to the right along the x-axis, so that at all times before noon my position is x₀ and at noon it is x₀ + 2. Suppose, further, that the teleportation is the only miracle God does. God doesn’t change any other properties of me besides position, and God lets nature take over at all times after noon.

What will happen to me after noon? Well, on the fixed Newtonian story, my velocity at noon is the left-derivative of position, i.e., lim_{h → 0−}(2 − 0)/(0 − h)= + ∞. Since there are no prevailing forces, my acceleration is zero, and so my velocity stays unchanged. Hence, at all times after noon, I have infinite velocity along the x-axis, and so at all times after noon I end up at distance infinity from where I was—which seems to make no sense at all!

So the left-derivative fix of the Newtonian story doesn’t seem right, either, at least in this miracle case.

My preference to both the original Newtonian story and the fixed story is to take velocity (or perhaps momentum) to be a fundamental physical quantity that is not defined as the derivative, or even left derivative, of position.

The rest is technicalities. Maybe we could now take Newton’s Second Law to be:

1. ∂_t⁺v(t)=F/m,

where ∂_t⁺ is the right (!) time-derivative, and add two new laws of nature:

2. ∂_t⁺x(t)=v(t), and

3. x(t) and v(t) are both left (!) continuous.

Now, (2) is an explicit law of nature about the interaction of velocity and position rather than a definition of velocity. On this picture, here’s what happens in the teleportation case. Before noon, my velocity is zero and my position is x₀. Because I supposed that the only thing that God miraculously affects is my position, my velocity is still zero at noon, even though my position is now x₀ + 2. And I think (by the answer to this), laws (1), (2) and (3) ensure that if there are no further miracles, I remain at x₀ + 2 in the absence of external forces. The miraculous teleportation violates (2) and (3) at noon and at no other times.

But of course this is all on the false premise of Newtonian mechanics.

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.

Movement

The following seem quite plausible:

1. It is possible for an object both (a) to have both a first and a last moment of its existence and (b) to be moving at every time during its existence.
2. It is not possible for an object (a) to exist at only one time and yet (b) be moving.

By (2), movement is not an instantaneous property: it is not a property an object has solely in virtue of how it is at one moment. By (1), however, movement is not a property defined in terms of the past and present states of an object (say, "an object moves at a time provided that it is a different location from where it was in the past"), since it can move at the first moment of its existence; nor is it a property defined in terms of the present and future states of the object since it can move at the last moment of its existence.

So what is movement? We could say that an object is moving at time t provided that there are arbitrarily close moments t* at which the object is in a different location. This would make sense of both (1) and (2). But this account falsifies the following intuition:

3. If a ball is thrown vertically into the air, then at the high point of its flight it is not moving.

(If it were moving, would it be moving upward or downward?) For at moments arbitrarily close to that top-point time, the ball is at different locations.

We could try to define movement in terms of there being a well-defined non-zero derivative of the position with respect to time, with the derivative being one-sided at the beginning and end of the object's existence. But then, given continuous time (which we need anyway to have time-derivatives), an object could continuously change location without ever moving, since there are continuous nowhere differentiable functions.

So what should we say? I think it is that the concept of "moving at time t" is underspecified, and specifications of it simply aren't going to cut nature at the joints. Being at different places at different times (at least relative to a reference frame) makes good and fairly precise sense. But moving (or changing) at a time does not. Zeno was right about that much.

Teleporting Zeno's arrow

Here are some plausible theses:

1. Necessarily, an object that is in the same place at time t as it has been for some non-zero period of time prior to t is not moving at t.
2. Necessarily, if an object is at one location at t₁ and at another at t₂ is moving at some time t at one of the two times or between them.
3. It is possible to have continuous time.
4. If it is possible to have continuous time, it is possible to have continuous time and instantaneous teleportation of the following sort: an object is in one place for some time up to and including t₁, then it is instantaneously teleported to a second place where it remains at all times after t₁ up to and including t₂.

These theses are logically incompatible. For, given (3) and (4), suppose we have a world with continuous time and instantaneous teleportation like in (4). Then by (2), this object moves at some time at or between the two times. But at t₁ the object is in the same place as it has been for some time, so by (1) it's not moving. And it's also not moving at any time after t₁ (up to t₂), since at any time after t₁, it's been sitting in the second location for some time.

In some ways, this is an improved version of Zeno's arrow paradox. Zeno had an implausibly strong version of (1) that implied that an object that stayed in the same place for an instant wasn't moving at that instant. That's implausible. But (1) is much weaker. The cost of this weakening is that we need to replace run-of-the-mill movement with teleportation.

Of the premises, I think (4) is the most secure, despite being the most complex. Surely God could teleport things. Here is an argument for (1). Whether an object is in motion at t should not be a future contingent at t. But if the answer to the question whether an object is in motion at t depends on what happens after t, then it would be a future contingent. So it only depends on what happens at or before t. Now if the object has been at the same place for some time prior to t, and is there at t, it should be possible (barring special cases like where God promised that the object will move) for the object to remain there for some time after t. In that case, the object would obviously not be moving at t. But since what happens after t is irrelevant to whether it's moving at t, we conclude that as long as the object has been standing in the same place for some time up to and including t, it's not moving at t.

That leaves (2) and (3). I am inclined to reject both of them myself, though of course the argument only requires one to reject one (given the reasons to believe (1) and (4)). Rejecting (2) seems to go hand-in-hand with seeing motion as something that doesn't happen at times, but only between times (the presentist may well have trouble with this).

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.