Wednesday, April 11, 2012

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 limh→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)=limh→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 mlimh→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.


Heath White said...

I am out of my depth here, but that never stopped me before. :-)

It seems to me that velocity is change in position over time. Consequently, velocity requires change and change requires time. There is no such thing, properly speaking, as velocity without duration; we can figure out velocity at an instant only in a larger context.

It is like (in fact, exactly like) noting that each point on a curve has a slope, and then asking what the slope of a single point, sans curve, would be. There is no answer to that question, because slope-at-a-point presumes the existence of a larger curve.

I actually don't know how this maps on to your larger discussion.

Dagmara Lizlovs said...

I'm out of my depth too on this one. However, today I read a news article on a physicist who used wrote a four page article using some similar arguements to get out of a ticket for going through a stop sign. Read here:

I wonder if Dmitri Krioukov, physicist from UCSD, is on to something. The judge seems to agree with him. Since Krioukov's paper is not available to us. I wonder if Zeno's arrow could be modified to succesfully contest moving violations in court?

Dagmara Lizlovs said...

Case I:

"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 . . . 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." No wonder I was unable to harvest a deer during archery season. It also seems that switching from a vertical bow to a crossbow hasn't helped much either. However, I did get a nice buck during firearms season back in '08 with my 20 gauge H&R.

Case II:

State Trooper - You know why I pulled you over?

Deer Hunter - Nope.

State Trooper - You just ran that red light.

Deer Hunter - Couldna done it.

State Trooper - Now how's that?

Deer Hunter (after he spits out his tobacco) - There's this paradox in my truck. It's like this every bit of time, my truck occupies one and the same spot. But an object that occupies one and the same spot over a time ain't moving at that time. So my truck ain't goin' nowhere. But the same just happens every time, so my truck ain't movin'. It can't move. So it couldna gone through that there light.

State Trooper - Sir, have you been drinking?

Deer Hunter - Nope.

State Trooper - Sir, get out of your vechicle. You're under arrest. You have the right to remain silent . . .

Deer Hunter - If I ain't been drinkin', what am I bein' busted for?

State Trooper - It's illegal to have a paradox in you vechicle.

Dagmara Lizlovs said...

There is something I would like to add to Zeno's paradox of the arrow while we are on the subject of arrows. I like shooting the vertical bow. I've had to switch to cross bow because of tendonitis and that with the scoped cross bow a humane kill on a game animal is far more easy to accomplish. The thing missing in this article is the straightness of the arrow. The most important thing an archer can have is an arrow that is perfectly straight. None of the physics arguements and reasonings in this post really mean anything if the arrow isn't straight. I was taught by Hunter Safety instructor, that if an arrow is straight, then one can use a broomstick for a bow and still hit the mark; however, if an arrow isn't straight, then no matter how good, expensive or sophisticated the bow is, the arrow will not hit its mark. Instead of thinking along the lines "Here's a way to, ahem, sharpen The Arrow." It is far more critical to think of straightening the arrow.