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.
5 comments:
In QT, the state of a particle can be described in the position representation or, differently by equivalently, in the momentum representation. This matches the intuition of your second-last paragraph. (For example, a wave packet centred at the origin travelling left could have the same probability distribution of position as one travelling right, but they would have different probability distributions of momentum.) The cost of this, from a classical point of view, is that it necessarily introduces uncertainty.
Right, though the position representation doesn't just represent the probability distribution of position, but also phase.
Pruss...
In your blurb concerning the 5 minute universe using your version of Aristotelianism-Lezbionism you reject the use of the Uncertainty Principle.... and now... you are using it!!!!!
Make up your mind man!!!!!!
Face it... you are lost in-the-woods...
It seems the act of "teleporting" the rock 1m west is undefined. What does teleportation consist of? It seems you are only explaining the term by the outcome - the rock instantaneously changes position. How has this occurred? Was the rock destroyed and re-created? But then how can we call it "the same rock". Was something like a wormhole created, altering the continuity of space in the rock's trajectory? Then surely it would continue with Northward trajectory. If by "miracle", we just mean that we don't know what happened, then I don't see how we can expect to anticipate its future trajectory.
I like this puzzle. In some science fiction that I read there was a conservation of momentum which meant that people had to be careful time-travelling, because they took their actual momentum (relative to the galaxy presumably) with them. I think that if you use "momentum" instead of "velocity" then your result is less interesting, especially if you define "momentum" so that it is not just velocity multiplied by mass, but is something prior to velocity, something that usually causes velocity. Usually "velocity" is defined to be the vector that is speed plus direction.
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