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.

Teleportation and time-travel

Let’s assume:

1. Faster-than-light travel is metaphysically possible in a special relativistic world.

And let’s assume:

2. In a special relativistic world, no (inertial) reference frames are metaphysically privileged.

Now, if faster-than-light travel occurs, then one travels from space point z₁ to point z₂ during a length of time t < d(z₁, z₂)/c according to some reference frame F₁. The arrival location then is not in the light-cone centered on the departure point (and light-cones do not depend on reference frames). But if the arrival point is not in the light-cone centered on the departure point, then there is a reference frame F₂ according to which the arrival is earlier than the departure. (For the forward light-cone centered on a point a is just the set of points of space-time that are later than or simultaneous with a according to all frames. So if you’re not in the forward light-cone centered on a, you are earlier than a according to at least one frame.)

But no frame is privileged by (2). Moreover, if faster than light travel is possible, then faster than light travel is possible at any finite speed, since anything else would be unacceptably ad hoc. So if faster than light travel is possible according to F₁, it is possible according to F₂. So let’s suppose that you traveled from z₁ to z₂ and arrived −δ units of time earlier according to F₂ (for some δ > 0). Then add another spot of faster than light travel from z₂ to z₁, at a speed high enough to ensure you arrive at z₁ in δ/2 units of time. Then according to F₂, you moved from z₁ to z₂ and back to z₁ and arrive −δ + δ/2 = −δ/2 units of time after the beginning of your journey.

So according to F₂ you time-traveled backwards at the same spatial location. But backwards time-travel at the same spatial location in one reference frame implies backwards time-travel according to all frames (because it implies going into one’s backwards light cone).

So, we’ve argued:

3. If (1) and (2) are true, then it is metaphysically possible to travel absolutely backwards in time,

where absolute backwards time-travel is time-travel backwards according to all inertial frames. (I assume that most people working in philosophy of time know this.)

And there is good reason to believe (1) and (2). Indeed, (2) seems definitional. And (1) seems pretty plausible, especially given an omnipotent God. After all, surely God could make you travel to alpha Centauri and back by Christmas of this year. Note, though, that a part of (1)—and perhaps this is the controversial part?—is the possibility of a special relativistic world.

I am not sure what to make of this.

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.

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).